ASYMPTOTICS OF THE TURAEV-VIRO INVARIANTS AND THEIR

CONNECTIONS IN LOW-DIMENSIONAL TOPOLOGY

By

Sanjay Lakshman Kumar

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics - Doctor of Philosophy

2021

ABSTRACT

ASYMPTOTICS OF THE TURAEV-VIRO INVARIANTS AND THEIR

CONNECTIONS IN LOW-DIMENSIONAL TOPOLOGY

By

Sanjay Lakshman Kumar

We study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invari-

ants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-

Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture

by Chen and Yang [13] which states that the asymptotics of the Turaev-Viro invariants

recover the hyperbolic volume of the manifold. Using topological tools, we are able to con-

struct infinite families of new examples of hyperbolic links in the 3-sphere which satisfy the

conjecture. Additionally, we show a general method for augmenting a link such that the

resulting link has hyperbolic complement which satisfies the conjecture.

As an application of the constructed links in the 3-sphere, we extend the class of known

examples which satisfy a conjecture by Andersen, Masbaum, and Ueno [3] relating the quan-

tum representations of surface mapping class groups to its Nielsen-Thurston classification.

Most notably, we provide explicit elements in the mapping class group for a genus zero

surface with n boundary components, for n > 3, as well as elements in the mapping class

group for any genus g surface with four boundary components such that the obtained el-

ements satisfy the conjecture. This is accomplished by utilizing the intrinsic relationship

between the Turaev-Viro invariants and the quantum representations shown by Detcherry

and Kalfagianni in [18].

Dedicated to Chung Ae Kim and Karla Gomez.

iii

ACKNOWLEDGEMENTS

I would first like to thank my advisor, Effie Kalfagianni, for her guidance throughout my

time in the mathematics PhD program at Michigan State University. I sincerely appreciate

the effort she has dedicated towards my growth as a mathematician.

I would also like to express my gratitude towards Charles Frohman and Rosemary Guz-

man who were at the beginning of my journey as an undergraduate. Because of them, I have

always felt supported in the community.

Additionally, I would like to extend my gratitude towards Robert Bell, Matthew Hedden,

Kristen Hendricks, and Ben Schmidt for their involvement and help during the graduate

program as well as Renaud Detcherry who provided insightful comments and suggestions on

some of the results appearing in this work.

Moreover, I would like to thank my family and friends whose positivity and support has

always been a valuable asset in my life.

As a further acknowledgement, the material appearing in this work is based on research

partially supported by NSF grants DMS-1404754, DMS-1708249, DMS-2004155 and by a

Dr. Paul and Wilma Dressel Endowed Scholarship from the Department of Mathematics at

Michigan State University.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Turaev-Viro Invariant Volume Conjecture . . . . . . . . . . . . . . . . .
1.2.1 Main Results From Chapter 3 . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Main Results From Chapter 4 . . . . . . . . . . . . . . . . . . . . . .
1.3 The Andersen-Masbaum-Ueno (AMU) Conjecture . . . . . . . . . . . . . . .
1.3.1 Main Results From Chapter 5 . . . . . . . . . . . . . . . . . . . . . .

Chapter 2: Preliminaries

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 The Turaev-Viro Invariants
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Surface Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Nielsen-Thurston Classification . . . . . . . . . . . . . . . . . . .
2.2.2 Braid Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Quantum Representations and the Turaev-Viro Invariants
. . . . . .
2.3 Shadows of 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Fundamental Shadow Links
. . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Planar Graph Constructions of Fundamental Shadow Links . . . . . .

1
3
3
6
7
10
12

15
15
20
22
23
26
30
35
37

Chapter 3: The Turaev-Viro Invariants for Once-Punctured Torus Bundles 40
40
40
43
51
54

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Canonical Triangulations . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
3.3 A Conjecture in Relation to the Stretch Factor . . . . . . . . . . . . . . . . .

3.2 A Closed Formula for the Turaev-Viro Invariants

3.2.1 Calculations of Turaev-Viro Invariants

Chapter 4: Realizing Fundamental Shadow Links in S3

. . . . . . . . . . . .
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Stein Factorization of Stable Maps of Links . . . . . . . . . . . .
4.1.2
Stein Surfaces and Shadows of 3-Manifolds . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Kirby Calculus
4.2 An Obstruction to Knots in S3
. . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Links in S3 up to 11-Crossings . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Families of Links in S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Augmenting Links in S3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 The Algorithm for Augmentation . . . . . . . . . . . . . . . . . . . .

Chapter 5: Applications Towards the AMU Conjecture . . . . . . . . . . . .
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58
58
59
61
63
64
66
69
77
82

85
85

v

5.2 Constructions of Pseudo-Anosov Elements

85
5.1.1 The Stallings Fibration . . . . . . . . . . . . . . . . . . . . . . . . . .
87
. . . . . . . . . . . . . . . . . . .
87
5.2.1 Applications of the Stallings Fibration . . . . . . . . . . . . . . . . .
93
5.2.2 Applications of Corollary 4.5.4 . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Explicit Pure Braids Satisfying the AMU Conjecture . . . . . . . . . 100
5.3 Cosets of the Pure Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . 104

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

vi

LIST OF TABLES

Table 1.1: The links up to 11-crossings in S3 which satisfy the Turaev-Viro in-
variant volume conjecture with volume 2v8. . . . . . . . . . . . . . . .

Table 4.1: The links with homeomorphic complements to a fundamental shadow
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

link.

Table 5.1: The fundamental shadow links up to 11-crossings written as the closure
of a homogeneous braid such that the fiber surface is given by the
Stallings fibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

67

88

Table 5.2: The fundamental shadow links from Theorem 4.3.2 which can be writ-

ten as the closure of a pure braid.

. . . . . . . . . . . . . . . . . . . . 106

vii

LIST OF FIGURES

Figure 1.1: The defined link families Lk, Jk, and Kk in S3. For the link family
{Lk} with (5k + 6) crossings, the number of components is (k + 2) for
k odd and (k + 3) for k even. For the link family {Jk} and {Kk} with
(5k + 5) and (5k + 4) crossings, the number of components is (k + 2)
. . . . . . . . . . . . . . . . . . . . . . . . .
and (k + 3), respectively.

Figure 2.1: The correspondence of an admissible colored tetrahedron to the quan-
tum 6j-symbol. The edges of the tetrahedron are labeled by the 6-tuple
(i, j, k, l, m, n), and the blue shaded region is a quadrilateral which cor-
responds to Q1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

17

Figure 2.2: The image of the blue curve under a Dehn twist of the annulus.

. . .

21

Figure 2.3: For the two leftmost diagrams, we have the generator σi and its inverse
σ−1
, respectively, for the braid group Bn where i ∈ {1, . . . , n − 1}.
On the rightmost diagram, we show the composition with the example
σ1σ−1
2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Figure 2.4: On the left diagram, we show the braid closure of an arbitrary braid
b ∈ Bn. On the right diagram, we show the diagram of the associated
. . . . . . . . . . . . .
braided link where we labeled the braid axis.

24

25

Figure 2.5: The local models of the simple polyhedron.

. . . . . . . . . . . . . . .

30

Figure 2.6: The procedure to replace the vertices of the graph. . . . . . . . . . . .

37

Figure 2.7: An example of constructing a fundamental shadow link from a planar
4-valent graph G. On the left, we begin with a planar 4-valent graph
of complexity 2 where the maximal tree T is colored blue. We obtain
the middle illustration by replacing the edges and vertices by 3-braids
and the 6 strands described in Figure 2.6, respectively. The right il-
lustration is obtained by encircling the 3-braids corresponding to the
. . . . . . . . . . . .
edges G\T by 0-framed unknotted components.

Figure 3.1: On the left, we have two triangulations of the once-punctured torus as

the quotient of R2\Z2. On the right, we have the tetrahedron obtained

from stacking the triangulations by identifying the edges with the same
slope. Here, the slopes of the triangulations are given in the illustra-
tions, the white vertices are the puncture, and the red and blue edges
correspond to the edges of the diagonal exchange.
. . . . . . . . . . .

38

44

viii

Figure 3.2: An ideal triangulation of the figure-eight knot. Here, we identify the
. . . .

black edges together, and we identify the blue edges together.

52

Figure 4.1: On the left, the Kirby I move which adds or removes an unlinked com-
ponent with framing ±1. On the right, the Kirby II move where any two
components with their framing curves contained in a possibly knotted
two-holed disk in S3 can be changed as shown. Here, the framed link is
colored blue, the two-holed disk may be linked with other components,
. . . . . .
and the new framing curves are still on the two-holed disk.

Figure 4.2: The six fundamental shadow links with complements homeomorphic to
. . . . . . . . . . . . . .

the complements of links up to 11-crossings.

Figure 4.3: The Borromean rings are the leftmost link, and the Borromean twisted
sisters are the two rightmost links. . . . . . . . . . . . . . . . . . . . .

Figure 4.4: On the left are the complexity k fundamental shadow links embedded
in connected sums of S2 × S1, and on the right are their associated
links in S3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.5: On the leftmost diagram, we have the shadow P with Sing(P ) the
dashed lines, ∂int(P ) colored blue, and ∂ext(P ) colored black. We cap
∂int(P ) with disks to obtain regions Rj shaded blue, and we mark
their centers with p1 and p2 to recover a new shadow P (cid:48). In this case,
2. P (cid:48) is the shadow of a link
the gleam of the new regions are both 1
complement in S3 such that the preimages of the points p1 and p2 are
the cores of the ∞-framed link in S3. Drilling out the ∞-framed link
gives a link complement in S3 homeomorphic to the complement of the
original fundamental shadow link.
. . . . . . . . . . . . . . . . . . . .

Figure 4.6: The planar 4-valent graph of complexity k used in the construction of
. . . . . . . . . . . . . . . . . . . . .

the link families Lk, Jk, and Kk.

Figure 4.7: The diagrams of the twist, belt, clasp, and mirrored clasp components
are illustrated, respectively. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: On the leftmost diagram, we have a planar 4-valent graph with com-
plexity 2. From the graph, we obtain the boundary-decorated shadowed
polyhedron shown in the middle illustration. Here, the edges were re-
placed by identity 3-braids, the outermost boundary is labeled C, the
internal decorated boundaries are blue, the external decorated bound-
aries are black, and the singular set is dashed. As in Theorem 4.4.2,
we cap the blue curves with disks and drill out the corresponding tori
to recover the link in the third diagram. . . . . . . . . . . . . . . . . .

64

67

69

70

71

72

76

78

ix

1 σ−1

Figure 4.9: As shown in the first two diagrams, we begin with the unreduced word
b = σ1σ−1
1 ∈ B2 which we embed into B3, and we take the closure
of the new braid with outermost component labeled C. The projection
of the second diagram onto S2 gives the third diagram where we label
the regions not touching C with Ri. Now each region Ri corresponds to
a link component parallel to the z-axis given in the final diagram. The
final diagram is of a hyperbolic link that has volume 6v8, satisfies the
Turaev-Viro invariant volume conjecture, and contains ˆb as a sublink.

Figure 5.1: The link L5,m where the box indicates 2m negative crossings. Here, the
. . . . . . . . . . . . .

Borromean rings are the blue colored sublink.

Figure 5.2: The complexity k fundamental shadow links realized in S3 obtained
from the braid family αk. On the left, for k even, there are (k + 4)
components, and on the right, for k odd, there are (k + 3) compo-
nents. Here, our eventual braid axis component is labeled as B, and
the component that bounds a (k + 1)-punctured disk is labeled L(cid:48).
.
Figure 5.3: On the leftmost diagram, we have the closure of the braid α2 = σ1σ−1
with the region labeled RB corresponding to the B labeled component
in the second diagram. From the braid, we can apply Corollary 4.5.4
to obtain the link in the second diagram where B will eventual be the
braid axis component. Through an isotopy, we obtain the link in the
third diagram.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Figure 5.4: The case of obtaining a fibration for k = 2. Performing a full-twist
on the 3-punctured disk shaded blue in the leftmost diagram results
in the middle diagram. Through an isotopy of the link, we obtain
a link fibering over S1 with fiber surface D5 as represented by the
last diagram. Here, the monodromy may be written explicitly as
(σ3σ2σ2
. . . . . . . . . . . . . . . . . . . . . . .

2 σ3)(σ4σ3σ2

1σ−1

2σ3σ4).

Figure 5.5: An isotopy of the Borromean twisted sister to a braided link where the
braid axis is colored black. The resulting manifold has fiber surface
D3 with monodromy the braid group element σ−2
2 ∈ B3 under the
. . . . . . . . . . . . . . . . . . . . . . .
surjective homomorphism Γ.

1 σ2

83

89

94

95

96

97

Figure 5.6: On the left, we define the boxed diagram as the identity 3-braid with
2s additional components. For a box labeled with 0, the diagram is the
identity 3-braid without additional components. On the right, we show
the link in S3 obtained from applying Corollary 4.5.4 to the element
(σ1σ−1
2 ) ∈ B3. Here, the braid axis is labeled B, the link
component that bounds a (3 + 2s)-punctured disk is labeled L(cid:48), and
the link contains (7 + 2s) components.

1 )s(σ2σ−1

. . . . . . . . . . . . . . . . . 102

x

Chapter 1

Introduction

In the 1980’s, Witten [60] provided a new interpretation of the Jones polynomials [30, 31]

from the perspective of physics leading to the construction of a sequence of complex-valued

3-manifold invariants. This led to the field of quantum topology, and the construction of a

wealth of new link and manifold invariants known as quantum invariants. This was first

formalized by Reshetikhin and Turaev [48] through the representation theory of quantum

groups which gave rise to the colored Jones polynomials. These are a sequence of poly-

nomial invariants of links parameterized by a positive integer n which are a generalization

of the Jones polynomial. Additionally, Reshetikhin and Turaev constructed a sequence of

3-manifold invariants calculated from the surgery description of the manifold [35] which

correspond to Witten’s invariants.

Although the quantum invariants can be used to distinguish links and manifolds, it is

conjectured that there are also strong connections between the hyperbolic geometry and

quantum topology of 3-manifolds. The most famous example being Kashaev’s volume con-

jecture [32, 33]. Kashaev introduced a sequence of quantum invariants for links in S3 using

quantum dilogarithm functions, and Kashaev also observed in a few examples, as well as

conjectured in the general case, that the absolute value of the invariants grows exponentially

for hyperbolic links where the growth rate is given by the hyperbolic volume of the comple-

ment. Later in [44], Murakami and Murakami showed that the values of the colored Jones

polynomials evaluated at a particular root of unity coincide with Kashaev’s invariants lead-

ing to a reformulation of the volume conjecture. As this conjecture is analogous to numerous

topics in this dissertation, we provide its statement here.

1

Conjecture 1.0.1 ([33, 44]). For a hyperbolic link L in S3, let Jn(L; q) be the n-th colored

Jones polynomial evaluated at q. Then

lim
n→∞

2π
n

log

√−1

2π

n

L; e

(cid:16)

(cid:12)(cid:12)(cid:12)Jn

(cid:17)(cid:12)(cid:12)(cid:12) = V ol(cid:0)S3\IntNbd(L)(cid:1)

where V ol (S3\IntNbd(L)) is the hyperbolic volume of the complement of L in S3.

The volume conjecture is one of the many proposed relations between hyperbolic geom-

etry and quantum topology. In this work, we will investigate two other conjectures which

further deepen the connection between the fields. Namely, we will explore the Turaev-Viro

invariant volume conjecture, stated by Chen and Yang in [13], and the Andersen-Masbaum-

Ueno (AMU) conjecture [3]. For the formal statements of the conjectures as well as main

results, see Sections 1.2 and 1.3, respectively.

Briefly, the Turaev-Viro invariant volume conjecture is analogous to the original volume

conjecture; however, in place of the colored Jones polynomials, the Turaev-Viro invariant

volume conjecture relates the exponential growth rate of a sequence of real-valued invariants

known as the Turaev-Viro invariants [57] to the volume of the hyperbolic manifold. As

opposed to the original volume conjecture which was stated for link complements in S3, the

Turaev-Viro invariant is a 3-manifold invariant defined on a triangulation of the 3-manifold.

Because of this, the Turaev-Viro invariant version of the volume conjecture can be stated

for more classes of hyperbolic 3-manifolds.

For the latter, the AMU conjecture asserts that the geometric properties of the elements

of the surface mapping class groups given by the Nielsen-Thurston classification [23, The-

orem 13.2] are detected by objects arising from the Witten-Reshetikhin-Turaev topological

quantum field theories known as the quantum representations [48, 56]. More explicitly, the

Nielsen-Thurston classification splits the mapping class group into either periodic, reducible,

or pseudo-Anosov elements, and in the reducible case, the elements may be further reduced

to smaller parts on which the Nielsen-Thurston classification again applies. The AMU con-

2

jecture states that there is a quantum representation of infinite order if and only if the

associated mapping class group element contains any parts that are pseudo-Anosov.

Notably, we will see in Subsection 2.2.3 that it is natural to study the two conjectures

jointly. This is due to the Turaev-Viro invariant volume conjecture implying the AMU

conjecture as shown by Detcherry and Kalfagianni in [18].

1.1 Organization of Dissertation

The main results of this dissertation along with the statements of the Turaev-Viro invariant

volume conjecture and the AMU conjecture can be found in Sections 1.2 and 1.3, respectively.

In Chapter 2, we will present necessary preliminary material involving the definition of

the Turaev-Viro invariant [57], an introduction to surface mapping class groups [23], an

introduction to Turaev’s shadow theory approach to 3-manifolds [56], and the construction

of a family of links known as the fundamental shadow links which were originally introduced

by Constantino and Thurston in [15]. In Chapter 3, we consider the Turaev-Viro invariants of

the once-punctured torus bundles and their possible relationship to quantities in hyperbolic

geometry. Lastly, in Chapters 4 and 5, we demonstrate positive results towards the Turaev-

Viro invariant volume conjecture and the AMU conjecture, respectively, through applications

of the fundamental shadow links.

1.2 The Turaev-Viro Invariant Volume Conjecture

In this section, we will write the statement of the Turaev-Viro invariant volume conjecture

followed by the main results found in Chapters 3 and 4.

The SO(3)-version of the Turaev-Viro invariants T Vr (M, q) [21, 57] of a compact 3-

manifold M are real-valued invariants of M that depend on an odd integer r ≥ 3 and a 2r-th
root of unity q. For a precise definition of the Turaev-Viro invariant, see Section 2.1. Similar

in form to the original volume conjecture, Chen and Yang proposed the following which

3

relates the exponential growth rate of T Vr (M, q) to the hyperbolic volume of the manifold

M , denoted by V ol (M ).

Conjecture 1.2.1 ([13]). Let M be a hyperbolic 3-manifold and q = exp

(cid:16) 2π

(cid:17)

√−1

r

. Then

lim
r→∞

2π
r

log |T Vr (M, q)| = V ol (M )

as r ranges along the odd natural numbers.

(cid:17)

(cid:16) π

r

Additionally in [13], Chen and Yang provided numerous computations supporting the

exponential growth rate of the Turaev-Viro invariant at this particular root of unity. It is
√−1

notable that this conjecture is in contrast to the evaluation of the invariant at q = exp

where the growth rate is expected to only be polynomial.

Currently, Conjecture 1.2.1, which we call the Turaev-Viro invariant volume conjecture,

has been proven for multiple 3-manifolds.

It has been shown for the complement of the

Borromean rings and the complement of the figure-eight knot by Detcherry, Kalfagianni,

and Yang in [21]. It was verified by Ohtsuki that the closed hyperbolic 3-manifolds arising

from integral Dehn surgeries on the figure-eight knot satisfy the conjecture in [45] which

was extended by Wong and Yang to rational surgeries in [62]. In [7], Belletti, Detcherry,

Kalfagianni, and Yang verified the conjecture for the complements of the fundamental shadow
links which are a family of links in connected sums of S2 × S1. Additionally in [6], Belletti
proved Conjecture 1.2.1 for families of links in connected sums of S2 × S1 which contain
examples that are not fundamental shadow links. Lastly in [61], Wong proved Conjecture

1.2.1 for the twisted Whitehead links as well as certain subfamilies of Whitehead chains

originally introduced by Roland van der Veen in [58].

Additionally in this work, it will be useful for us to generalize the Turaev-Viro invariant

volume conjecture for the cases when the manifold is not necessarily hyperbolic. For this,

we will write the Turaev-Viro invariant volume conjecture in terms of the simplicial volume.

We will only state the relationship between the simplicial volume and the hyperbolic volume

4

of a manifold; however, more details may be found in [27] by Gromov and in [54, Section

6.5] by Thurston.

We will denote the simplicial volume for a manifold M by (cid:107)M(cid:107). If M is a hyperbolic

manifold of finite volume, then

V ol (M ) = v4 (cid:107)M(cid:107)

where v4 ≈ 1.0149 is the volume of a regular ideal tetrahedron. For manifolds which are not
hyperbolic, the simplicial volume can be thought of as measuring the sum of the volumes

of the hyperbolic pieces. Because of this, the following conjecture, which is formally stated

by Detcherry and Kalfagianni in [19], is a natural extension of the Turaev-Viro invariant

volume conjecture.

Conjecture 1.2.2 ([13]). Let M be a compact and orientable 3-manifold with empty or

toroidal boundary. Then for q = exp

(cid:16) 2π

√−1

r

(cid:17)

,

LT V (M ) = lim sup

r→∞

2π
r

log |T Vr (M, q)| = v4 (cid:107)M(cid:107)

as r ranges along the odd natural numbers.

In [19], the authors show that the extended Turaev-Viro invariant volume conjecture

holds for certain manifolds of simplicial volume zero. In addition, the authors show that the

extended Turaev-Viro invariant volume conjecture is preserved under gluings by manifolds

known as the invertible cabling spaces. Furthermore, Detcherry shows that the conjecture is

preserved under (p, 2)-cabling for p odd in [17].

In the following subsections, we provide results from this work in relation to the Turaev-

Viro invariant volume conjecture.

5

1.2.1 Main Results From Chapter 3

In Chapter 3, we consider the Turaev-Viro invariants of once-punctured torus bundles.

Briefly, the mapping class group M od(Σ1

1) is the group of isotopy classes of orientation-

preserving self-homeomorphisms of the surface. More information on mapping class groups

may be found in Section 2.2. From the elements of M od(Σ1
punctured torus bundles. For φ ∈ M od(Σ1
as the mapping torus M (φ) = Σ1

1), the once-punctured torus bundle is constructed
1 × [0, 1]/(x, 0) ∼ (φ(x), 1) such that M (φ) has monodromy
In the case of hyperbolic once-punctured torus bundles, the monodromy φ admits a

1), we may form the once-

φ.

canonical presentation in terms of generators R and L which is related to the canonical tri-

angulation of the bundle. From this canonical triangulation, we construct a closed formula

for the Turaev-Viro invariants which depends, in a trivial way, on the form of the canonical
presentation of the monodromy φ ∈ M od(Σ1
or the canonical triangulations, see Subsections 2.2.1 and 3.1.1, respectively.

1). For more details on either the mapping tori

We will now proceed with the closed formula for the Turaev-Viro invariants of the hy-

perbolic once-punctured torus bundles. Although understanding the formula involves the

introduction of additional notation found in Section 3.2, we will still first state the formula

here as follows. We remark that an admissible coloring is a set of criteria, dependent on the

odd and positive integer r, which the terms ENk

ik−1,jk−1

known as the quantum 6j-symbols will

have to satisfy. See Section 2.1 for the precise definition.

Theorem 3.2.1. Let φ = RN1LN2 ··· RN2n−1LN2n ∈ M od(Σ1

2n, and let N =(cid:0)(cid:80)2n

(cid:1). Then the mapping torus M (φ) has Turaev-Viro invariant
(cid:17)(cid:35)

(cid:34)(cid:32)N +3(cid:89)

1) where n, Ni > 0 for 1 ≤ i ≤

(cid:33)(cid:16)

|ak|

EN1
i0,j0 · EN2

i1,j1 · . . . · EN2n

i2n−1,j2n−1

i=1 Ni

T Vr (M (φ) , q) = 2b2−b0(cid:88)

adm

k=4

where the sum is over all admissible colorings, a1 = a(N +2), a2 = a(N−N2n+2), and a3 = a(N +3).

Additionally, the indices ik and jk may be computed inductively from (i0, j0) = (2, 3) where

6

ik (cid:54)∈ {ik−1, jk−1} and with

ik−1 For Nk odd

jk−1 For Nk even

.

jk =

In Subsection 3.2.1, we provide examples of explicit calculations of the Turaev-Viro in-

variants for once-punctured torus bundles using Theorem 3.2.1.

Lastly, we end Chapter 3 with a conjecture relating the Turaev-Viro invariants of hy-

perbolic manifolds to a geometric quantity of the mapping class group known as the stretch

factor. The conjecture is stated in the following where pseudo-Anosov elements are defined

in Subsection 2.2.1.

Conjecture 3.3.3. Let φ ∈ M od(Σ1
Then

1) be a pseudo-Anosov element with stretch factor λφ.

LT V (M ) = lim sup

r→∞

2π
r

log |T Vr (M (φ), q)| ≤ 2π log (λφ)

(cid:16) 2π

√−1

r

(cid:17)

.

where r runs along the odd integers and q = exp

As the goal of the Turaev-Viro invariant volume conjecture is to relate quantum topology

to hyperbolic geometry, Conjecture 3.3.3 further supports the claim. As we will discuss in

Section 3.3, a similar bound involving the hyperbolic volume and the stretch factor is known.

1.2.2 Main Results From Chapter 4

We first note that a large part of the results stated here are based on the work found in

[36] by the author of this dissertation. An important observation is that the Turaev-Viro

invariant volume conjecture is preserved under homeomorphisms of manifolds. This is due to

the Turaev-Viro invariant being a 3-manifold invariant as well as the hyperbolic volume being

a topological invariant by Mostow-Prasad rigidity [43, 46]. By utilizing this property, we are

able to prove the Turaev-Viro invariant volume conjecture for many link complements in S3

through explicit homeomorphisms to the complements of the fundamental shadow links. In

this sense, we say that the fundamental shadow links are realized as links in S3.

7

The precise definition of the fundamental shadow links may be found in Subsection 2.3.1;

however, in summary, the fundamental shadow links are a family of links in connected sums
of S2 × S1 with useful properties. More specifically, they are hyperbolic links with explicit
hyperbolic volumes constructed from even multiples of regular ideal hyperbolic octahedra

along with relatively simple Turaev-Viro invariants from which the Turaev-Viro invariant

volume conjecture can be directly verified [7]. Additionally, they are “universal” in the sense

that every orientable 3-manifold with empty or toroidal boundary can be realized as the

manifold obtained from Dehn fillings along some subset of the boundary components for

the complement of some fundamental shadow link. See Subsection 4.1.3 for more details on

Dehn fillings.

Using LinkInfo [39] by Livingston and Moore, we begin by classifying links up to 11-

crossings in S3 with complements homeomorphic to the complement of a fundamental shadow

link. Here, we denote v8 ≈ 3.6639 as the volume of a regular ideal hyperbolic octahedron.

L6a4
L10n36
L10n108 L11n287 L11n376 L11n378 L11n385

L8n5
L10n70

L8n7
L10n84

L9n25
L10n87

L9n26
L10n32
L10n97 L10n105

Table 1.1: The links up to 11-crossings in S3 which satisfy the Turaev-Viro invariant volume
conjecture with volume 2v8.

Theorem 1.2.3. Each of the links in Table 1.1 is hyperbolic with volume 2v8 and satisfies

the Turaev-Viro invariant volume conjecture.

The given notation L∗∗∗ is such that the first index indicates the minimum number of

crossings, the second index indicates whether it is alternating or non-alternating, and the

last index indicates that it is the k-th link with the same first two indices. We note that in

Section 4.3, we compile a list of all possible links in LinkInfo which could be a fundamental

shadow link with hyperbolic volume 2v8. With the exception of the link L10n59, all of the

possible links are fundamental shadow links realized in S3. For the case of L10n59, our method

gave an inconclusive result; however, we believe that the link does not have complement

8

Lk

Jk

k + 1

k + 1

Kk

k + 1

Figure 1.1: The defined link families Lk, Jk, and Kk in S3. For the link family {Lk} with
(5k + 6) crossings, the number of components is (k + 2) for k odd and (k + 3) for k even.
For the link family {Jk} and {Kk} with (5k + 5) and (5k + 4) crossings, the number of
components is (k + 2) and (k + 3), respectively.

homeomorphic to the complement of a fundamental shadow link.

Additionally, we construct infinite link families in S3 which satisfy the Turaev-Viro in-

variant volume conjecture.

Theorem 4.4.2. Given an integer k ≥ 1, let Lk, Jk, and Kk be the links in S3 shown in
Figure 1.1. The complement of each Lk, Jk, and Kk is a hyperbolic manifold with volume

2kv8 and satisfies the Turaev-Viro invariant volume conjecture.

In the sense that they correspond to k = 1, the complements of these links should be seen

as a generalization of the complements of the Borromean rings and the Borromean twisted

sisters. For an illustration of the Borromean rings and the Borromean twisted sisters, see

Figure 4.3.

As we will see in Subsection 2.3.2, the fundamental shadow links may be constructed

from planar 4-valent graphs. In Section 4.5, we show that at least one of the fundamental

shadow links corresponding to a planar 4-valent graph can be realized as a link in S3. Here,

the complexity is the number of vertices of the graph. We remark that the following theorem

is a generalization of Theorem 4.1 first appearing in [36].

Theorem 4.5.1. For any G, a planar 4-valent graph of complexity k, there exists a corre-

9

1sponding fundamental shadow link realized in S3 with hyperbolic volume 2kv8.

Now as an immediate corollary, we obtain the result that every link in S3 may be

augmented by adding additional link components such that the resultant link satisfies the

Turaev-Viro invariant volume conjecture.

Corollary 4.5.2. Every link J ⊂ S3 is a sublink of a link L ⊂ S3 such that the link
complement S3\IntNbd(L) is homeomorphic to a fundamental shadow link complement and
satisfies the Turaev-Viro invariant volume conjecture.

We note that Corollary 4.5.2 should be compared to a result of Baker [5, Theorem 1] and

Roland van der Veen [59, Corollary 3] which states that all links are sublinks of arithmetic

links. In particular, Corollary 4.5.2 is similar in application to Corollary 1 in [59]. By using

augmented knotted trivalent graphs, the author showed that any link may be augmented to

satisfy an SO(3)-version of the original volume conjecture.

At the end of Chapter 4, we note that the augmentations of the links can be described

explicitly. We outline the general method for augmenting a link in S3, and we provide a

specific example of the augmentation for an unknot.

1.3 The Andersen-Masbaum-Ueno (AMU) Conjecture

In this section, we will write the statement of the Andersen-Masbaum-Ueno conjecture in

addition to the main results regarding the conjecture found in Chapter 5.

We will begin with a brief introduction to the quantum representations. We first fix

an odd integer r ≥ 3 which we refer to as the level. We now let Σg,n be a surface with
genus g and n boundary components, and we let M od(Σg,n) be its mapping class group

which fixes the boundaries. For a more in-depth discussion of surface mapping class groups,

see Section 2.2. Now let |∂(Σg,n)| denote the set of boundary components of Σg,n, and we
define Ir = {0, 1, . . . , r−5
2 } to be the set of non-negative integers less than or equal to

2 , r−3

10

r−3
2 . Lastly, given a primitive 2r-th root of unity and an assignment c : |∂Σg,n| → Ir, the
SO(3)-Witten-Reshetikhin-Turaev TQFT [10, 48, 56] gives a projective representation

ρr,c : M od(Σg,n) → P GLdr,c(C)

called the SO(3)-quantum representation where P GLdr,c(C) is the projective general linear

group with dr,c an integer dependent on the level r and the choice of c. Here, we will often

refer to the assignment c as a coloring of the boundaries. More intuitively, for a choice of r

and c, we can associate a matrix to every element of M od(Σg,n).

Now the AMU conjecture relates the Nielsen-Thurston classification of elements of the

mapping class group for a compact surface Σg,n to its quantum representations. For more de-

tails on the Nielsen-Thurston classification, see Subsection 2.2.1. Under the Nielsen-Thurston

classification, there are two types of infinite order elements in the mapping class group. These

are the reducible and pseudo-Anosov elements. For the reducible case, a power of the element

fixes a collection of disjoint curves on the surface such that we may consider the induced

self-homeomorphisms on the individual components obtained from cutting the surface along

the fixed curves. If at least one of the induced self-homeomorphisms is pseudo-Anosov, we

say that the mapping class group element contains pseudo-Anosov parts.

As we will see in the following statement, the AMU conjecture asserts that the asymp-

totics of the quantum representations detect the pseudo-Anosov parts in the mapping class

group.

Conjecture 1.3.1 ([3], AMU conjecture). An element f ∈ M od(Σg,n) has pseudo-Anosov
parts if and only if, for any big enough level r, there is a choice of coloring c of the elements

of ∂Σg,n such that ρr,c(f ) has infinite order.

We remark that one direction of the AMU conjecture is already known. If there exists

a ρr,c(f ) with infinite order, then f has pseudo-Anosov parts. Because of this, whenever we

find a ρr,c(f ) with infinite order, then we know that the associated element f satisfies the

11

conclusion of the AMU conjecture. Although, in this work, we still consider cases where the

elements can additionally be shown to be pseudo-Anosov and not reducible.

The AMU conjecture was first stated in [3] where Andersen, Masbaum, and Ueno proved

the conjecture for the mapping class group of the four-holed sphere. Using a similar method,

Santharoubane also proved the conjecture for the one-holed torus in [51]. Although for the

following surfaces the AMU conjecture is still unknown, certain elements in the mapping

class group have been shown to satisfy the conjecture. Egsgaard and Jorgensen in [22]

and Santharoubane in [52], by using different approaches, produced elements of M od(Σ0,2n)

satisfying Conjecture 1.3.1.

For the higher genus cases, there have also been results. March´e and Santharoubane in

[40] construct finitely many conjugacy classes of pseudo-Anosov elements for M od(Σg,1), and

Detcherry and Kalfagianni construct cosets of M od(Σg,1), for sufficiently large g, that satisfy

the AMU conjecture in [20]. Additionally, Detcherry and Kalfagianni in [18] find infinitely

many pseudo-Anosov elements in M od(Σg,2) for g ≥ 3 and in M od(Σg,n) where g ≥ n ≥ 3.
In the following subsection, we provide results from this work which further progress the

Andersen-Masbaum-Ueno conjecture.

1.3.1 Main Results From Chapter 5

Similar to Subsection 1.2.2, we remark that a large part of the results stated here are also

based on the work found in [36] by the author of this dissertation. As we will see in Subsection

2.2.3, the asymptotics of the Turaev-Viro invariants can be utilized in proving the AMU

conjecture. Our results in Chapter 5 are a consequence of this correlation. We first begin

with the following definition.

Definition 1.3.2. We say that two elements f, g ∈ M od(Σg,n) are independent if there is
no h ∈ M od(Σg,n) such that both f and g are conjugate to non-trivial powers of h.

As explained in [20], the motivation behind the definition of independent is if f and g

are not independent, then f satisfies the AMU conjecture if and only if g does. Because

12

of this, we need a less trivial definition for having infinitely many elements in the mapping

class group which satisfy the AMU conjecture. We now state the following theorem which

is based on techniques developed by Detcherry and Kalfagianni in [18].

Theorem 5.2.1. Given g ≥ 0, there exists a pseudo-Anosov element in M od(Σg,4) that
satisfies the AMU conjecture. Furthermore, for g ≥ 3, there are infinitely many pairwise
independent pseudo-Anosov elements in M od(Σg,4) that satisfy the AMU conjecture.

As a byproduct of the proof of Theorem 5.2.1, we also construct examples in M od(Σn−1,n)
and M od(Σn−2,n), for n ≥ 5, as well as examples in M od(Σ1,3) and M od(Σ3,3) which satisfy
the AMU conjecture.

For our next result, we construct explicit elements in the mapping class group for a genus

zero surface with boundary components that satisfy the AMU conjecture. In particular, to

the author’s knowledge, we obtain the first known examples of pseudo-Anosov elements

satisfying the AMU conjecture in the mapping class group for a genus zero surface with an

odd number of boundary components. We will now state the result as follows where the

elements ωn ∈ M od(Σ0,n) are defined in Subsection 5.2.3.

Corollary 5.2.9. For n ≥ 4, the element ωn ∈ M od(Σ0,n) satisfies the AMU conjecture.
Additionally, if n (cid:54)∈ {5, 7}, then ωn is a pseudo-Anosov element.

We note that the elements ωn ∈ M od(Σ0,n) can be written explicitly as elements in the
pure braid group. For a summary of the braid group, see Subsection 2.2.2. Moreover, we

remark that the elements appearing in Corollary 5.2.9 have corresponding 3-manifolds which

also satisfy the Turaev-Viro invariant volume conjecture for when n (cid:54)∈ {5, 7}. In the case
when n ∈ {5, 7}, the corresponding 3-manifolds satisfy the extended Turaev-Viro invariant
volume conjecture.

At the end of Chapter 5, we discuss a method, originally stated in [18, Remark 5.7], to

construct cosets of the pure braid group which satisfy the AMU conjecture. The general

idea utilizes Artin’s normal form for the pure braid group [4]. Artin demonstrates that every

13

pure braid group of n-strands, denoted by P Bn, can be realized as the semidirect product of

a subgroup Wn in P Bn with P Bk where k < n. In particular, every element in P Bn may be

written as a product β · ω for β ∈ P Bk and ω ∈ Wn. By a method stated in Section 5.3, we
show, for certain β, that any pure braid group element of the form β · ω satisfies the AMU
conjecture. In the following statement, the set of elements P is explicitly defined in Section
5.3; however, they are the collection of pure braids obtained from the links in Theorem 1.2.3

as well as the elements of Corollary 5.2.9.

Corollary 5.3.2. For a fixed n, let β ∈ P Bs be an element of P with n > s. Now for
any b ∈ P Bn with normal form b = β · ω where ω ∈ Wn, the corresponding element b in
M od(Σ0,n+1) satisfies the AMU conjecture.

14

Chapter 2

Preliminaries

In this chapter, we will detail relevant material that will be used throughout this work. In

Section 2.1, we introduce definitions for the particular version of the Turaev-Viro invariant

studied in this dissertation. In Section 2.2, we introduce definitions for the mapping class

groups, the statement of the Nielsen-Thurston classification, an introduction to braid groups,

and relations between the quantum representations and the Turaev-Viro invariants. Lastly,

in Section 2.3, we introduce the shadow theory approach towards studying 3-manifolds as

well as the construction of the fundamental shadow links.

2.1 The Turaev-Viro Invariants

In this section, we will recall the basic definitions of the SO(3)-version of the Turaev-Viro

invariants [21, 57].

For the remainder of this section, we fix an odd r ≥ 3, and we let Ir = {0, 1, . . . , r−5

be the set of non-negative integers less than or equal to r−3

2 , r−3
2 }
2 where the set Ir is commonly

referred to as the coloring set. As mentioned in Section 1.2, the Turaev-Viro invariant

T Vr (M, q) is a real-valued invariant defined on a triangulation of the 3-manifold M which

depends on a 2r-th root of unity q. In order to construct the invariant, we will first need to

define additional sets of complex numbers.

Let q ∈ C be a root of unity such that q2 is a primitive root of unity of order r. For an

15

integer n, we let the quantum integer [n] be the number defined by

(cid:1)
2 sin(cid:0) 2nπ
(cid:1) .
2 sin(cid:0) 2π

r

r

[n] =

qn − q−n
q − q−1 =

(cid:81)n

Analogous to the factorial for integers, we can also define a quantum factorial by [n]! =

i=1[i] where we define [0]! = 1. Using the quantum integers, we can build the necessary

data set for the invariant.

Definition 2.1.1. We say that a triple (i, j, k) ∈ I 3

r is admissible if

• i + j + k ≤ (r − 2),
• i ≤ j + k, j ≤ i + k, and k ≤ i + j.
Given an admissible triple (i, j, k) ∈ I 3

∆(i, j, k) =

with the convention that √x =(cid:112)

(cid:19) 1

2

r , we define the number ∆(i, j, k) by

(cid:18)[i + j − k]![j + k − i]![k + i − j]!
|x|√−1 when the real number x is negative.

[i + j + k + 1]!

Additionally, we say that a 6-tuple (i, j, k, l, m, n) ∈ I 6

r is admissible if each of the triples

(i, j, k), (j, l, n), (i, m, n), and (k, l, m) are admissible.

Definition 2.1.2. For a given admissible 6-tuple (i, j, k, l, m, n) ∈ I 6
6j-symbol by

r , we define the quantum

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i

j

k

l m n

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =(cid:0)√−1(cid:1)2(i+j+k+l+m+n)
min{Q1,Q2,Q3}(cid:88)

∆(i, j, k)∆(j, l, n)∆(i, m, n)∆(k, l, m)

z=max{T1,T2,T3,T4}

[z − T1]![z − T2]![z − T3]![z − T4]![Q1 − z]![Q2 − z]![Q3 − z]!

(−1)z [z + 1]!

where T1 = i + j + k, T2 = j + l + n, T3 = i + m + n, T4 = k + l + m, Q1 = i + j + l + m,

Q2 = i + k + l + n, and Q3 = j + k + m + n. Furthermore, the quantum 6j-symbol has the

16

i

k

j

m

n

l

Figure 2.1: The correspondence of an admissible colored tetrahedron to the quantum 6j-
symbol. The edges of the tetrahedron are labeled by the 6-tuple (i, j, k, l, m, n), and the blue
shaded region is a quadrilateral which corresponds to Q1.

following symmetry relations

j

k

i k

j

l m n

m l n

l n m

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i k

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i m n

k

l

j

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l m k

j n

i

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l

j n

i m k

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .

Although the definition of the quantum 6j-symbol is quite complicated, it is easier to

memorize using the geometric association described in Definition 2.1.3.

Definition 2.1.3. An admissible coloring of a tetrahedron is an assignment, or coloring, of
the six colors of an admissible 6-tuple (i, j, k, l, m, n) ∈ I 6
such that the three colorings assigned to the edges of each face form the corresponding

r to the edges of the tetrahedron

admissible triples of the admissible 6-tuple.

As a memory aid, we will construct an admissible coloring of a tetrahedron from a given

admissible 6-tuple (i, j, k, l, m, n). We begin by coloring the edges of an arbitrary face of the

tetrahedron with i, j, and k, and we also color the edges on the opposite sides of i, j, and k

with l, m, and n, respectively. Now each of the T1, T2, T3, and T4 corresponds to a face of the

tetrahedron, and each of the Q1, Q2, and Q3 corresponds to a quadrilateral separating two

pairs of vertices of the tetrahedron. This labeling can be seen in Figure 2.1 where the blue

shaded area is a quadrilateral which corresponds to Q1. Notice that for a given admissible

17

1coloring of a tetrahedron, there are multiple associated quantum 6j-symbols which accounts

for the symmetry relations.

Lastly, in order to define our invariant, we need a triangulation of our manifold M . If M

is a closed manifold, we consider triangulations in the usual sense where a triangulation is

a finite collection of tetrahedra and a collection of homeomorphisms between pairs of faces

such that the space obtained from gluing along the homeomorphisms is homeomorphic to

M .

For compact manifolds M with boundary, we allow (partially) ideal triangulations where

some of the vertices are truncated such that the truncated faces triangulate the boundary

of M by triangles. When defining the invariant on a triangulation of M , we will need to

additionally consider the sets V and E. The set V is the collection of vertices not lying

on ∂M , and the set E are the collection of edges of the triangulation excluding the edges

coming from the truncation of the vertices.

Definition 2.1.4. For a triangulation τ of a manifold M , an admissible coloring c at level r

of the pair (M, τ ) is an assignment of colors by c to the edges of τ such that each tetrahedron

has an admissible coloring. Furthermore, we will consider Ar(τ ) to be the set of all possible

admissible colorings at level r.

Now for a given coloring c at level r of the pair (M, τ ), an edge e ∈ E, and a tetrahedron

∆ ∈ τ , we consider the numbers

|e|c = (−1)2c(e) [2c(e) + 1]

and |∆|c where |∆|c is the quantum 6j-symbol associated to the admissible coloring of the
tetrahedron ∆ obtained by the assignment of c.

We are now ready to define the Turaev-Viro invariant as a state-sum.

Theorem 2.1.5 ([57]). Let M be a compact and orientable manifold that is either closed or
with boundary. Let b2 and b0 denote the Z2-Betti number and the number of closed connected

18

components in M , respectively. Then for ηr =

, the following state-sum

(cid:33)(cid:32)(cid:89)

(cid:33)(cid:35)

|e|c

|∆|c

∆∈τ

T Vr(M, q) = 2b2−b0η2|V |

r

2 sin( 2π
r )√

r

(cid:34)(cid:32)(cid:89)

(cid:88)

c∈Ar(τ )

e∈E

is a topological invariant independent of the partially ideal triangulation.

Now lastly, we will end the section with two propositions both appearing in [19] by

Detcherry and Kalfagianni which relate to the changes of the asymptotics of the Turaev-

Viro invariant under the drilling and gluing of tori.

Proposition 2.1.6 ([19], Corollary 5.3). Let M(cid:48) be a compact and oriented 3-manifold with
(possibly empty) toroidal boundary, and let M be a manifold obtained from M(cid:48) by drilling

out tori. Then

where

lT V (M(cid:48)) ≤ lT V (M )

for r running over odd integers and q = exp

lT V (M ) = lim inf
r→∞

2π
r

(cid:16) 2π

r

log |T Vr(M, q)|
√−1

(cid:17)

.

Proposition 2.1.6 states that drilling out tori from our manifold does not decrease the

growth rate of the Turaev-Viro invariants. This will be a particularly useful property when

dealing with the AMU conjecture in Chapter 5.

Now for the second proposition, we will first define a family of manifolds known as

the invertible cabling spaces. Although the invertible cabling spaces have a more general

definition [19, Definition 8.3], we will only be concerned with the one provided.

Definition 2.1.7. We define an invertible cabling space Tp to be the complement in the solid

torus of p ≥ 2 parallel copies of the core. Here, we note that there is a distinguished outer
boundary component arising from the original solid torus.

19

A key property of the invertible cabling spaces is that they all have simplicial volume zero.

As the following proposition states, gluings of these manifolds will preserve the extended

Turaev-Viro invariant volume conjecture.

Proposition 2.1.8 ([19], Corollary 8.4). Let M be a 3-manifold with toroidal boundary such

that the extended Turaev-Viro invariant volume conjecture holds, and let Tp be an invertible
cabling space with distinguished torus boundary T . Now the manifold M(cid:48) obtained by gluing

a component of ∂(Tp)\T to a component of ∂M satisfies the extended Turaev-Viro invariant
volume conjecture such that

lim
r→∞

2π
r

log |T Vr(M, q)| = lim
r→∞

2π
r

log |T Vr(M(cid:48), q)| = v4(cid:107)M(cid:107) = v4(cid:107)M(cid:48)
(cid:16) 2π

√−1

(cid:107)

where r runs over the odd integers and q = exp

(cid:17)

.

r

Given a manifold which satisfies the extended Turaev-Viro invariant volume conjecture,

Proposition 2.1.8 may be used to generate additional examples satisfying the conjecture. We

will see an application of this using Theorem 5.2.5 towards both the extended Turaev-Viro

invariant volume conjecture and the AMU conjecture.

2.2 Surface Mapping Class Groups

In this section, we will introduce the necessary definitions and concepts from surface mapping

class groups. Although the mapping class groups for surfaces can be defined for non-compact

surfaces, we will mostly be concerned with the compact and oriented case with the exception

of the once-punctured torus.

Definition 2.2.1. For a compact and oriented surface Σg,n of genus g with n removed open

disks, we define the mapping class group of Σg,n, denoted by M od(Σg,n), to be the group

of isotopy classes of orientation-preserving homeomorphisms of Σg,n that fix the boundary

20

f

Figure 2.2: The image of the blue curve under a Dehn twist of the annulus.

components where the group operation is composition of homeomorphisms. Additionally,

elements of M od(Σg,n) are called mapping classes.

Example 2.2.2. As a quick introductory example, we consider the mapping class group

of the sphere Σ0,0. Notice that any self-homeomorphism of Σ0,0 may be isotoped to fix a

single point x ∈ Σ0,0. We may identify the space Σ0,0\{x} with the plane R2. In R2, every

homeomorphism may be isotoped to the identity by considering the straight line homotopy,

therefore, every self-homeomorphism of Σ0,0 is isotopic to the identity map. This implies

that M od(Σ0,0) is isomorphic to the trivial group.

In order to understand more complicated examples, it is useful to consider the mapping

class group of the annulus Σ0,2.

Definition 2.2.3. For an annulus A = S1 × [0, 1] where we identify S1 as a circle of radius
1 about the origin in C, we define a Dehn twist to be the self-homeomorphism

such that f(cid:0)exp(cid:0)π√−1θ(cid:1) , t(cid:1) =(cid:0)exp(cid:0)π√−1 (θ + 2πt)(cid:1) , t(cid:1). See Figure 2.2 for an illustration

f : S1 × [0, 1] → S1 × [0, 1]

of the Dehn Twist.

Now the Dehn twist of the annulus generates the mapping class group with M od(Σ0,2) =

(cid:104)[f ](cid:105) ∼= Z.

21

1Remark 2.2.4. In general, every surface contains an annulus where we can perform a Dehn

twist. Oftentimes, we describe the annulus by a curve on the surface which is its core. The

Dehn twists are particularly useful because they can be used to generate the mapping class

group of surfaces. In this work, an important example is the mapping class group of the

once-punctured torus which we cover in Chapter 3. The surface is not compact; however,

we can treat the mapping class group of the once-punctured torus as the mapping class

group of the one-holed torus where we define a Dehn twist tδ on a curve δ parallel to the

boundary to be trivial. In this case, the mapping class group is generated by Dehn twists

along the meridian curve α and the longitude curve β. Additionally, the mapping class group
is isomorphic to the group SL2(Z) with




0

1 1
 1

0 1

−1 1

tα →

and

tβ →

where tα and tβ are the Dehn twists along the curves α and β, respectively. As a supple-

mentary remark, the trivial Dehn twist tδ can be realized as (tαtβ)6. This can be seen as the
identity matrix under the identification with SL2(Z).

2.2.1 The Nielsen-Thurston Classification

In this subsection, we give a brief introduction into the Nielsen-Thurston classification which,

as the name suggests, classifies the elements of the surface mapping class groups.

Definition 2.2.5. We consider three characterizations of elements in M od(Σg,n).

i. An element f ∈ M od(Σg,n) is periodic if there exists some power of f which is isotopic

to the identity.

22

ii. An element f ∈ M od(Σg,n) is reducible if there exists a finite nonempty set of simple
In this case, we may cut our surface

closed curves in Σg,n which are fixed under f .

Σg,n along the collection of simple closed curves, and we can consider the induced self-

homeomorphisms on the resulting parts.

iii. An element f ∈ M od(Σg,n) is pseudo-Anosov if it has infinite order and is not reducible.
As stated in the following theorem, we see that the three characterizations of the elements

in the surface mapping class groups are a complete classification.

Theorem 2.2.6 (Nielsen-Thurston classification). For any g, n ≥ 0, each f ∈ M od(Σg,n) is
either periodic, reducible, or pseudo-Anosov.

An important motivation for the Nielsen-Thurston classification is the relationship with

the geometric structures of the following family of 3-manifolds.

Definition 2.2.7. For any f ∈ M od(Σg,n), we consider the mapping torus M (f ) to be the
3-manifold defined as

M (f ) = Σg,n × [0, 1]/(x, 0) ∼ (f (x), 1)

where f is called the monodromy of M (f ).

In [55], Thurston showed the following deep connection between the Nielsen-Thurston

classification of surface mapping class groups and the geometric structures of the mapping

torus.

Theorem 2.2.8 ([55]). An element f ∈ M od(Σg,n) is pseudo-Anosov if and only if the
mapping torus M (f ) is hyperbolic.

2.2.2 Braid Groups

In this subsection, we introduce briefly the braid groups as well as their relationship with

the surface mapping class groups. For a more comprehensive survey, one may refer to [23,

Chapter 9] by Farb and Margalit.

23

1

i

n

1

i

n

σ1

σ−1

2

σi

σ−1

i

σ1σ−1

2

Figure 2.3: For the two leftmost diagrams, we have the generator σi and its inverse σ−1
,
respectively, for the braid group Bn where i ∈ {1, . . . , n− 1}. On the rightmost diagram, we
show the composition with the example σ1σ−1
2 .

i

The braid group was first rigorously defined by Artin in [4]. There are multiple equivalent

ways to define the braid group; however, we will first choose the intuitive definition of braids

as a composition of n-strands.

Definition 2.2.9. For a given integer n ≥ 2, a braid of n-strands is a composition by vertical
stackings of standard generators σi for 1 ≤ i ≤ n − 1 and their inverses σ−1
. The standard

i

generators, their inverses, and the order of the compositions are shown in Figure 2.3. In the

cases for b ∈ Bn where each strand begins and ends in the same position, we call b a pure
braid.

Definition 2.2.10. The braid group of n-strands, denoted by Bn, is the group of isotopy

classes of all n-stranded braids where the group operation is stacking. Similarly, we define

the pure braid group of n-strands, denoted by P Bn, to be the subgroup of Bn comprised of

n-stranded pure braids.

Now for a given b ∈ Bn, we can define two operations on b such that we obtain a link in

S3.

Definition 2.2.11. For b ∈ Bn, we define the braid closure of b, denoted by ˆb, to be the
link in S3 obtained from connecting the endpoints of the strands as in the left diagram of

Figure 2.4. Additionally, we may define a braided link, denoted by b, to be the link in S3

obtained from the braid closure with the addition of an unknotted component which bounds

an n-punctured disk called the braid axis as shown on the right diagram of Figure 2.4.

24

1b

b

b

b

Braid Axis

Figure 2.4: On the left diagram, we show the braid closure of an arbitrary braid b ∈ Bn. On
the right diagram, we show the diagram of the associated braided link where we labeled the
braid axis.

The braided links will be of particular use to us in this work. Our braided links in S3

will always fiber over S1 where the fiber surface is the n-punctured disk bounded by the

braid axis. We will see that this is closely related to the mapping class group of a genus zero

surface which we will utilize in finding elements which satisfy the AMU conjecture.

Definition 2.2.12. For a surface with punctures Σ, we will define M CG(Σ) to be the

group of isotopy classes of orientation-preserving self-homeomorphisms of Σ where the group

operation is composition of homeomorphisms.

Now for a given n, there exists a surjective homomorphism Γ from the braid group of

n-strands to the mapping class group of the n-punctured disk M CG(Dn) such that

Γ : Bn → M CG(Dn).

This map can be defined explicitly as sending the standard generators σi ∈ Bn, for i ∈
{1, . . . , n − 1}, to a positive half-twist on an arc between the i-th puncture to the (i + 1)-
th puncture of Dn. Furthermore, the kernel of the map Γ is generated by the element

(σ1σ2 . . . σn−1)n which corresponds to a full-twist along the boundary of Dn.

For a given Γ(b) ∈ M CG(Dn), we can construct the associated mapping torus M (Γ(b)) =
Dn× [0, 1]/(x, 0) ∼ (Γ(b)(x), 1) with monodromy Γ(b). This mapping torus is homeomorphic
to the complement of the braided link b. Now from our setup, we can note that every braid

b ∈ Bn corresponds to a braided link b which, up to homeomorphism, can be realized as the

25

1mapping torus M (Γ(b)) with monodromy Γ(b).

Lastly, we may use this correspondence with the pure braid group to study the mapping

tori for genus zero surfaces with boundary. By replacing the boundary of Dn by an additional

puncture, we induce an injective homomorphism from M CG(Dn) to M CG(S2

n+1) where S2

n+1

is the (n + 1)-punctured sphere. Under this homomorphism, we may identify every element

of M CG(Dn) as an element in M CG(S2

n+1). Now consider the case for an element b in the
pure braid group P Bn. The element Γ(b) ∈ M CG(Dn) fixes the n punctured points of Dn
which corresponds to fixing the associated punctures in S2
n+1. Because of this, we may also

realize the element Γ(b) as an element in M od(Σ0,n+1) in which we defined the boundary

components to be fixed. From the induced injective homomorphism, the complement of the

mapping torus M (Γ(b)) is homeomorphic to the mapping torus of the corresponding element

in M od(Σ0,n+1). Since our mapping torus M (Γ(b)) may be understood in terms of the braided

link b, we can similarly approach the corresponding mapping torus of M od(Σ0,n+1) with the

complement of the braided link.

2.2.3 Quantum Representations and the Turaev-Viro Invariants

In this subsection, we will write necessary properties of the quantum representations. In

particular, we will see that the quantum representations of a surface mapping class group

are closely related to the Turaev-Viro invariants of the corresponding mapping tori. Because

of this connection, it can be shown that the Turaev-Viro invariant volume conjecture implies

the AMU conjecture.

As noted in Section 1.3, for a given surface Σg,n, an odd r ≥ 3, and a coloring c of ∂(Σg,n)
2 }, we have an associated SO(3) quantum representation

by elements in Ir = {0, 1, . . . , r−5

2 , r−3

ρr,c : M od(Σg,n) → PGLdr,c(C).

Although it is not focused on in this work, we will briefly mention that the traces of the

26

individual quantum representations themselves are significant in that they are related to

another well-known quantum invariant. We denote the Reshetikhin-Turaev invariant for

closed manifolds with an embedded colored link (L, c) by RTr(M, (L, c)) where c is a coloring

of the components of L with elements of Ir. For more information on the Reshetikhin-Turaev

invariant, one may refer to [9, 48]. Now the traces of the quantum representations of the

mapping class groups are related to the Reshetikhin-Turaev invariant in the following way.

Theorem 2.2.13 ([10]). For r ≥ 3 and odd, let Σg,n be a compact and oriented surface with
a coloring of the boundary from elements of Ir. Define Σg,n to be the surface obtained from

Σg,n by capping the components of ∂Σg,n by disks. For f ∈ M od(Σg,n), let f ∈ M od(Σg,n) be
the mapping class group element of the extension of f from capping the disks by the identity.

Now let L ⊂ M (f ) whose components consists of the cores of the tori in M (f ) arising from
the capping disks on the surface. Then

T r(ρr,c(f )) = RTr(M (f ), (L, c))

where T r(ρr,c(f )) is the trace of the quantum representation.

From Theorem 2.2.13, we can see that we can study quantum invariants through the

traces of the quantum representations. Although we omit the details here, the Reshetikhin-

Turaev invariant for closed manifolds are part of a larger set of compatible invariants for

compact 3-manifolds known as a TQFT [9, 10]. In this sense, the Reshetikhin-Turaev in-

variant may be defined for manifolds M with toroidal boundary; however, in these cases,

RTr(M ) is a vector.

Now the Turaev-Viro invariant can be recovered from the Reshetikhin-Turaev invariant

which was originally shown for closed 3-manifolds by Roberts in [49] and extended to mani-

folds with boundary by Benedetti and Petronio in [8]. Although Benedetti and Petronio state

their result for the SU (2)-version of the Turaev-Viro invariant, Detcherry and Kalfagianni

adapt the proof in the setting of the SO(3)-version in [21]. The relation is as follows.

27

Theorem 2.2.14 ([8], Theorem 3.2). Fix r ≥ 3 and odd. For M an oriented and compact
3-manifold with either empty or toroidal boundary, we have

√−1

2π

r

T Vr(M, e

) = (cid:107)RTr(M, e

√−1

π

r

)(cid:107)2.

As a consequence of Theorems 2.2.13 and 2.2.14, we can similarly study the Turaev-Viro

invariant in terms of quantum representations. In the proof of Theorem 1.2 in [18], Detcherry

and Kalfagianni show the following relation.

Proposition 2.2.15 ([18], Theorem 1.2). Let f ∈ M od(Σg,n) and M (f ) the mapping torus
of f , then

(cid:88)

c

T Vr(M (f ), q) =

|T r(ρr,c(f ))|2

where c runs over all possible colorings of the boundary components of Σg,n by Ir.

Although our definition of the Turaev-Viro invariant in Section 2.1 follows the original

construction, we can equally calculate the invariant in terms of the quantum representations.

This is especially advantageous when dealing with the Turaev-Viro invariant of mapping tori.

Even though ρr,c are projective representations, we note that the scaling coefficients are roots

of unity such that the magnitude of the trace is well-defined.

We end this subsection by showing that the Turaev-Viro invariant volume conjecture

implies the AMU conjecture. This result was first shown by Detcherry and Kalfagianni in

[18].

Theorem 2.2.16 ([18], Theorem 1.2). For a compact and oriented surface Σg,n, let f be

an element in M od(Σg,n) with pseudo-Anosov parts. For the mapping torus M (f ) of the

element f , if lT V (M (f )) > 0, then f satisfies the AMU conjecture where

for r running over odd integers and q = e

lT V (M ) = lim inf
r→∞

(cid:17)

log |T Vr(M, q)|

.

(cid:16) 2π

2π
r

√−1

r

28

Proof. Let f ∈ M od(Σg,n) such that it contains pseudo-Anosov parts where we define M (f )
to be its corresponding mapping torus. Now suppose that

lT V (M ) = lim inf
r→∞

2π
r

log |T Vr(M, q)| > 0.

Therefore, our sequence of Turaev-Viro invariants with respect to r is growing exponentially.

Now from [10], it is known that the dimensions of the vector spaces of the quantum represen-

tations for these surface mapping class groups grow at most polynomially. By Proposition

2.2.15, we note that

T Vr(M (f ), q) =

|T r(ρr,c(f ))|2 .

(cid:88)

c

Here, the left-hand side grows exponentially with r, and the dimensions of the quantum

representations on the right-hand side grow at most polynomially. This implies that for a

large enough r, there exists at least one coloring c with quantum representation ρr,c(f ) such

that the modulus of T r(ρr,c(f )) exceeds the dimension of ρr,c(f ).

Now since the modulus of the trace exceeds the dimension, then there must exist at

least one eigenvalue whose modulus is strictly larger than 1. This implies that the quantum

representation ρr,c(f ) is infinite order which is the conclusion of the AMU conjecture for the

given f .

Notice if T Vr(M (f ), q) has exponential growth rate, then lT V (M (f )) > 0. This is a

more general condition; however, it states that the Turaev-Viro invariant volume conjecture

implies the AMU conjecture for mapping tori. Theorem 2.2.16 shows that as long as we can

represent our manifold as a mapping torus, we can study the Turaev-Viro invariant volume

conjecture and the AMU conjecture simultaneously by looking at the asymptotics of the

Turaev-Viro invariant.

Lastly, in conjunction with Theorem 2.2.16, we consider Proposition 2.1.6. Proposition

2.1.6 states that drilling out tori from our manifold does not decrease the growth rate of

our Turaev-Viro invariants. Together with Theorem 2.2.16, once we have a mapping torus

29

Region

Edge

Figure 2.5: The local models of the simple polyhedron.

Vertex

with exponential growth rate, we can continue to drill out tori to obtain new manifolds

with exponential growth rate from which Theorem 2.2.16 applies again. In general, the new

manifolds we obtain may not be hyperbolic; however, as we will see in Proposition 5.2.3, it

may be clear in certain circumstances which tori we may drill out such that the resulting

manifold is a hyperbolic mapping torus. By Theorem 2.2.8, our corresponding element in

the mapping class group will be pseudo-Anosov, and we know that it will also satisfy the

AMU conjecture.

2.3 Shadows of 3-Manifolds

In this section, we introduce Turaev’s shadow theory approach towards 3-manifolds [56].

Briefly, shadows are a way to study 3- and 4-manifolds using 2-dimensional polyhedra en-

dowed with topological data. We will use the theory to construct the family of fundamental

shadow links which will have direct applications towards the Turaev-Viro invariant volume

conjecture and the AMU conjecture.

We will begin by defining the theory for oriented, closed, and connected 3-manifolds.

Definition 2.3.1. A simple polyhedron P is a compact topological space which locally is

homeomorphic to one of the models in Figure 2.5. The set of points with local models of the

right two types form a graph called the singular set, denoted by Sing(P ). Additionally, the

set of points locally homeomorphic to the third model are known as vertices, and the number

of vertices of P is called the complexity of P . The set of points locally homeomorphic to the

30

1second model are known as edges, and we define the regions of P to be the set of connected

components of P\Sing(P ). Lastly, the set of points with local models corresponding to the
boundaries of Figure 2.5 are known as the boundary of P , denoted by ∂P , and if P has no

boundary components, then we say P is closed.

Definition 2.3.2. Let W be a piecewise linear, compact, and oriented 4-manifold, and let

P be a closed simple polyhedron. We say that P ⊂ W is a shadow of W if the following two
conditions are satisfied.

i. W collapses onto P as a deformation retract.

ii. Every point x in P has a neighborhood Ux ⊂ W which can be piecewise-linearly embed-
ded in R4 such that Ux ∩ P ⊂ R3 ⊂ R4. This condition will be referred to as P ⊂ W

being locally flat.

Definition 2.3.3. A closed simple polyhedron P is a shadow of an oriented 3-manifold M

if ∂(W ) = M where P is the shadow of the 4-manifold W .

Theorem 2.3.4 ([56], Theorem IX.2.1.1). Every closed, oriented, and connected 3-manifold

has a shadow.

We can see from Theorem 2.3.4 that every closed, oriented, and connected 3-manifold

has a shadow; however, the simple polyhedron P may not be unique. There could possibly

be non-homeomorphic 3-manifolds with the same shadow.

In order to uniquely describe the 3-manifolds with a shadow, we need to endow the shadow

with the additional topological information known as the gleam. In this work, we will define

the gleam of a region in its simplest case where the closure of the region is embedded in the

simple polyhedron; however, one can refer to [56] for a more detailed discussion.

Definition 2.3.5. Let D be an embedded region in P where the closure D is embedded

with ∂D ⊂ Sing(P ), and let N (D) be a regular neighborhood of ∂D ⊂ P\Int(D). We say
that the Z2-gleam of the region D is 1 if N (D) collapses over a M¨obius strip, otherwise, the
Z2-gleam is 0.

31

Definition 2.3.6. We define the gleam of a simple polyhedron to be an assignment of 1
2

Z

to all regions of the polyhedron such that the assignment is an integer if and only if the
Z2-gleam is 0. Now a simple polyhedron equipped with a gleam will be called a shadowed

polyhedron.

The idea behind the gleam is the following. When we thicken our polyhedron to a 4-
manifold, the regions of the polyhedron thicken to copies of B2 × B2 where B2 is a disk.
These copies of B2 × B2 may be glued to the rest of the thickened polyhedron in different
ways resulting in possibly non-homeomorphic manifolds. The gleam of the region is the

topological data necessary to convey the gluing information.

Now we will extend our definitions to include shadows for manifolds M containing links

where M possibly has toroidal boundary components.

In order to achieve this, we will

consider decorations of the boundary components for non-closed simple polyhedron.

Definition 2.3.7. A boundary-decorated simple polyhedron P is a simple polyhedron such

that each boundary component is either labeled as internal, external, or false. Respectively,

we will denote these subsets as ∂int(P ), ∂ext(P ), and ∂f (P ).

The intuition behind these labelings can be thought of as follows. We consider the in-

ternal labeled boundaries to be components of the link in the manifold, the external labeled

boundaries correspond to drilling out solid tori contributing to the toroidal boundary com-

ponents of the manifold, and the false labeled boundaries are to be ignored.

Now we can define the boundary-decorated simple polyhedron acting as a shadow for the

pair (M, L) where M is an oriented 3-manifold, possibly with toroidal boundary components,

and L is a link embedded in M .

Definition 2.3.8. For a 4-manifold W with ∂W = M , a boundary-decorated simple poly-

hedron P is a shadow for the pair (M, L) if the following three conditions are satisfied.

i. W collapses onto P .

32

ii. P ⊂ W is locally flat.

iii. (M, L) = (∂W\IntNbd(∂ext(P ), ∂W ), ∂int(P )) where IntNbd(∂ext(P ), ∂W ) is an interior

neighborhood of the subset ∂ext(P ) in ∂W .

Similar to before, we need to assign a gleam to the regions of the boundary-decorated

simple polyhedron in order to distinguish non-homeomorphic thickenings of the polyhedron.

Definition 2.3.9. For a boundary-decorated simple polyhedron P , an assignment by half-

integers of the regions of P which do not touch ∂f (P )∪∂ext(P ) is called a gleam. A boundary-
decorated simple polyhedron equipped with a gleam is called a boundary-decorated shadowed

polyhedron.

Now in [56], Turaev showed that if a simple polyhedron embedded in a compact oriented

smooth 4-manifold W is a shadow of ∂W , then there is an induced gleam of the regions of

the polyhedron. Otherwise stated, every shadow P of a manifold M inherits a gleam such

that P may be identified with a shadowed polyhedron of M . Conversely, Turaev showed

that given a boundary-decorated shadowed polyhedron P , one can construct a compact

and oriented 4-manifold W with an embedding of P such that the induced gleam of the

embedding corresponds to the initial gleam of P . This correspondence allows us to study

3-manifolds from the viewpoint of the boundary-decorated shadowed polyhedron.

In the following, we will show a general sketch of constructing a 3-manifold from a

boundary-decorated shadowed polyhedron. For a more in-depth discussion, one may consult

Turaev’s proof in [56].

Theorem 2.3.10 ([56], Section IX.6.1). For a given boundary-decorated shadowed polyhedron

P , there exists a pair (WP , P ) such that WP is a smooth, compact, and oriented 4-manifold,

and P is embedded such that P is a boundary-decorated shadowed polyhedron of the 3-manifold

∂WP . Additionally, the induced gleam of the regions from the embedding coincides with the

initial gleam of P .

33

Proof. We begin with the case when P is a shadowed polyhedron constructed from a closed

simple polyhedron. For when Sing(P ) is empty, WP is simply a 2-disk bundle over P with

Euler class equal to the gleam of the region. Now let us consider the case when the singular

set Sing(P ) is non-empty. We first take a neighborhood of the singular set in P , denoted

by S(P ) = N bd(Sing(P ), P ). The set S(P ) can be viewed as a collection of compositions of

the edge and vertex local models. We will now proceed to thicken this set to a 3-manifold N

such that S(P ) is properly embedded in N where the gluing of the components is induced

from the shadowed polyhedron P . Next, we further thicken the manifold N to an orientable

4-manifold W0.

The previous steps explained how to thicken the composition of the edge and vertex local

models to a 4-manifold. In the following, we will also thicken the regions of the shadowed

polyhedron to a 4-manifold, and we will glue the resulting collections together. We will

only consider one region R since adding additional regions follows similarly. As in the W0

case, we may thicken the region R into a manifold WR. When R is an orientable surface,
WR = R × B2 where B2 is a disk. In order to discuss the gluing of the manifolds WR and
W0, notice the following. From the intersection of R ∩ ∂(S(P )), we obtain a simple closed
curve C. In the 3-dimensional thickening of the collection R and S(P ), the curve C will be

either the core of an annulus or a M¨obius band. Our topological data of the gleam from P is

used as the framing of the core when identifying the components of the annuli and M¨obius

bands together.

Lastly, we will consider the case when P has boundary. For the results in this work,
we will assume that the boundaries are copies of S1. We let {Li} be the collection of
boundary components such that L is their union. We now thicken L to be the framed

link L × [−1, 1] such that L × {0} = ∂(P ). This results in a new polyhedron P where
Sing(P ) = Sing(P ) ∪ ∂(P ), and we may follow the same construction as before using
Sing(P ) to obtain a 3-manifold containing the framed link L × [−1, 1]. Now by repeating
the rest of the construction, we end with a 4-manifold WP such that L×[−1, 1] ⊂ ∂(WP ).

34

2.3.1 Fundamental Shadow Links

In this subsection, we will construct the fundamental shadow links. The family was originally

introduced and discussed by Constantino and Thurston in [15, Section 3.6]. The fundamental
shadow links are a family of links in connected sums of S2 × S1 with certain advantageous
properties. In particular, the Turaev-Viro invariants of the family are relatively simple, and

the hyperbolic volumes of the complements are well-understood. These two properties were

utilized to prove the following theorem in [7] by Belletti, Detcherry, Kalfagianni, and Yang.

Theorem 2.3.11 ([7], Theorem 1.1). For any fundamental shadow link L, the link comple-

ment of L satisfies the Turaev-Viro invariant volume conjecture.

Additionally, these links are “universal” in the sense that every orientable 3-manifold

with empty or toroidal boundary can be realized as the manifold obtained from Dehn fillings

along some subset of the boundary components for the complement of some fundamental

shadow link. For the definition of a Dehn filling, see Subsection 4.1.3. As a consequence, it

may be possible to prove the Turaev-Viro invariant volume conjecture by showing that the

conjecture is preserved under Dehn fillings of the boundary tori for the complements of the

fundamental shadow links.

Now as discussed in Subsection 1.2.2, the Turaev-Viro invariant volume conjecture is

preserved under homeomorphisms. This implies that if two manifolds are homeomorphic

where ones satisfies the Turaev-Viro invariant volume conjecture, then both manifolds satisfy

the conjecture. This will be our main motivation in considering the fundamental shadow

link complements.

In order to construct this family, we will be using Theorem 2.3.10 on a particular family

of boundary-decorated shadowed polyhedron. We will begin by considering a closed simple

polyhedron P which contains at least one vertex in the singular set Sing(P ) where Sing(P )

is also connected. Now let S(P ) be a regular neighborhood of Sing(P ) in P such that S(P )

can be viewed as a composition of the edge and vertex local models. From S(P ), we define

35

P (cid:48) to be the boundary-decorated shadowed polyhedron such that S(P ) is the 2-dimensional
simple polyhedron with the boundary components of P (cid:48) decorated as external boundaries.

We will now build a 3-manifold MP (cid:48) with boundary by applying Theorem 2.3.10 on
P (cid:48). We can better understand the manifold MP (cid:48) by studying the equipped collapsing map
π : MP (cid:48) → P (cid:48) obtained from the construction. As shown in Proposition 3.32 from [15], the
preimages of the edge local models will be a product of a pair of pants with [−1, 1], and the
preimages of the vertex local models are genus 3 handlebodies. The components glue along

pairs of pants, and the resulting manifold MP (cid:48) has boundary a union of tori. In Subsection

2.3.2, we will show an explicit representation of the manifold in terms of the complement of

a surgery presentation of a link.

Now in the following, we can see that the manifold MP (cid:48) has a decomposition into regular

ideal hyperbolic octahedra, therefore, an explicit hyperbolic volume.

Proposition 2.3.12 ([15], Proposition 3.33). For a given P (cid:48) as above, the manifold MP (cid:48)

can be equipped with a complete hyperbolic metric with volume 2kv8. Here, k is the number

of vertices of P (cid:48), and v8 ≈ 3.6639 is the volume of a regular ideal hyperbolic octahedron.

Moreover, the manifold MP (cid:48) can be realized as the complement of a link embedded in
connected sums of S2 × S1. This can be achieved using the same simple polyhedron S(P );
however, instead of decorating the boundary components as external boundaries, we choose

to decorate the boundary components as internal boundaries.

Definition 2.3.13. Let P (cid:48)(cid:48) be the boundary-decorated shadowed polyhedron with simple

polyhedron S(P ) of complexity k and with boundary components decorated as internal
boundaries. Using Theorem 2.3.10, the 4-manifold WP (cid:48)(cid:48) obtained from P (cid:48)(cid:48) corresponds to

the pair

(MP (cid:48)(cid:48), LP (cid:48)(cid:48)) = (∂(WP (cid:48)(cid:48)), ∂int(P (cid:48)(cid:48))) = (#k+1(S2 × S1), LP (cid:48)(cid:48)).

These pairs of links and 3-manifolds are known as the fundamental shadow links. In addition,

the 3-manifold MP (cid:48) is homeomorphic to the link complement of LP (cid:48)(cid:48) in the manifold MP (cid:48)(cid:48).

36

Vertex

Figure 2.6: The procedure to replace the vertices of the graph.

2.3.2 Planar Graph Constructions of Fundamental Shadow Links

Although the fundamental shadow links can be constructed using boundary-decorated shad-

owed polyhedron, it is more intuitive to build them from planar 4-valent graphs. In particu-

lar, the fundamental shadow links of complexity k can be defined in terms of 0-surgeries on

a set of (k + 1) unknotted link components of a link in S3 obtained from the planar 4-valent

graphs. For a brief summary of a 0-surgery on a link component, see Subsection 4.1.3.

Before we begin, we will first define the following.

Definition 2.3.14. We define a planar 4-valent graph G to be a connected graph embedded

on the plane in S3 such that each vertex is incident to 4 edges where loops are edges which

are counted twice. Additionally, the complexity of G is the number of vertices of the graph.

Definition 2.3.15. For a graph G, a tree is a connected subgraph without any cycles.

Furthermore, a maximal tree is a tree with a maximal set of edges, or equivalently, a tree

which includes all vertices of G.

We will now state the construction which was first discussed as Proposition 3.35 in [15]

and written explicitly in terms of the planar 4-valent graphs in [14, Section 2.3].

• Let G be a planar 4-valent graph of complexity k embedded in S3, and let T be a

maximal tree of the graph G.

• We replace each edge of G by any 3-braid, and we replace each vertex of G by the 6

strands in Figure 2.6 or its mirror.

37

1Figure 2.7: An example of constructing a fundamental shadow link from a planar 4-valent
graph G. On the left, we begin with a planar 4-valent graph of complexity 2 where the
maximal tree T is colored blue. We obtain the middle illustration by replacing the edges
and vertices by 3-braids and the 6 strands described in Figure 2.6, respectively. The right
illustration is obtained by encircling the 3-braids corresponding to the edges G\T by 0-framed
unknotted components.

• Next, for each 3-braid corresponding to an edge of G\T , we encircle the 3-braid by a

0-framed unknotted component.

• Lastly, the union of the 3-braids, the 6 strands obtained from Figure 2.6, and the

0-framed unknotted link give a surgery presentation in S3 of a fundamental shadow

link.

As in the case of the boundary-decorated shadowed polyhedron, the hyperbolic volume

of the resulting fundamental shadow link from a graph of complexity k is given by 2kv8.

In Figure 2.7, we provide an explicit example of constructing a fundamental shadow link

given a graph G of complexity 2. In this case, our maximal tree consists of a single edge

which is colored blue in the leftmost diagram of Figure 2.7. Shown in the middle diagram

of Figure 2.7, we decide to replace three of the four edges with the identity 3-braid and the

fourth edge with a non-trivial 3-braid. On the rightmost diagram of Figure 2.7, we encircle

by 0-framed unknotted components the 3-braid components which correspond to G\T .

Remark 2.3.16. For a given planar 4-valent graph G, we have multiple choices of 3-braids

for each edge of the graph. The choice of 3-braids may result in different non-homeomorphic

fundamental shadow link complements such that the graph G does not have a unique as-

sociated fundamental shadow link. However, it is to be noted that 3-braids which induce

the same permutation on the 3-strands result in homeomorphic fundamental shadow link

38

0001complements. Therefore, there are only finitely many fundamental shadow links for each

given complexity k. Because there are only finitely many hyperbolic manifolds of the same

given volume [54, Chapters 5 and 6], this is consistent with the fact that the fundamental

shadow links are hyperbolic links.

Remark 2.3.17. From the viewpoint of constructing a fundamental shadow link from a

boundary-decorated shadowed polyhedron, we can see that the strands of the 3-braids are

the boundary of a simple polyhedron S(P ). For the simple polyhedron S(P ), the singular

set Sing(P ) is the planar 4-valent graph.

It is from this correspondence that we obtain

the planar 4-valent graph construction from the boundary-decorated shadowed polyhedron

construction.

39

Chapter 3

The Turaev-Viro Invariants for Once-

Punctured Torus Bundles

In this chapter, we define a closed formula for the Turaev-Viro invariants of hyperbolic once-

punctured torus bundles which depends, in a trivial way, on the monodromy. Additionally,

we introduce a conjecture which relates the Turaev-Viro invariants of the mapping torus to

the stretch factor of the corresponding pseudo-Anosov element in the mapping class group.

3.1 Background

In this section, we introduce the canonical triangulations for hyperbolic once-punctured torus

bundles.

3.1.1 Canonical Triangulations

For a reference, see [24] by Floyd and Hatcher as well as [25, Section 3] by Futer and

Gu´eritaud. Given a pseudo-Anosov element in the mapping class group of the once-punctured

torus, we will highlight the construction of the canonical triangulation for the corresponding

mapping torus.

From the isomorphism between M od(Σ1

1) and SL2(Z) described in Section 2.2, we define

the following matrices

R =

1 1

0 1



40

and

1 0

1 1

 .

L =

Now consider the following well-known proposition. For a sketch of the proof, we refer

to [25, Proposition 2.1].

Proposition 3.1.1. Let φ be an element in SL2(Z) with two distinct positive eigenvalues.
Then the conjugacy class of φ ∈ SL2(Z) contains an element of the form

RN1LN2RN3LN4 . . . RN2n−1LN2n

where n, Ni > 0 for all i. Conversely, any product of such form is an element in SL2(Z)

with two distinct positive eigenvalues.

Since mapping tori constructed with conjugate monodromies are homeomorphic, we may

always consider our element in the mapping class group to be represented in this form. We

now begin with an arbitrary matrix

RN1LN2RN3LN4 . . . RN2n−1LN2n =

From the given monodromy, we consider the following vectors

 .

c d

a b
 .

a

 ,

c

b

 ,

d

a + b

c + d

and

When viewing the once-punctured torus Σ1

1 as the quotient of the space R2\Z2, we can form

a triangulation of the space by lines connecting the punctures where the slopes of the lines

are the rational slopes of the vectors. Now note that when we multiply our monodromy

on the right by either R or L, one of the corresponding vectors will be fixed by either the

41

column

of R or the column

1

0


0

1

of L. This implies that after multiplication, we will have a new triangulation of Σ1

1 using the

slopes associated to the fixed vector, the vector

a + b

c + d

 ,

and a new vector which is the sum of the other two vectors. Because of this, the new

triangulation will always switch exactly one edge. This principle will be known as a diagonal

exchange.

Now given a sequence of triangulations of Σ1

1 related by diagonal exchanges, we can form

a collection of ideal tetrahedra in the sense that the vertices are removed by the puncture

in the triangulation of Σ1

1. This is obtained by considering the 2-dimensional triangulation

formed by pairwise gluing edges with identical slopes from two consecutive triangulations

of Σ1

1. The resultant figure will bound our desired tetrahedron where two opposite pairs of

edges are identified together. The remaining two edges which are not identified correspond

to the diagonal exchange.

Now lastly, our collection of tetrahedra are stacked on top of each other from the previous

gluings such that the bottommost and topmost triangulations of Σ1

1 are unaffected. These

final two triangulations are glued together by the self-homeomorphism from the mapping

class group. The resulting 3-dimensional triangulation is homeomorphic to the mapping

torus obtained from the corresponding mapping class group element.

42

3.2 A Closed Formula for the Turaev-Viro Invariants

In this section, we calculate a closed formula for the Turaev-Viro invariants of the hyperbolic

once-punctured torus bundles as well as compute explicit examples. This will be achieved by

considering the canonical triangulations of the manifolds along with the original definition

of the Turaev-Viro invariant stated in Section 2.1.

As mentioned in Subsection 3.1.1, for a given monodromy which is pseudo-Anosov, we

can construct a triangulation of the manifold through diagonal exchanges. Now notice the

following. Every tetrahedron of the manifold is constructed from a diagonal exchange on a

triangulation of the once-punctured torus where each diagonal exchange corresponds to either
the element R or L. Because of this, for a given monodromy φ = RN1LN2 ··· RN2n−1LN2n
in M od(Σ1

1), the number of tetrahedra for the mapping torus M (φ) is N =(cid:0)(cid:80)2n

(cid:1). On

i=1 Ni

the triangulations of the once-punctured torus, we have two pairs of opposite sides that are

identified, and the sides corresponding to the diagonal exchange will be opposite of each

other on the formed tetrahedron. Therefore, every 6j-symbol is of the following form.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3

a4 a2 a3

E1 =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , E2 =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3

a1 a4 a3

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3

a1 a2 a4

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

or E3 =

where Ei is the 6j-symbol such that the side opposite of the side associated to the color ai is

the side of the diagonal exchange. In this case, we chose to color the new side obtained from

the diagonal exchange with a4. For an illustration of the triangulation of the once-punctured

torus bundles, see Figure 3.1 where we demonstrate the first tetrahedron obtained from the

diagonal exchanges. In this example, as the quotient of R2\Z2, the once-punctured torus
may be triangulated by edges with slopes 1, 0, and ∞, shown on the leftmost diagram, which

43

0

1

∞

1

1

0

0

0

1

1
2

0

1
2

1

∞

1

0

Figure 3.1: On the left, we have two triangulations of the once-punctured torus as the

quotient of R2\Z2. On the right, we have the tetrahedron obtained from stacking the trian-

gulations by identifying the edges with the same slope. Here, the slopes of the triangulations
are given in the illustrations, the white vertices are the puncture, and the red and blue edges
correspond to the edges of the diagonal exchange.

corresponds to the vectors given by the identity matrix

0 1


1 0
1 1

 .

R =

0 1

as well as edges with slopes 1, 0, and 1

2, shown on the second diagram, which corresponds to

vectors given by the matrix

The two resulting triangulations may then be identified to obtain the tetrahedron of the

rightmost diagram where each edge of the same slope is glued together. Since every tetra-

hedron of the triangulation is formed in a similar way, we see more directly that, depending

on the monodromy, each tetrahedron will correspond to one of the defined 6j-symbols E1,

E2, and E3.

To simplify the notation, we will now define a specific product, denoted by ·, on the 6j-
symbols Ei. We will think of the operation Ei· Ej as the standard product of the 6j-symbols
where we additionally “glue” the bottom row of Ei to the top row of Ej. Because of this,

we refer to the product as a gluing of 6j-symbols. This is best illustrated with an example

44

1such as

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3

a4 a2 a3

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a2 a3

a4 a5 a3

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a5 a3

a4 a6 a3

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a6 a3

a4 a6 a7

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .

E1 · E2 · E2 · E3 =

This product corresponds with the general principle of obtaining the triangulation of

the once-punctured torus bundles where we stack the tetrahedra on top of each other. For

convenience, we will take the convention of labeling the 6j-symbols such that the first 6j-

symbol in the product has the first row labeled as

(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3

(cid:12)(cid:12)(cid:12)(cid:12) ,

and we label any edge obtained from the diagonal exchange with the subsequent index such

as in the example of E1 · E2 · E2 · E3. Additionally, it is worth noting that any gluing of k
6j-symbols will have (k + 3) labelings by colors ai.

Before we state the closed formula, we will introduce more useful notation using the

monodromy RN1 as our example. By nature of the construction, the triangulation associated

to RN1 will always fix one of the edges. As mentioned in Subsection 3.1.1, this is due to the

first column of R being the vector

1
 .

0

Now because of this, the diagonal exchanges are going to alternately occur on the other

two non-fixed edges. In terms of the Ei’s, this corresponds to N1 alternating gluings of Ei0
and Ej0 where i0 (cid:54)= j0. We will denote this alternating gluing as EN1
the column which is not either i0 nor j0 will always remain a fixed color under the gluings.

where we note that

i0,j0

Additionally, we can see that depending on whether N1 is odd or even, the final Ei in the

alternating gluings of EN1
i0,j0

will either be Ei0 or Ej0, respectively. We will denote this final

45

index by j1 such that

gluing ENk

ik−1,jk−1

, the final index is

jk =

j1 =

.

j0 For N1 even

i0 For N1 odd
ik−1 For Nk odd

jk−1 For Nk even

.

Furthermore, we can generalize this notation to all RNk such that for any alternating

Analogously for LNk, we can define the same alternating gluing ENk

ik−1,jk−1

with jk defined in

the same manner.

For the once-punctured torus Σ1

1, we will now state the closed formula for the correspond-
ing mapping torus of a pseudo-Anosov element where we define |ai| = (−1)2ai[2ai + 1] for a
color ai.

2n, and let N =(cid:0)(cid:80)2n

Theorem 3.2.1. Let φ = RN1LN2 ··· RN2n−1LN2n ∈ M od(Σ1

(cid:1). Then the mapping torus M (φ) has Turaev-Viro invariant
(cid:17)(cid:35)

(cid:34)(cid:32)N +3(cid:89)

1) where n, Ni > 0 for 1 ≤ i ≤

(cid:33)(cid:16)

|ak|

EN1
i0,j0 · EN2

i1,j1 · . . . · EN2n

i2n−1,j2n−1

i=1 Ni

T Vr (M (φ) , q) = 2b2−b0(cid:88)

adm

k=4

where the sum is over all admissible colorings, a1 = a(N +2), a2 = a(N−N2n+2), and a3 = a(N +3).

Additionally, the indices ik and jk may be computed inductively from (i0, j0) = (2, 3) where
ik (cid:54)∈ {ik−1, jk−1} and with

ik−1 For Nk odd

jk−1 For Nk even

.

jk =

46

Proof. From Theorem 2.1.5, the Turaev-Viro invariant may be defined as

T Vr(M, q) = 2b2−b0η2|V |

r

(cid:34)(cid:32)(cid:89)

(cid:88)

(cid:33)(cid:32)(cid:89)

c∈Ar(τ )

e∈E

|e|c

|∆|c

∆∈τ

(cid:33)(cid:35)

where c runs over all admissible colorings of the edges. Since our triangulation is ideal, then

|V | = 0 such that

T Vr(M, q) = 2b2−b0 (cid:88)

c∈Ar(τ )

(cid:34)(cid:32)(cid:89)

e∈E

(cid:33)(cid:32)(cid:89)

∆∈τ

|e|c

(cid:33)(cid:35)

|∆|c

.

Notice that our triangulation will have N distinct edges such that each edge is colored by ak

where 4 ≤ k ≤ N + 3. The edges corresponding to the colors a1, a2, and a3 are not included
since they will be identified to other edges under the homeomorphism φ. We now obtain

T Vr(M (φ), q) = 2b2−b0(cid:88)

(cid:34)(cid:32)N +3(cid:89)

(cid:33)(cid:32)(cid:89)

(cid:33)(cid:35)

adm

k=4

|ak|

|∆|c

.

∆∈τ

Next, to obtain the 6j-symbols, we need to consider the triangulations of the manifold.

As mentioned previously, each RNk and LNk(cid:48) corresponds to the alternating gluings of 6j-

symbols ENk

ik−1,jk−1

and ENk(cid:48)

ik(cid:48)−1,jk(cid:48)−1

. We consider the case when we multiply RNk on the right

by L where the case for multiplying LNk(cid:48) on the right by R is similar. This corresponds to a

diagonal exchange on the edge that is fixed by RNk such that we introduce a gluing on the

right by the 6j-symbol Eik where ik (cid:54)∈ {ik−1, jk−1}. Now if we continue multiplying on the
right by L, we will once again have an edge which is fixed such that we have an alternating

gluing of 6j-symbols ENk+1
ik,jk
diagonal exchange of ENk

ik−1,jk−1

such that

for some jk. This jk will correspond to the column of the final

ik−1 For Nk odd

jk−1 For Nk even

.

jk =

47

This implies that we can rewrite(cid:0)(cid:81)
T Vr(M (φ), q) = 2b2−b0(cid:88)

such that

∆∈τ |∆|c

(cid:34)(cid:32)N +3(cid:89)

(cid:1) in terms of alternating gluings of 6j-symbols
(cid:33)(cid:16)

(cid:17)(cid:35)

EN1
i0,j0 · EN2

i1,j1 · . . . · EN2n

i2n−1,j2n−1

.

|ak|

adm

k=4

Lastly, we will consider the identification of the colors under the gluing of the edges

by the map φ. We will note that each once-punctured torus is triangulated by lines with

rational slopes between the punctures. For any given monodromy conjugated to the form
RN1LN2 ··· RN2n−1LN2n with matrix

 ,

b(cid:48)
c(cid:48) d(cid:48)

a(cid:48)
 ,
b(cid:48)
a(cid:48) + b(cid:48)

c(cid:48) + d(cid:48)

d(cid:48)

and



the slopes of the lines are determined by the slopes of the vectors

 ,

a(cid:48)

c(cid:48)

a(cid:48) + b(cid:48)

c(cid:48) + d(cid:48)

 .

Here, the vector

corresponds to the new edge obtained from the diagonal exchange. From this viewpoint, it

is more clear to understand the alternating gluings. For example, for φRn, we have that

a(cid:48) a(cid:48)n + b(cid:48)

c(cid:48)n + d(cid:48)

c(cid:48)



φRn =

a(cid:48)

 =
1 n

such that the edge corresponding to the vectora(cid:48)


b(cid:48)
c(cid:48) d(cid:48)

0 1

c(cid:48)

48

is always fixed. Now if we multiply φRn by Lm, then we have

a(cid:48) a(cid:48)n + b(cid:48)

c(cid:48)n + d(cid:48)

c(cid:48)



φRnLm =

such that the new fixed edge corresponds to the vector



0

m 1

c(cid:48)n + d(cid:48)

c(cid:48) + m(c(cid:48)n + d(cid:48))

 1
 =
a(cid:48) + m(a(cid:48)n + b(cid:48)) a(cid:48)n + b(cid:48)
a(cid:48)n + b(cid:48)
 .
a(cid:48)(n + 1) + b(cid:48)

c(cid:48)n + d(cid:48)



c(cid:48)(n + 1) + d(cid:48)

Notice that when m = 1, then the first column of φRnL is

which corresponds to the edge obtained by the final diagonal exchange of φRn. This implies

that by gluing the tetrahedra corresponding to Lm, our gluings of 6j-symbols are alternating

between the Ei’s associated to the previously fixed column of φRn and the Ej’s associated

to the final diagonal exchange of φRn .

We will now continue with our consideration of gluing the edges of the top and bottom

tetrahedra. If we consider the triangulation of the once-punctured torus by

0 1

1 0
 ,
0
 ,

1

then our slopes are determined by the vectors

1
 ,

0

v1 =

v2 =

1
 .

1

and v3 =

In order to see which edges are identified, we only need to check the mapping of the vectors

49

v1, v2, and v3 under φ.

Let us consider the final multiplication of the monodromy RN1LN2 ··· RN2n−1LN2n−1 by

L. This may be viewed as the matrix

φ = RN1LN2 ··· RN2n−1LN2n−1L =

such that we have the vectors

a + b

c + d

 , w2 =

b

d

w1 =

and w3 =



b

c + d d



c d

a b
 ,

1 1

1 0
 =
a + b
 .
a + 2b

c + 2d

First, we note that φ(v3) = w3 where w3 corresponds to our final diagonal exchange.

Since v3 is associated to the color a3, then we are identifying a3 with a(N +3) which is the

color of the final edge added by the diagonal exchange.

Next, we note that φ(v1) = w1 where w1 corresponds to the previous diagonal exchange.

Similar to before, since v1 is associated to the color a1, then we are identifying a1 with a(N +2)

which is the color of the edge added by the previous diagonal exchange.

In the last case, φ(v2) = w2 where w2 corresponds to the fixed edge under LN2n. Here,

v2 is associated to the color a2 which we are identifying to the color of the fixed column of
. This edge is obtained from the gluing of R(N2n−1)−1 which

the alternating gluing EN2n

i2n−1,j2n−1

has the color a(N−N2n+2).

Although the following is not reliant on Theorem 3.2.1, it is still worth noting to demon-

strate the similarities between the hyperbolic volume of the manifold and the asymptotics of

the Turaev-Viro invariants. In [7], Belletti, Detcherry, Kalfagianni, and Yang showed a rela-

tionship between the hyperbolic volume of a regular ideal octahedron and the asymptotics

of the 6j-symbols at q = exp

(cid:16) 2π

(cid:17)

.

√−1

r

(cid:16) 2π

(cid:17)

√−1

r

and any admissible 6-tuple

Theorem 3.2.2 ([7], Theorem 1.2). For q = exp

50

(i, j, k, l, m, n) ∈ I 6

r , we have that

lim sup

r→∞

2π
r

log

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i

j

k

l m n

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ v8

where r runs over the odd integers and with v8 the volume of a regular ideal octahedron.

Using Theorem 3.2.2, we have the direct calculation that

LT V (M (φ)) = lim sup

where φ = RN1LN2 ··· RN2n−1LN2n, N = (cid:0)(cid:80)2n

r→∞

2π
r

log |T Vr (M (φ), q)| ≤ N v8

(cid:1), and q = exp

(cid:16) 2π

i=1 Ni

(cid:17)

. For the steps

√−1

r

involved in the calculation, see the proof of Theorem 1.3 in [19] by Detcherry and Kalfagianni.

On the hyperbolic geometry side, a similar, although more precise, inequality has been

obtained by Agol in [1, Theorem 2.4], and separately by Futer and Gu´eritaud in [25, Theorem

B.1], where the inequality is given by

V ol(M (φ)) < 2nv8.

The similarities of the two results further support the claim of the Turaev-Viro invariant

volume conjecture. Furthermore, it also suggests that much sharper bounds on the asymp-

totics of the Turaev-Viro invariants for the once-punctured torus bundles can be obtained

through closer analysis.

3.2.1 Calculations of Turaev-Viro Invariants

In this subsection, we will calculate some examples of the Turaev-Viro invariants for once-

punctured torus bundles using Theorem 3.2.1.

Example 3.2.3. We will begin with the small example of RL ∈ M od(Σ1
complement of the mapping torus is homeomorphic to the complement of the figure-eight

1). For context, the

51

Figure 3.2: An ideal triangulation of the figure-eight knot. Here, we identify the black edges
together, and we identify the blue edges together.

knot in S3 with hyperbolic volume 2v4 ≈ 2.0299.

By Theorem 3.2.1, we have that φ = RN1LN2 where N1 = N2 = 1 such that N =

N1 + N2 = 2. Now the indices ik and jk may be computed inductively from (i0, j0) = (2, 3).

Since i1 (cid:54)∈ {i0, j0}, then i1 = 1, and since N1 = 1 is odd, then j1 = i0 = 2. This implies that

T Vr(M (φ), q) = 2b2−b0(cid:88)
= 2b2−b0(cid:88)

adm

(cid:2)(|a4||a5|)(cid:0)E1
(|a4||a5|)

adm

T Vr(M (φ), q) = 2b2−b0(cid:88)
= 2b2−b0(cid:88)

adm

adm

Additionally, we can compute that a1 = a4, a2 = a3, and a3 = a5. Using the symmetry

relations of the 6j-symbols in Section 2.1, we have that

1,2

[(|a4||a5|) (E2 · E1)]

adm

a1 a4 a3

a5 a4 a3

2,3 · E1

 .

(cid:1)(cid:3) = 2b2−b0(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a4 a3
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a1 a2 a3
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a5 a5
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a4 a5
(|a4||a5|)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a4 a4 a5
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(|a4||a5|)
2 .

a5 a4 a5

a4 a4 a5

a4 a5 a5



(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

This Turaev-Viro invariant corresponds with the known ideal triangulation of the figure-eight

knot given in Figure 3.2.

52

1Example 3.2.4. In this example, we will consider the element RN1LN2 ∈ M od(Σ1
case, we have that N = N1 + N2, and the indices change depending on the parity of N1. We

1). In this

still have that i0 = 2, j0 = 3, and i1 = 1; however, we will either have that j1 = 2 or j1 = 3

for N1 odd or even, respectively. For the gluings of the edges, we have that a1 = aN +2,

a2 = aN1+2, and a3 = aN +3. This gives us either

T Vr(M (φ), q) = 2b2−b0(cid:88)
T Vr(M (φ), q) = 2b2−b0(cid:88)

adm

(cid:34)(cid:32)N +3(cid:89)
(cid:34)(cid:32)N +3(cid:89)

k=4

(cid:33)(cid:0)EN1
(cid:33)(cid:0)EN1

(cid:1)(cid:35)
(cid:1)(cid:35)

|ak|

2,3 · EN2

1,2

|ak|

2,3 · EN2

1,3

adm

k=4

for N1 odd, or

for N1 even.

Example 3.2.5. As our last example, we will consider the element (RL)n ∈ M od(Σ1
mapping torus of this element has complement homeomorphic to the complement of the

1). The

n-fold cyclic cover of the figure-eight knot. Because of this, the complement has hyperbolic

volume 2nv4.

We consider φ = RN1LN2RN3LN4 . . . RN2n−1LN2n such that each Ni = 1 with N = 2n.

Now for the indices, notice the following. Since every Ni is odd, then jk = ik−1 for every
k. Also notice that ik (cid:54)∈ {ik−1, jk−1} = {ik−1, ik−2}. Since initially i0 = 2 and i1 = 1, then
i2 = 3. Inductively, this implies that ik cycles between the numbers 2, 1, and 3 starting at

i0 = 2. For the gluings, we have that a1 = a2n+2, a2 = a2n+1 and a3 = a2n+3.

Now our Turaev-Viro invariants will be 2n gluings of 6j-symbols and fall into one of the

three following cases:

T Vr(M (φ), q) = 2b2−b0(cid:88)

(cid:34)(cid:32)2n+3(cid:89)

adm

k=4

(cid:33)

(cid:35)

|ak|

(E2 · E1 · E3 · . . . · E2 · E1 · E3)

53

for n ≡ 0 mod 3.

T Vr(M (φ), q) = 2b2−b0(cid:88)

(cid:34)(cid:32)2n+3(cid:89)

adm

k=4

for n ≡ 1 mod 3.

T Vr(M (φ), q) = 2b2−b0(cid:88)

(cid:34)(cid:32)2n+3(cid:89)

adm

k=4

for n ≡ 2 mod 3.

(cid:33)

(cid:35)

|ak|

(E2 · E1 · E3 · . . . · E3 · E2 · E1)

(cid:33)

(cid:35)

|ak|

(E2 · E1 · E3 · . . . · E1 · E3 · E2)

Remark 3.2.6. The Turaev-Viro invariant volume conjecture is known for the complement

of the figure-eight knot [21]; however, the proof utilized the relationship between the Turaev-

Viro invariants and the colored Jones polynomials. This relationship is a special case of

Theorem 2.2.15 which related the traces of the quantum representations to the Turaev-Viro

invariants. It would be of interest to prove the Turaev-Viro invariant volume conjecture for

the figure-eight knot from the viewpoint of the triangulation given in Example 3.2.3. This

may give further insight on proving the Turaev-Viro invariant volume conjecture for the

n-fold cyclic cover of the figure-eight knot using the triangulation given in Example 3.2.5.

3.3 A Conjecture in Relation to the Stretch Factor

In this section, we will state a conjecture which relates the asymptotics of the Turaev-Viro

invariants of a mapping torus with its stretch factor. This conjecture is supported by known

results in hyperbolic geometry.

Briefly, we will introduce the stretch factor for the example of the once-punctured torus.

For more details, see Chapter 14 in [23]. Now for the once-punctured torus Σ1

1, the mapping

class group M od(Σ1

1) is generated by the Dehn twists tα and tβ along the canonical meridian

and longitude, respectively. As stated in Section 2.2, there is also an isomorphism between

54

M od(Σ1

1) and SL2(Z) which sends

tα →

 = R
1 1
 1
 = L−1

0 1

and

tβ →

0

−1 1

where R and L−1 are the generators of SL2(Z). Now for certain elements in M od(Σ1

1), we

can define the stretch factor.

Definition 3.3.1. For a given element φ ∈ M od(Σ1
unique real-valued eigenvalue λ > 1 is defined as the stretch factor. Furthermore, φ is

1) ∼= SL2(Z) with |T race(φ)| > 2, the

pseudo-Anosov if and only if |T race(φ)| > 2.

It is easier to understand the stretch factor when viewing the once-punctured torus as

a quotient of R2\Z2. Homeomorphisms of Σ1
1 may be viewed as the corresponding matrices
of SL2(Z) acting on R2\Z2 followed by the quotient of the space. Because we have that
|T race(φ)| > 2, any curve in Σ1
1 will be stretched by λ along parallel copies of one eigenvector
and will contract by 1

λ along parallel copies of the other eigenvector. These dilations and

contractions are the geometric interpretation of the stretch factor. In fact, a pseudo-Anosov

map is more accurately defined as a generalization of this principle.

Now we will discuss the following open question posed by Andersen, Masbaum, and Ueno

in [3].

Question 3.3.2 ([3], Question 1.1). Can one compute the stretch factor for pseudo-Anosov

mapping classes from the quantum representations?

In [3], the authors show that this is true for the four-punctured sphere with certain

colorings of the punctures. In regards to Question 3.3.2, we consider the case for the once-

punctured torus where there is only one puncture to color. In [51], Santharoubane showed

55

that, for any dimension N , the sequence of quantum representations for a fixed dimension

converge to a matrix with known eigenvalues. In particular, the eigenvalues of the convergent

matrix are powers of the stretch factor which implies that Question 3.3.2 is also true for the

once-punctured torus.

Now the following is not rigorous and only supported through small computational exam-

ples. For a fixed r ≥ 3 with r odd, there is one quantum representation ρN
N for 1 ≤ N ≤ r−1

2 where c(N, r) is a chosen coloring of the puncture such that the quantum

r,c(N,r) of dimension

representation has dimension N . For an explicit formulation of the quantum representations,

one may refer to [26] by Gilmer and Masbaum as well as [51] by Santharoubane. Now given

a pseudo-Anosov element φ with stretch factor λφ, we make the assumption that, for any

N , the eigenvalues λi,N of the quantum representation ρN

r,c(N,r)(φ) has the property that

. The motivation behind assuming this property is that it appears that the

|λi,N| ≤ λ
eigenvalues of the quantum representations converge to the eigenvalues of the convergent

φ

r−3
2

matrix from below. By Proposition 2.2.15, we have the following calculation which would

imply the subsequent conjecture.

r−1

N =1
r−1

2(cid:88)
2(cid:88)
2(cid:88)

N =1
r−1

=

r−1

2(cid:88)
(cid:12)(cid:12)T r(ρN
r,c(N,r)(φ))(cid:12)(cid:12)2
2(cid:88)
(cid:2)N(cid:0)
|λ1,N|2 + ··· + |λN,N|2(cid:1)(cid:3)
2(cid:88)
(cid:1) r−1
(cid:1)(cid:3) =(cid:0)λr−3
(cid:2)N 2(cid:3) =
(cid:2)N 2(cid:0)λr−3

≤

N =1

r−1

N =1

φ

φ

N =1

N =1

T Vr(M (φ)) =

≤

=

|λ1,N + ··· + λN,N|2

(cid:34)

N

(cid:32) N(cid:88)

(cid:33)(cid:35)

λr−3

φ

λr−3

φ

i=1

(r3 − r)
24

.

Conjecture 3.3.3. Let φ ∈ M od(Σ1
Then

1) be a pseudo-Anosov element with stretch factor λφ.

LT V (M ) = lim sup

r→∞

2π
r

log |T Vr (M (φ), q)| ≤ 2π log (λφ)

(cid:16) 2π

(cid:17)

.

√−1

r

where r runs along the odd integers and q = exp

Although we could write a more general conjecture for all surface mapping class groups,

56

the arguments for the four-punctured sphere and the once-punctured torus depended on

exploiting their convergence to the homological representation which does not generalize to

other cases. Because of this, it is not known to what extent the quantum representations

determine the stretch factors of the mapping class group.

The main motivation of Conjecture 3.3.3 is still to relate the asymptotics of the Turaev-

Viro invariants to the hyperbolic volume of the manifold. The term log(λφ), known as the

entropy, has significance in hyperbolic geometry. By the works of Brock, Mazur, and Minsky

in [11, 12, 42] and Kin, Kojima, and Takasawa in [34], the authors have shown a relationship

between the hyperbolic volume of the mapping torus for a pseudo-Anosov element and the

corresponding entropy. In particular, it is known that the volume of a hyperbolic manifold is

bounded by a constant multiplied by the entropy where the constant depends on the surface

of the mapping class group. Because of this, Conjecture 3.3.3 would further support that

the Turaev-Viro invariant volume conjecture is true. Additionally, it may provide another

method in calculating the constants for the bounds.

57

Chapter 4

Realizing Fundamental Shadow Links

in S3

In this chapter, we construct infinite families of links in S3 whose complements satisfy the

Turaev-Viro invariant volume conjecture. Additionally, we provide a general method in

augmenting a link such that the resultant link also has complement which satisfies the

Turaev-Viro invariant volume conjecture. The method involves finding a homeomorphism

of the complement of the link with the complement of a fundamental shadow link where the

conjecture is already known. Since the Turaev-Viro invariant and the hyperbolic volume are

preserved under homeomorphisms, this provides link complements in S3 which satisfy the

conjecture. We note that the results for Sections 4.3, 4.4, and 4.5 first appeared in [36] by

the author of this dissertation; however, in Section 4.5, we consider a more general approach.

4.1 Background

In this section, we will introduce the Stein factorization of stable maps of links followed by

their connection with shadows of links in S3. Under this correspondence, we are able to

understand the collapsing map given by Turaev’s shadow theory approach to 3-manifolds

through the explicit map given by the Stein factorization. Additionally, we briefly discuss

Kirby calculus which we will use in an alternate proof of Theorem 4.5.1. The method

involving Kirby calculus provides a direct approach based purely on the surgery presentation.

58

4.1.1 The Stein Factorization of Stable Maps of Links

In this subsection, we will introduce the Stein factorization of stable maps. We remark

that the definitions stated here are written formally; however, it is only necessary to have

a loose conceptual understanding of this subsection in order to follow the results appearing

later in Chapter 4. As discussed in Subsection 2.3.2, the fundamental shadow links may

be viewed as a surgery presentation in S3. In this case, one can realize the fundamental

shadow links in S3 through the widely used Kirby calculus. An alternate proof using this

technique will be provided in this work; however, by Constantino and Thurston [15], we will

also see that there is a close connection between the Stein factorization of stable maps of

links and boundary-decorated shadowed polyhedron. Through the explicit maps, we have a

more natural approach to studying the complements of the fundamental shadow links.

We will begin with a brief outline. For a 3-manifold M , the Stein factorization of a stable

map f = g ◦ h : M → R2 decomposes f into a map h : M → Wf of connected fibers and
a map g : Wf → R2 which is finite-to-one where Wf is a 2-dimensional polyhedron. For

our cases, Wf will be a simple polyhedron in the same manner as the shadows in Section

2.3. Analogous to the 3-manifolds obtained from Turaev’s shadow construction, the generic

points of Wf have preimages under the map h which are circle bundles inside a 3-manifold.

At these circle bundles, we can glue disks which is similar to the 4-thickening process. This

results in a 4-manifold which collapses onto the simple polyhedron along the disks where the

boundary is given by the 3-manifold M .

Much of this construction is close to the boundary-decorated shadowed polyhedron view-

point of a manifold. The main difference is that we do not state the topological data of the

gleam explicitly in the Stein factorization. We will proceed with the basic definitions of the

Stein surface. One can find more detailed information in [37] by Levine and in [50] by Saeki.

We will now begin to define the Stein factorization for a map on a 3-manifold with

(possibly empty) boundary of tori.

59

Definition 4.1.1. Let M be a closed 3-manifold and f a smooth map from M to an ori-

entable 2-manifold Σ. Now suppose that f satisfies the following conditions. For every

p ∈ M and f (p), there exist local coordinates centered at p such that f can be described as
one of the following:

i. (u, x, y) (cid:55)→ (u, x).
ii. (u, x, y) (cid:55)→ (u, x2 + y2).
iii. (u, x, y) (cid:55)→ (u, x2 − y2).
iv. (u, x, y) (cid:55)→ (u, y2 + ux − x3).
For the previous conditions (i), (ii), (iii), and (iv) at a point p, we say p is a regular point,

a definite fold point, an indefinite fold point, or a cusp point, respectively. Furthermore, if

S(f ) are the singular points of f , then suppose f also satisfies these global conditions:

v. f−1 ◦ f (p) ∩ S(f ) = {p} for a cusp point.

vi. The restriction of f to the singular points which are not cusp points is an immersion

with normal crossings.

We say that a map f satisfying all of the conditions is a stable map.

Definition 4.1.2. Suppose that M is a compact and orientable 3-manifold with (possibly

empty) boundary of tori. We say that a smooth map f of M into an orientable 2-manifold

Σ is an S-map if the restriction to the interior of M is a stable map, and each p ∈ ∂M can
be described locally by the map (u, x, y) (cid:55)→ (u, x) such that x = 0 corresponds to points on
the boundary.

Definition 4.1.3. For a given S-map, we define the space Wf to be the quotient of the

space M along the connected fibers of f with map h : M → Wf . Now we let g : Wf → Σ be
the map such that f = g ◦ h. This factorization of the map f is the Stein factorization with
corresponding Stein surface Wf .

60

Lastly, we can further extend this definition such that M contains a link L.

Definition 4.1.4. Let M be a compact and orientable 3-manifold M with (possibly empty)

boundary of tori and a (possibly empty) link L. Let f be an S-map of the manifold M into

an orientable 2-manifold Σ. We define f to be an S-map of the pair (M, L) if the link L is

contained in the set of definite fold points of f . In the case when M is a closed 3-manifold,

the map f is called a stable map of the pair (M, L).

4.1.2 Stein Surfaces and Shadows of 3-Manifolds

In this subsection, we will construct a boundary-decorated shadowed polyhedron for the

manifold pair (S3, L). We will then see that the collapsing map in Turaev’s construction

from shadows can be realized as an explicit map given by the Stein factorization of the pair.

We let D be a flat oriented disk in the 4-ball B4, and we define the closure of D minus

a neighborhood of the boundary to be the set D(cid:48). Now we have a clear map π : S3 → D

given by collapsing B4 onto the disk D. From this collapsing map, we can assume that
the preimage of D(cid:48) under the map π is a solid torus T , and additionally, the remaining
T (cid:48) = S3\Int(T ) is also a solid torus.

Now in order to consider a link L in S3 of n-components, we proceed in the following way.

We begin by isotoping the link L such that it is contained in D(cid:48)

× [−
, 
] for some 
 where
the projection under the collapsing map π|T is generic with respect to the link L. Here, we
define generic in the usual sense where the projection contains no cusps, no crossings contain

more than two strands, and there are no self-tangencies of the projection. This property can

always be obtained through small isotopies of the link.

Now we will begin to define the boundary-decorated shadowed polyhedron for the pair

(S3, L). Consider the mapping cylinder obtained from the quotient

P (cid:48)
L = (L × [0, 1] (cid:116) D(cid:48))/((x, 0) ∼ π(x)).

61

From P (cid:48)
L, we define the boundary-decorated shadowed polyhedron PL as having the simple
polyhedron obtained from collapsing the region touching the boundary ∂(D(cid:48)) onto the com-

ponents of π(L). Here, we decorate the boundary ∂(PL) = L × {1} as internal components
where the gleam is explained as follows. For the regions not touching the boundary of PL,

there is a choice of gleam such that PL is a boundary-decorated shadowed polyhedron cor-

responding to the pair (S3, L). In this work, the gleam for such regions R will be VR

2 where

VR denotes the number of vertices touching R. More details can be found in Section 3.2 of

[15].

Now given the boundary-decorated shadowed polyhedron PL, we can construct a stable

map f : (S3, L) → R2 such that the corresponding Stein surface Wf with map h : (S3, L) →

Wf can be identified in a natural way to PL. In [29], Ishikawa and Koda discuss the explicit

map h in terms of preimages of points in Wf for links inside compact and orientable 3-

manifolds with (possibly empty) toroidal boundary. Additionally, in Section 4 of [29], the

authors construct the map h for the specific case of the pair (S3, L). Their result is provided

as follows where the details are discussed in-depth in the proof of Theorem 3.5 in [29].

Proposition 4.1.5 ([29], Theorem 3.5). Let L be a link in S3. There exists a stable map

f : (S3, L) → R2 with Stein surface Wf and Stein factorization f = g ◦ h such that the

preimage of a point in Wf can be viewed as one of the five following:

i). If x1 is a point on PL\(L × [0, 1]), then h−1(x1) is the regular fiber {x1} × S1 ⊂ T .
ii). If x2 is a point on L × (0, 1), then h−1(x2) is a meridian of the corresponding link

component of L × {1}.

iii). If x3 is a point on L × {1}, then h−1(x3) is a point on the corresponding link in S3.

iv). Let x4 be a point from the singular set S(P ) which are not vertices. Locally around x4,

there are three regions separated by the singular set. There exists a triple of points, one

from each region, such that their preimages are circles which are the boundary to a pair
of pants with h−1(x4) as its spine.

62

v). Let x5 be a point from the set of vertices of the singular set S(P ). As in case iv), there

exist preimages of points from the regions around the vertex which pullback to circles
such that they bound a surface with spine h−1(x5).

4.1.3 Kirby Calculus

In this subsection, we give a brief summary of Kirby calculus. For an introduction, one may

refer to Chapter VI in [47] by Prasolov and Sossinsky.

Similar to how isotopic links in S3 may be realized through Reidemeister moves of the

link diagrams, Kirby calculus gives a method for studying homeomorphic 3-manifolds from

their surgery presentation.

We recall that a 3-manifold may be viewed as an integer framing of a link in S3. Here, the

framed link carries the topological data to construct the 3-manifold through Dehn surgeries.

Definition 4.1.6. A Dehn surgery is the procedure of removing a toroidal neighborhood

of the link component followed by a Dehn filling where we glue back the meridian of the

removed solid torus to the boundary torus. For a given integer framing n, the gluing sends

the meridian of the removed torus to a simple closed curve γ in the homology class of [nα+β]

where α and β, respectively, are the standard generating meridian and longitude curves of

the first homology group of the torus.

The Dehn surgery operation may be defined for rational surgeries; however, we will

mostly only see either 0-framed surgeries or ∞-framed surgeries in this work. For the 0-
framed surgeries, the solid torus is removed and glued back such that the meridian of the

removed torus is glued as the longitude of the boundary torus. For the ∞-framed surgeries,
we consider this case to be the identity Dehn surgery. The removed solid torus is glued back

to the boundary torus such that the meridian is still identified with the meridian.

Now given a surgery presentation of a 3-manifold, we consider the two moves shown in

Figure 4.1. In this work, we will often refer to the Kirby II move as a handleslide. Now

63

±1

Kirby I

Kirby II

Figure 4.1: On the left, the Kirby I move which adds or removes an unlinked component with
framing ±1. On the right, the Kirby II move where any two components with their framing
curves contained in a possibly knotted two-holed disk in S3 can be changed as shown. Here,
the framed link is colored blue, the two-holed disk may be linked with other components,
and the new framing curves are still on the two-holed disk.

by the following theorem, the Kirby moves may be used to relate surgery presentations of

homeomorphic manifolds.

Theorem 4.1.7 ([35]). Two framed links in S3 represent the same oriented 3-manifold by

surgery if and only if they are related by a sequence of Kirby I and Kirby II moves.

Theorem 4.1.7 will be essential in our alternate proof of Theorem 4.5.1. By using the

handleslides on the surgery presentation of the fundamental shadow links given in Sub-

section 2.3.2, we can obtain a simpler surgery presentation for a homeomorphic manifold.

From this new surgery presentation, we can explicitly find a link in S3 with a complement

homeomorphic to the complement of a fundamental shadow link.

4.2 An Obstruction to Knots in S3

In the following sections of the chapter, we will see that every obtained hyperbolic link in S3

which satisfies the Turaev-Viro invariant volume conjecture contains at least two components.

In this section, we provide an explanation by showing that, up to homeomorphisms of the

knot complements, there are only finitely many knots in S3 that can have complement

homeomorphic to the complement of a fundamental shadow link with one component.

Proposition 4.2.1. Let K be a knot in S3, and let K c ⊂ #c+1 (S2 × S1) be a complexity

64

LLt1c fundamental shadow link with one link component. For any c ≥ 2, the complements of K
and K c are not homeomorphic.

Proof. This proof is a simple consequence of the Mayer–Vietoris sequence. For more infor-

mation, see Hatcher [28, Section 2.2]. For a knot K in S3, it is well-known through a quick
application of the Mayer–Vietoris sequence that the first homology group H1(S3\IntNbd(K))
is isomorphic to Z [38, Theorem 1.5]. In a similar way, we will use the Mayer–Vietoris se-

quence for the first homology group of the complement of the fundamental shadow link. We

will first define the spaces

X = #c+1(cid:0)S2 × S1(cid:1) , A = D2 × S1, B = X\A,

and Y = A ∩ B = S1 × S1

where D2 is the two-dimensional closed disk. Now from the Mayer–Vietoris sequence, we

have the long exact sequence

··· → H1(Y ) → H1(A) ⊕ H1(B)

f

−→ H1(X)

g

−→

→ H0(Y ) → H0(A) ⊕ H0(B) → H0(X) → 0.

The zeroth homology group is generated by the number of path connected components such

that we have

··· → H1(Y ) → H1(A) ⊕ H1(B)

→ Z → Z ⊕ Z → Z → 0.

f

−→ H1(X)

g

−→

In this case, from the exactness of the sequence, g is the zero map. Therefore, we can consider

the following long exact sequence

··· → H1(Y ) → H1(A) ⊕ H1(B)

f

−→ H1(X)

g

−→ 0.

65

We will compute the homologies of H1(A) and H1(X). For H1(A), there is a deformation
retraction of the space A onto the space S1 such that H1(A) ∼= Z. Now the manifold X is

obtained from the connected sum of closed and oriented 3-manifolds. In the case of finitely

generated abelian groups, the first homology group of the connected sum may be computed

as the direct sum of the first homology groups shown as

(cid:0)#c+1(cid:0)S2 × S1(cid:1)(cid:1)

∼=

c+1(cid:77)

H1

(cid:0)S2 × S1(cid:1)

H1

∼= Zc+1.

Replacing our isomorphisms into the long exact sequence, we obtain

··· → H1(Y ) → Z ⊕ H1(B)

f

−→ Zc+1 g

−→ 0.

Now since g is the zero map, then the kernel of g is the entire set Zc+1. By exactness
of the sequence, the kernel of g must be equal to the image of f such that Z ⊕ H1(B) must

have at least (c + 1) generators. This implies that H1(B) will have at least c generators. For

the complement of a fundamental shadow link B, in the case when c ≥ 2, the first homology
group H1(B) has more generators than the first homology group of the knot complement

in S3. This implies that their first homology groups cannot be isomorphic, and the two

manifolds must be distinct.

4.3 Links in S3 up to 11-Crossings

In this section, we will classify, up to homeomorphisms of their complements, the hyper-

bolic links of volume 2v8 with at most 11-crossings which are complexity one fundamental

shadow links. As discussed in Subsection 1.2.2, since the Turaev-Viro invariant and the hy-

perbolic volume are preserved under homeomorphisms, this provides new examples of links

in S3 which satisfy the Turaev-Viro invariant volume conjecture. We will obtain the results

66

0

0

0

0

FSL 1

FSL 2

0

0

0

0

0

0

FSL 3

0

0

FSL 4

FSL 5

FSL 6

Figure 4.2: The six fundamental shadow links with complements homeomorphic to the
complements of links up to 11-crossings.

using the link database of LinkInfo [39] in combination with computations from SnapPy

[16]. In Figure 4.2, we provide the six fundamental shadow links which have complements

homeomorphic with at least one of the complements of links up to 11-crossings.

Remark 4.3.1. From our calculations, the only candidate that we were not able to determine

was a fundamental shadow link was the link L10n59. Although SnapPy determined the link

was not a fundamental shadow link, this is not rigorous and may be due to a numerical error.

To prove whether or not it is a fundamental shadow link would require a different method.

Fundamental Shadow Link

LinkInfo Name

FSL 1
FSL 2
FSL 3
FSL 4
FSL 5
FSL 6

L10n32
L10n36

L6a4, L9n25, L11n287, L11n378

L10n84, L10n87

L8n5, L9n26, L10n70, L11n376, L11n385

L8n7, L10n97, L10n105, L10n108

Table 4.1: The links with homeomorphic complements to a fundamental shadow link.

Theorem 4.3.2. The links given in Table 4.1 have volume 2v8 and satisfy the Turaev-Viro

invariant volume conjecture.

67

1Proof. We will prove Theorem 4.3.2 using the following procedure:

• Since we know that we are looking for complexity one fundamental shadow links, then

the volume of all of our links will be 2v8 ≈ 7.3277. From using LinkInfo, this will allow
us to narrow down all links up to 11-crossings whose volume is approximately 2v8.

• The previous step gives us the list of links: L6a4, L8n5, L8n7, L9n25, LL9n26, L10n32,

L10n36, L10n59, L10n70, L10n84, L10n87, L10n97, L10n105, L10n108, L11n287, L11n376, L11n378,

and L11n385 where, as a reminder, the first number indicates the number of crossings,

the letter indicates whether it is alternating or non-alternating, and the second number

indicates it is the n-th link with the same first two parameters.

• As mentioned in Remark 2.3.16, the complements of the fundamental shadow links

obtained by replacing edges by 3-braids of single vertex planar 4-valent graphs are

homeomorphic if they induce the same permutation on the 3-strands, therefore, there

are only finitely many fundamental shadow links with volume 2v8. This implies that we

can compare the finite list of links up to 11-crossings with the finite list of fundamental

shadow links of volume 2v8.

• As demonstrated in Subsection 2.3.2, the complements of the fundamental shadow
links in connected sums of S2 × S1 can be realized as the complement of a surgery
presentation of a link in S3. From this, we define each of the surgery presentations of

the fundamental shadow links into SnapPy as well as our candidates from the list of

links up to 11-crossings.

• Lastly, we use SnapPy to test the finitely many cases to see which manifolds are

homeomorphic to each other. Of the possible fundamental shadow links, the links in

Figure 4.2 are the only fundamental shadow links which SnapPy recognizes as having

homeomorphic complements with a link of at most 11-crossings.

68

Figure 4.3: The Borromean rings are the leftmost link, and the Borromean twisted sisters
are the two rightmost links.

Remark 4.3.3. From LinkInfo, all links up to 11-crossings have volume strictly less than 6v8.

This implies that the only other links in this set that have homeomorphic complements with

the fundamental shadow links must have complexity two. Similar to the proof of Theorem

4.3.2, we could also look for candidates by looking for links with volume 4v8 in LinkInfo.

From doing this, the only possible links are the links L11n387 and L11n388; however, currently,

it has not been shown that if either of the links have complements homeomorphic to the

complement of a complexity two fundamental shadow link.

4.4 Families of Links in S3

In this section, we will construct infinite families of links which satisfy the Turaev-Viro

invariant volume conjecture. The motivation for this comes from the Borromean rings and

the Borromean twisted sisters depicted in Figure 4.3. These are links with non-homeomorphic

complements; however, they each have the same hyperbolic volume of 2v8. Although the

conjecture is already known for the Borromean rings, it was not previously verified for both

of the Borromean twisted sisters. In pursuing a positive result for these links, we generalize

the link complements to infinite families where each link complement satisfies the conjecture.

Before the statement of the main theorem in this section, we will need the following

technical lemma.

69

10

0

k + 1

0

0

k + 1

0

0

0

0

0

Lk

Jk

Kk

k + 1

k + 1

k + 1

k + 1

Figure 4.4: On the left are the complexity k fundamental shadow links embedded in con-
nected sums of S2 × S1, and on the right are their associated links in S3.

Lemma 4.4.1 ([15], Lemma 3.25). Let P be a boundary-decorated shadowed polyhedron of
(M, L) where L is a framed link and with M an oriented 3-manifold. Then the manifold M(cid:48)
obtained from Dehn filling L has a shadow P (cid:48) given by capping components of ∂P by disks.

Proof. We will begin by letting P be a boundary-decorated shadowed polyhedron of a 4-

dimensional manifold W with the boundary ∂(W ) = M and the link L = ∂int(P ). Performing
a Dehn filling along L in the manifold ∂(W ) corresponds to gluing 2-handles B2 × B2 to
W . Without loss of generality, we can assume that the copies of B2 × B2 collapse onto
the disks B2 × {pt} such that the boundary of each disk S1 × {pt} is a link component of
∂int(P ). After the gluings, we obtain a new simple polyhedron P (cid:48) which is the result of the
disk components B2 × {pt} being attached to the original P . On the simple polyhedron P (cid:48),

the gleam of the new regions are determined by the framing of the link L.

Now with Lemma 4.4.1, we will proceed to state and prove the main result of the section.

70

1∞

∞

p1

p2

Shadow P

Shadow P (cid:48)

S3\L

S3\(L (cid:116) L0)

Figure 4.5: On the leftmost diagram, we have the shadow P with Sing(P ) the dashed lines,
∂int(P ) colored blue, and ∂ext(P ) colored black. We cap ∂int(P ) with disks to obtain regions
Rj shaded blue, and we mark their centers with p1 and p2 to recover a new shadow P (cid:48). In
2. P (cid:48) is the shadow of a link complement
this case, the gleam of the new regions are both 1
in S3 such that the preimages of the points p1 and p2 are the cores of the ∞-framed link
in S3. Drilling out the ∞-framed link gives a link complement in S3 homeomorphic to the
complement of the original fundamental shadow link.

First, we will define the generalizations of the complements of the Borromean rings and the

Borromean twisted sisters as the complements of the link families appearing in Figure 4.4,

denoted by Lk, Jk, and Kk. The links in S3 on the right of Figure 4.4 are the same as the

families in S3 appearing in Figure 1.1.

Theorem 4.4.2. Given an integer k ≥ 1, let Lk, Jk, and Kk be the links in S3 shown in
Figure 1.1. The complement of each Lk, Jk, and Kk is a hyperbolic manifold with volume

2kv8 and satisfies the Turaev-Viro invariant volume conjecture.

Proof. Each of the links shown on the right of Figure 4.4 can be constructed using the

following steps. As an example, one can reference Figure 4.5 which demonstrates the case

for K1. As a quick note, in Remark 4.4.4, we discuss a proof using Hopf link pairs which

are sublinks of the surgery presentation. Although this approach leads to a more immediate

discovery of the homeomorphisms, the method we choose to highlight here allows us to

consider a more general case stated later in this work.

From the procedure of constructing a fundamental shadow link from a graph in Sub-

section 2.3.2, we let S(P ) be the simple polyhedron of the fundamental shadow link of

complexity k which corresponds to the planar 4-valent graph of Figure 4.6. Here, we choose

the construction such that the link can be represented as one of the links in connected sums

71

1Figure 4.6: The planar 4-valent graph of complexity k used in the construction of the link
families Lk, Jk, and Kk.

of S2 × S1 illustrated by the surgery presentations on the left side of Figure 4.4.

We now consider P to be the boundary-decorated shadowed polyhedron obtained from

S(P ) by decorating the innermost (k +1) boundaries of S(P ) as internal boundaries with the

remaining boundary components decorated as external boundaries. In the leftmost diagram

of our example of Figure 4.5, this is demonstrated as the boundary-decorated shadowed

polyhedron with singular set the dashed lines, with internal boundaries colored blue, and

with external boundaries colored as black.

Now we consider the boundary-decorated shadowed polyhedron P (cid:48) with newly introduced

regions and gleam obtained by capping the internal boundaries with disks. We will label this

set of regions by Rj. From Lemma 4.4.1, gluing disks to P corresponds to performing Dehn

fillings along the internal boundary components where the new gleam is determined by the
initial framing of the link. In order for P (cid:48) to be a boundary-decorated shadowed polyhedron
VRj
2

for the link complement in S3, we can choose a framing such that the gleam of Rj is

where VRj is the number of vertices of the singular set touching Rj.

boundary components by blue shaded disks with gleam 1

In the second diagram from the left of Figure 4.5, this is shown by capping the blue colored
2. As shown in Subsection 4.1.2, P (cid:48) is
of the form of a boundary-decorated shadowed polyhedron which can be used in constructing
a link complement S3\IntNbd(L) where the link L corresponds to the boundary components
of P (cid:48). Note that in Subsection 4.1.2, we constructed a pair (S3, L) from a boundary-decorated

shadowed polyhedron; however, in this case, the link complement comes from decorating the
boundary of P (cid:48) as external as opposed to internal.

Now from Proposition 4.1.5, we have a map f with a Stein surface Wf and Stein factor-

72

1ization f = g◦h such that Wf can be identified with P (cid:48) where the preimages of the points are

as described in Proposition 4.1.5. Now by using the map h, we can find a homeomorphism

between the fundamental shadow link and a specific link in S3.

Previously, the Dehn fillings corresponded to gluing copies of the 2-handles B2 × B2 to
the 4-thickening of our boundary-decorated shadowed polyhedron P . We let pj ∈ Rj be the
points in the 2-handles which correspond to the center of the capping disks. Under the map

h, the preimages of the set of centers {pj} correspond to cores of tori. From Proposition
4.1.5, we can explicitly see that the preimages of {pj} form an unknotted link {pj} × S1,
denoted by L0, embedded in S3\IntNbd(L) such that L0 runs parallel to the z-axis. In our
manifold S3\IntNbd(L), the link L0 can be viewed as having framing ∞. In Figure 4.5, the
link complement S3\IntNbd(L) with the ∞-framed link is illustrated in the third diagram
from the left.

Next, we will remove the 2-handles which contain the ∞-framed link. This will add
an additional toroidal boundary for each component of the link L0. Our resultant mani-
fold is homeomorphic to the link complement S3\IntNbd(L (cid:116) L0) with boundary-decorated
shadowed polyhedron P (cid:48)(cid:48) where P (cid:48)(cid:48) can be obtained from decorating all of the boundary

components of the simple polyhedron S(P ) as external boundaries.

As stated in Subsection 2.3.1, P (cid:48)(cid:48) is a boundary-decorated shadowed polyhedron for the

complement of a fundamental shadow link. This implies that the union of the links L and

L0 in S3 form the realization of the fundamental shadow link in S3. This is shown as the

right diagrams of Figure 4.4 as well as the rightmost diagram in Figure 4.5. Now since

the complement of a fundamental shadow link of complexity k is a hyperbolic manifold

of volume 2kv8 which satisfies the Turaev-Viro invariant volume conjecture, then through
homeomorphisms, the complement of the link L (cid:116) L0 in S3 is also a hyperbolic manifold
satisfying the Turaev-Viro invariant volume conjecture with volume 2kv8.

Remark 4.4.3. Notice for the cases of the link families Jk and Kk, we can reduce our

73

fundamental shadow links of complexity k to one of complexity (k − 1) through certain ∞-
framed Dehn surgeries. This can be done by performing the Dehn surgery on the rightmost

of the (k + 1)-unknotted components bounding the twice punctured disks.

Remark 4.4.4. For the link families of Figure 4.4, we implicitly used the following property

of the surgery presentation of the link in S3. Notice that each 0-framed unknotted component

could be paired with a component of the fundamental shadow link to form a Hopf link pair
in the surgery presentation. In the manifold of connected sums of S2 × S1, the component of
the Hopf link pair coming from a fundamental shadow link component is isotopic to the core

of the tori obtained from the 0-framed surgery on the unknot component. This implies that

we can isotope the fundamental shadow link component such that the removed torus in the

complement corresponds to removing the torus from the 0-framed unknot surgery. In terms

of the surgery diagrams on the left of Figure 4.4, we can obtain our link complements repre-

sented on the right by simply drilling out tori with cores the 0-framed unknot and deleting

their corresponding fundamental shadow link components from the link. This procedure can

work for any fundamental shadow link as long as we can pair as Hopf links every 0-framed

unknotted component with a fundamental shadow link component such that all of the Hopf

link pairs are disjoint.

Using the viewpoint of the Hopf link pairs on the surgery presentation, we immediately

obtain the results of Theorem 4.4.2; however, this is generally not the case. As we will see

in Theorem 4.5.1, from the surgery presentation, the Hopf link pairs may not always be

disjoint from each other. This may be remedied through the use of Kirby calculus which is

provided as an alternate proof; however, we can eliminate this technical issue by utilizing

the Stein surface approach. Because of this, the Stein surface approach is a more natural

method for generalizing our results. From the map equipped to the Stein surface, we are

able to see that each capped disk corresponds to a Hopf link pair. Here, the boundary of

the disk is the fundamental shadow link component, and the preimage of the center of the

disk corresponds to our 0-framed unknotted component. In order to utilize this method, we

74

only need a boundary-decorated shadowed polyhedron of a fundamental shadow link which

can be augmented by capping disks to a boundary-decorated shadowed polyhedron of the

form PL as defined in Subsection 4.1.2. From the explicit map, the 0-framed unknotted

components always run parallel to the z-axis, therefore, all of the Hopf links pairs will be

disjoint from each other.

By using Theorems 4.3.2 and 4.4.2, we can now provide an alternate proof to the Turaev-

Viro invariant volume conjecture for the Borromean rings as well as extend the result to

our initial motivation of the Borromean twisted sisters. Additionally, we can see that the

complements of the link families Lk, Jk, and Kk are a generalization of the complements of

the Borromean rings and the Borromean twisted sisters.

Corollary 4.4.5. The Borromean rings and the Borromean twisted sisters shown in Fig-

ure 4.3 are hyperbolic links with volume 2v8 and satisfy the Turaev-Viro invariant volume

conjecture.

Proof. For the case of k = 1 for Theorem 4.4.2, we can see that the links L1, J1, and

K1 correspond to the links FSL 3, FSL 5, and FSL 6 in Figure 4.2, respectively. Since

the Turaev-Viro invariant volume conjecture is preserved under homeomorphisms, then any

links from Table 4.1 homeomorphic to the links FSL 3, FSL 5, and FSL 6 also satisfy the

conjecture. From this, we can see that L1, J1, and K1 have complements homeomorphic

to the complements of the Borromean rings and the Borromean twisted sisters in Figure

4.3, respectively.

In this sense, the complements of the link families Lk, Jk, and Kk are

a generalization of the complements of the Borromean rings and the Borromean twisted

sisters.

We will end this section by demonstrating that the links in Theorem 4.4.2 provide new

examples of links in S3 which satisfy the Turaev-Viro invariant volume conjecture. Of the

known examples of links in S3 which satisfy the conjecture, the Whitehead chains are the

only other infinite family. As detailed by Roland van der Veen in [58], the Whitehead chains

75

Twist

Belt

Clasp

Mirrored Clasp

Figure 4.7: The diagrams of the twist, belt, clasp, and mirrored clasp components are
illustrated, respectively.

Wa,b,c,d are a family of links that can be constructed as the closure of the composition of

a-twists, b-belts, c-clasps, and d-mirrored clasps shown in Figure 4.7. The Whitehead chains

have the particularly nice property that when b = 1, the complement is hyperbolic with

V ol(Wa,1,c,d) = (c + d)v8. We will let W be this set of hyperbolic links.

Proposition 4.4.6. For an integer k ≥ 3, no link of Theorem 4.4 has complement homeo-
morphic to the complement of an element of W.

Proof. For every element in W, the complement has volume (c + d)v8 where c and d are
the number of clasps and mirrored clasps, respectively. Since we want to compare the links

in W with the fundamental shadow links, we will always assume that the volume is an
even multiple of v8. Notice that the number of components of the Whitehead chain will be

(c + d + 1) because the clasps and mirrored clasps each add one component with the addition

of the belt component.

Now for a fundamental shadow link of complexity k, the volume of the complement will

be 2kv8. From the link families in Theorem 4.4.2, the number of components of the link will

either be (k + 2) or (k + 3). In order for the Whitehead chains to have the same volume

of 2kv8, they would need (2k + 1) link components. This implies that when k ≥ 3, every
element of W with volume 2kv8 will have complement with more boundary components
than the complements of the links Lk, Jk, and Kk in Theorem 4.4. Therefore, for k ≥ 3,

76

1no element of W has complement homeomorphic to the complement of a link appearing in
Theorem 4.4.

As we will see in Section 4.5, we can construct many links in S3 which satisfy the Turaev-

Viro invariant volume conjecture. By following a similar proof to Proposition 4.4.6, many of

the links in Section 4.5 can also be shown to be new examples of links in S3 which satisfy

the Turaev-Viro invariant volume conjecture.

4.5 Augmenting Links in S3

In this section, we will generalize our methodology used in the proof of Theorem 4.4.2 to

show that every planar 4-valent graph corresponds to a fundamental shadow link which can

be realized as a link in S3. This is accomplished by considering the fundamental shadow link

obtained by replacing each edge by the identity 3-braid. Since every link may be projected

as a planar 4-valent graph, we will see that this implies that every link is a sublink of a

fundamental shadow link realized in S3. Therefore, we show that every link in S3 may be

augmented to a link whose complement satisfies the Turaev-Viro invariant volume conjecture.

Theorem 4.5.1. For any G, a planar 4-valent graph of complexity k, there exists a corre-

sponding fundamental shadow link realized in S3 with hyperbolic volume 2kv8.

Proof. As noted in Remark 4.4.4, the surgery presentation of a fundamental shadow link

generally does not contain disjoint Hopf link pairs. By using the Stein surface approach, we

only need to augment our boundary-decorated shadowed polyhedron by capping with disks

to one that corresponds to the boundary-decorated shadowed polyhedron PL in Subsection

4.1.2. By utilizing a particular construction of our graph G to a fundamental shadow link,

we can guarantee that such an augmentation is always possible. As stated before, this is an

advantage in using the Stein surface method; however, we also provide an alternate proof

involving Kirby calculus which remedies the case when the Hopf link pairs are not disjoint.

77

C

C

Figure 4.8: On the leftmost diagram, we have a planar 4-valent graph with complexity 2.
From the graph, we obtain the boundary-decorated shadowed polyhedron shown in the mid-
dle illustration. Here, the edges were replaced by identity 3-braids, the outermost boundary
is labeled C, the internal decorated boundaries are blue, the external decorated boundaries
are black, and the singular set is dashed. As in Theorem 4.4.2, we cap the blue curves with
disks and drill out the corresponding tori to recover the link in the third diagram.

We will begin with the proof where a small example of the general method can be found

in Figure 4.8. Let G be a planar 4-valent graph of complexity k. From the graph G, we

can consider the boundary-decorated shadowed polyhedron P obtained in the following way.

We let P have the simple polyhedron S(P ) where the boundary is obtained from replacing

all of the edges of G by the identity 3-braid and by replacing the vertices by the 6-stranded

figure or its mirror. To recall the 6-stranded figure, see Figure 2.6. Notice that since we have

chosen the identity 3-braids, there exists an outermost boundary component C in the sense

that it bounds a disk D where D × (−
, 
) contains the remaining components of ∂(S(P ))
for some 
.

Now the boundaries of P are decorated such that the outermost boundary component is

an external boundary, the boundaries which convey crossing information are external bound-

aries, and the remaining boundaries are internal boundaries. The decorations are chosen as

such since we will be gluing disks onto the internal boundaries. This can be seen as the two

leftmost illustrations of Figure 4.8. The leftmost illustration is an arbitrary planar 4-valent

graph, and the following illustration shows the associated boundary-decorated shadowed

78

1polyhedron P in S3. Depicted in the figure, the outermost component is labeled C, the

external boundaries are colored black, the internal boundaries are colored blue, and the sin-

gular set of P is dashed. It is worth noting that since we chose the identity 3-braids, the

internal boundary components are in direct correspondence to all of the initial regions of the

graph G excluding the outermost region.

Now since ∂(P ) contains an outermost region, we can obtain a boundary-decorated shad-

owed polyhedron for a link complement in S3. This can be achieved in the same way as in

the proof of Theorem 4.4.2 by gluing disks along the internal boundary components. By our

initial choices, this will lead to a boundary-decorated shadowed polyhedron of the form PL

as in Subsection 4.1.2. As mentioned in Remark 4.4.4, from the map arising from the Stein

surface of our specific boundary-decorated shadowed polyhedron, we can obtain a collection

of Hopf link pairs where one component comes from the fundamental shadow link and the

other from a 0-framed unknot component. Again, from the explicit map, the 0-framed un-

knot component runs parallel to the z-axis such that each Hopf link pair is disjoint, therefore,

each capped region corresponds to adding an additional link component.

Finally, the collection of the link components coming from the external decorated bound-

ary components along with the link components arising from the Hopf link pairs give a

hyperbolic link in S3 that is the realization of a fundamental shadow link with volume 2kv8.

This can be seen in the rightmost illustration of Figure 4.8 where each blue colored inter-

nal decorated boundary component from the previous illustration now corresponds to a link

component which runs parallel to the z-axis.

Alternatively, we will provide a proof of Theorem 4.5.1 using a combination of the surgery

presentation of the fundamental shadow link along with Kirby calculus.

Alternate proof of Theorem 4.5.1. We will let G be a planar 4-valent graph of k vertices,

and we pick a maximal tree T . For convenience, we will consider G as a graph on S2 such

that the outermost region is defined to contain the point ∞. We will now define two regions
of our graph to be adjacent if there exists an edge in G\T which is adjacent to both of them.

79

Notice that since T is a maximal tree, then T contains no cycles. This implies that any two

regions of G are related by a sequence of adjacent regions. In particular, we have that every

region is related to the outermost region through a sequence of adjacent regions. We also

note that any two adjacent regions are adjacent by exactly one edge of G\T .

We will now consider the surgery presentation construction from a planar 4-valent graph

given in Subsection 2.3.2. We begin by replacing the edges of the graph by identity 3-braids,

replacing the vertices of the graph by the 6-stranded figure or its mirror, and encircling the

3-braids corresponding to the edges of G\T by 0-framed unknots. From this construction,
we can see that every region of the graph which is adjacent to an edge G\T will be punctured
by at least one 0-framed unknot. Also notice that each region of G will correspond to an

unknotted fundamental shadow link component obtained from replacing the edges by the

trivial 3-braid. Excluding the component arising from the outermost region which we denote

by C, the link components corresponding to the regions will eventually form part of the

Hopf link pairs mentioned in Remark 4.4.4. With the exception of the link component C,

we will denote the collection of the disks these fundamental shadow link components bound

by {Di}i∈I with corresponding regions of G denoted by {Si}i∈I.

We will now perform a sequence of handleslides such that each disk Di is punctured by

exactly one 0-framed unknot. This will ensure that our Hopf link pairs will all be disjoint.

Now from the construction, given any two adjacent regions, there exists a 0-framed unknot

which corresponds to the edge between them. We first consider the collection of regions

{Si}i∈I1 of G where I1 is a subset of I such that the regions are adjacent to the outermost
region. As the corresponding 0-framed unknots only puncture the collection of disks {Di}i∈I1,
we will leave them alone.

Next, we consider the collection {Si}i∈I2 of regions of G which are adjacent to the col-
lection {Si}i∈I1. Since this collection of corresponding 0-framed unknots also puncture the
previous collection {Di}i∈I1, we can see that the Hopf link pairs are currently not disjoint.
Now by performing handleslides of the 0-framed unknots arising from the edges between

80

{Si}i∈I1 and {Si}i∈I2 over the 0-framed unknots corresponding to the edges between {Si}i∈I1
and the outermost region, each Di∈I1 will be punctured by exactly one 0-framed unknot.

Additionally, for clarity, we can isotope the two sets of 0-framed unknots such that each

0-framed unknot is parallel to the z-axis and punctures exactly one element of {Di}i∈I1∪I2.
We can now see that the Hopf link pairs corresponding to the disks {Di}i∈I1 are each disjoint.
Next, we consider the collection of regions {Si}i∈I3 which are adjacent to {Si}i∈I2 with
I3 ∩ I1 = ∅. Similar to before, the corresponding 0-framed unknots arising from the edges
between the regions {Si}i∈I3 and {Si}i∈I2 will puncture the disks {Di}i∈I3 as well as also
puncturing the collection {Di}i∈I2. This will create Hopf link pairs which are not disjoint.
We can again perform handleslides on the 0-framed unknots which puncture both the disks

{Di}i∈I2 and {Di}i∈I3 over the adjacent previous collection of 0-framed unknots. By our
isotopy of the previous collection of 0-framed unknots, the 0-framed unknots corresponding to

the edges between {Si}i∈I2 and {Si}i∈I3 will also be parallel to the z-axis after the handleslide.
Once again, our three sets of 0-framed unknots will each only puncture one element of

{Di}i∈I1∪I2∪I3.

Now lastly, since any two regions are related by a sequence of adjacent regions, we can

perform these handleslides until every disk of {Di}i∈I is punctured exactly once by a 0-
framed unknot which is parallel to the z-axis. Notice that in the end, we obtain a surgery

presentation which is the same as the one obtained from the Stein surface approach.

In

this case, each of the Hopf link pairs are disjoint from each other, and we can proceed as in

Remark 4.4.4. Thus, this concludes the alternate proof of Theorem 4.5.1 using the surgery

presentation of the fundamental shadow link along with Kirby calculus.

Now as a simple application of Theorem 4.5.1, we obtain the following result.

Corollary 4.5.2. Every link J ⊂ S3 is a sublink of a link L ⊂ S3 such that the link
complement S3\IntNbd(L) is homeomorphic to a fundamental shadow link complement and
satisfies the Turaev-Viro invariant volume conjecture.

81

Proof. Let J be a link in S3. Under isotopies of the link, we can obtain a projection G of

J such that G is a planar 4-valent graph where we keep track of the crossing information at

each vertex. Note that we may need to introduce crossings of the link through isotopies such

that G is connected. Now we follow the same fundamental shadow link construction from

Theorem 4.5.1; however, we choose to replace the vertices of our graph by the 6-stranded

figure or its mirror such that the crossing information is consistent with the original link J.

Notice that since we kept track of the crossing information while replacing the vertices, the

link components which correspond to the set ∂ext(P )\{C} form our initial link J.

Lastly, we can see that J is now a sublink of the link L which is the realization of the

fundamental shadow link in S3 constructed from Theorem 4.5.1. Therefore, we have that
J ⊂ L ⊂ S3 such that the link complement S3\IntNbd(L) satisfies the Turaev-Viro invariant
volume conjecture.

4.5.1 The Algorithm for Augmentation

Corollary 4.5.2 tells us that we can always augment our link to eventually satisfy the Turaev-

Viro invariant volume conjecture. Moreover, the augmentation can be seen explicitly through

following the proofs of Theorem 4.5.1 and Corollary 4.5.2; however, for convenience, we will

summarize the explicit augmentation in terms of the closure of braids.

Definition 4.5.3. For a braid in the braid group b ∈ Bn, we say b is an unreduced word if
we do not cancel pairs of elements of the form σiσ−1
i σi. Additionally, we define the

or σ−1

i

length of b to be the number of standard generators appearing in the unreduced word b.

The following corollary requires that each generator appears in the braid b at least once;

however, since the result is stated in terms of the unreduced word, we can obtain a new
unreduced word by adding the elements σiσ−1

i σi without changing the link in the

or σ−1

i

closure of the braid. By a classical result of Alexander [2] which states that every link is the

closure of a braid, Corollary 4.5.4 also implies that every link can be augmented to a link

with more components that satisfies the Turaev-Viro invariant volume conjecture.

82

C

C

C

R1

R2

R3 R4

1 σ−1

Figure 4.9: As shown in the first two diagrams, we begin with the unreduced word b =
σ1σ−1
1 ∈ B2 which we embed into B3, and we take the closure of the new braid with
outermost component labeled C. The projection of the second diagram onto S2 gives the
third diagram where we label the regions not touching C with Ri. Now each region Ri
corresponds to a link component parallel to the z-axis given in the final diagram. The
final diagram is of a hyperbolic link that has volume 6v8, satisfies the Turaev-Viro invariant
volume conjecture, and contains ˆb as a sublink.

Corollary 4.5.4. Let b ∈ Bn be an unreduced word of length k with each generator appearing
at least once. Then the braid closure ˆb is a sublink of a link L in S3 where the complement
S3\IntNbd(L) satisfies the Turaev-Viro invariant volume conjecture and has volume 2kv8.
Proof. Since ˆb is a link in S3, the result is almost immediate by Corollary 4.5.2; however,

we require that each generator appears at least once so that the associated planar 4-valent

graph is connected. Additionally, notice that each generator contributes a vertex to the

planar 4-valent graph. Therefore, the complexity of the planar 4-valent graph is equal to the

length of b. This implies that our resulting hyperbolic manifold has volume 2kv8 where k is

the length of the unreduced word b.

We will proceed to wrap up the subsection by summarizing the link augmentation algo-

rithm from a braid group element. A simple example of the augmentation can be seen in
Figure 4.9 for the unknot as the closure of the unreduced word σ1σ−1

1 ∈ B2. Since our
initial braid has length 3, then the resulting hyperbolic volume of the link complement will

1 σ−1

be 6v8 by Corollary 4.5.4.

Let L be a link, and let ˆb for b ∈ Bn be its braid closure where b has at least one of every

83

1generator. We begin by embedding b into the larger braid group Bn+1 which we denote by
b(cid:48) such that each generator σi is sent to σi+1. This can be viewed as adding a new individual

strand in front of the original braid b. This new strand will correspond to the outermost

component C in the closure.

Now from the braid closure of ˆb(cid:48), we will add new link components to obtain our link L.
Consider the projection p(ˆb(cid:48)) onto the surface S2 such that each crossing is now a vertex.

Excluding the regions touching C, each region of the projection will correspond to a link

component in the augmentation.

We will proceed to count the resulting regions. Since p(ˆb(cid:48)) is a graph in S2, the Euler
characteristic χ(S2) = 2 = v − e + r where v, e, and r are the number of vertices, edges,
and regions, respectively. Now each vertex corresponds to a generator such that v = k, and

since we have a 4-valent graph, then e = 2k. Through a quick calculation which includes

the additional region obtained from C, we see that r = (k + 3).

Lastly, since we do not need to add link components to the two regions touching C, we

can obtain our link L by adding components parallel to the z-axis in each of the remaining
(k + 1) corresponding regions of ˆb(cid:48).

84

Chapter 5

Applications Towards the AMU Con-

jecture

In this chapter, we construct elements in the surface mapping class groups which satisfy the

AMU conjecture. This is achieved by using the intrinsic relationship between the Turaev-Viro

invariant and the quantum representations as stated in Theorem 2.2.16. The links obtained

in Chapter 4 can be realized as mapping tori such that the corresponding monodromies

satisfy the AMU conjecture. With the exception of Section 5.3, we note that the results of

this chapter first appeared in [36] by the author of this dissertation.

5.1 Background

In this section, we will briefly discuss the Stallings fibration which has a useful application in

constructing elements of the surface mapping class groups which satisfy the AMU conjecture.

5.1.1 The Stallings Fibration

We will begin with the following definitions.

Definition 5.1.1. An element b ∈ Bn is a homogeneous braid if each standard generator
appearing in b always appears with the same sign.

Definition 5.1.2. For an oriented link L in S3, a Seifert surface is a connected, compact,

and oriented surface contained in S3 such that the oriented boundary is the oriented link L.

85

Seifert showed algorithmically that every oriented link in S3 has a Seifert surface. A

brief introduction of Seifert surfaces may be found in [38, Chapter 2]. Although we will not

provide the procedure in this work, we will note that the equation for calculating the genus

of the Seifert surface for the closure of a braid b ∈ Bs constructed with Seifert’s algorithm is

g =

2 + #(Crossings) − #(Components) − s

2

.

(5.1)

Now in [53], Stallings showed that the complement of the closure of any homogeneous

braid is fibered over S1. In these cases, the fiber surface is Σg,n where the surface is obtained

using Seifert’s algorithm. In particular, the complement of the closure of the homogeneous

braid is a mapping torus of the Seifert surface Σg,n for some f ∈ M od(Σg,n). In addition,
Stallings showed that there is an operation which takes a fibered link and transforms it into

another fibered link with the same fiber surface.

Definition 5.1.3. A Stallings twist is the operation of performing a 1/m-surgery on a simple

closed curve of the fiber surface where the framing of the curve is induced by the normal

vector of the surface.

In general, the Stallings twist may be trivial in the sense that the newly obtained link

complement may still be homeomorphic to the complement of the original link. In [18], the

authors demonstrate a criterion in obtaining non-trivial Stallings twists. They show that

as long as the braid contains either the elements σ2

i σ−2

i+1 or σ−2

i σ2

i+1, then the closure of the

braid admits non-trivial Stallings twists on the fiber surface.

As mentioned in Subsection 1.3.1, we recall that two elements f, g ∈ M od(Σg,n) are
independent if there is no h ∈ M od(Σg,n) such that both f and g are conjugate to non-
trivial powers of h. An important utility of the Stallings twist is that we can obtain infinitely

many pairwise independent pseudo-Anosov mapping class group elements as stated in the

following.

86

Theorem 5.1.4 ([18], Theorem 5.4). Let L be a hyperbolic fibered link with fiber Σ and mon-
odromy f . Suppose that L contains a sublink K with lT V (S3\K) > 0. Suppose, moreover,
that the fiber Σ admits a non-trivial Stallings twist along a curve c ⊂ Σ such that the interior
of the twisting disc D intersects K at most once geometrically. Let τc denote the Dehn twist
of Σ along c. Then the family {f ◦ τ m
c }m of homeomorphisms give infinitely many pairwise
independent pseudo-Anosov mapping class group elements in M od(Σ) that satisfy the AMU

conjecture.

Later in Subsection 5.2.1, we will use Theorem 5.1.4 on a particular family of links to

construct infinitely many pairwise independent pseudo-Anosov mapping class group elements

which satisfy the AMU conjecture.

5.2 Constructions of Pseudo-Anosov Elements

In this section, we construct elements in the mapping class group of the compact and ori-

entable surface Σg,n. We will accomplish this using a combination of techniques involving

the Stallings fibration, Theorem 4.3.2, and Corollary 4.5.4. At the end of the section, we

will define an explicit family of pure braids all of which satisfy the AMU conjecture.

5.2.1 Applications of the Stallings Fibration

In this subsection, we will provide new results of pseudo-Anosov elements in surfaces Σg,n

which satisfy the AMU conjecture. The main result of the subsection can be stated as

follows.

Theorem 5.2.1. Given g ≥ 0, there exists a pseudo-Anosov element in M od(Σg,4) that
satisfies the AMU conjecture. Furthermore, for g ≥ 3, there are infinitely many pairwise
independent pseudo-Anosov elements in M od(Σg,4) that satisfy the AMU conjecture.

The consequence of Theorem 5.2.1 is that we can find a pseudo-Anosov element for every

surface with four boundary components such that it satisfies the AMU conjecture. As it

87

is a direct consequence of the other results of this subsection, we will reserve the proof of

Theorem 5.2.1 until the end.

We will begin with a simple application of the Stallings fibration to the links in S3 up to

11-crossings which are the realization of the fundamental shadow links of complexity one.

LinkInfo Name Braid Group Bs Homogeneous Braid

L6a4
L6a4
L8n7
L10n87
L10n97
L10n108
L11n385

s = 3
s = 3
s = 4
s = 3
s = 4
s = 4
s = 4

(σ1σ−1
2 )3
(σ−1
1 σ2)3
σ1σ−2
2 σ1σ−1
3 σ−2
2 σ−1
1σ2
σ3
1σ2
2σ2
2σ1
1 σ−2
σ−3
2 σ−1
2 σ−1
1 σ−1
3 σ−2
3
(σ−1
1 σ−1
2 )3σ3σ−2
2 σ3
1 σ−1
(σ−1
2 σ3σ−1
2 )3σ3σ−2
2

3

Fiber Surface

Σ1,3
Σ1,3
Σ1,4
Σ3,3
Σ2,4
Σ2,4
Σ3,3

Table 5.1: The fundamental shadow links up to 11-crossings written as the closure of a
homogeneous braid such that the fiber surface is given by the Stallings fibration.

Corollary 5.2.2. The monodromy of the closure of the braids in Table 5.1 obtained from

the Stallings fibration is pseudo-Anosov and satisfies the AMU conjecture.

Proof. From Theorem 4.3.2, the link complements appearing in Table 5.1 are homeomor-

phic to the complement of a fundamental shadow link. Therefore, they all are hyperbolic

and satisfy the Turaev-Viro invariant volume conjecture. Since they are hyperbolic and by

Theorem 2.2.8, any corresponding monodromy as a mapping torus will be pseudo-Anosov.

Since they satisfy the Turaev-Viro invariant volume conjecture and by Theorem 2.2.16, then

their monodromies will also satisfy the AMU conjecture.

To end the proof, we will compute the fiber surfaces of the links in Table 5.1. Because

the fiber surface is obtained from Seifert’s algorithm on a braid closure, Equation (5.1) will

allow us to compute the genus as

g =

2 + #(Crossings) − #(Components) − s

2

.

Here, #(Crossings) is the length of the homogeneous braid, and #(Components) is the

88

2m

Figure 5.1: The link L5,m where the box indicates 2m negative crossings. Here, the Bor-
romean rings are the blue colored sublink.

number of components of the closure of the braid. This quick computation leads to the fiber

surfaces shown on the rightmost column of Table 5.1.

Next, we will construct infinitely many pairwise independent pseudo-Anosov mapping

class group elements using Corollary 5.2.2. The techniques utilized here were developed

in [18]. In their work, the authors build a 2-parameter family of links which contains the

figure-eight knot as a sublink in order to create their examples. Their method relies on

Theorem 5.1.4, and thus, it is necessary for their sublink to have an exponentially growing

Turaev-Viro invariant. By using the links in Corollary 5.2.2 where the Turaev-Viro invariant

volume conjecture is known, we can consider similar steps to construct a 2-parameter family

of links which also may be used to find examples satisfying the AMU conjecture. In this

subsection, we will only focus on using the Borromean rings. This specific choice of link will

be made clear in Remark 5.2.4.

We will begin to define our 2-parameter family of links Ln,m for n ≥ 4 and m ≥ 1 where

89

1n is the number of components. Necessary for the proof of our next proposition, we will

build Ln,m such that it is alternating and the closure of a homogeneous braid. We first let L

be the Borromean rings which corresponds to the link L6a4 in Tables 4.1 and 5.1. As noted

in Corollary 5.2.2, the Borromean rings have the property that they may be represented as
the closure of a homogeneous braid (σ−1
2 σ1)3 ∈ B3. We will now proceed to define Ci to
be the component obtained as the closure of the i-th strand of the homogeneous braid such

that C1 ∪ C2 ∪ C3 = L.

We will begin by defining the link for the case L5,m which is the union of L with the addi-

tional components C4 and C5 shown in Figure 5.1. In Figure 5.1, the parameter m indicates

the number of negative full-twists, or equivalently, 2m negative crossings. Additionally, we

can notice that our braid contains the subword σ2

4 . As mentioned in Subsection 5.2.1,

3σ−2

this implies that we will be able to later utilize a non-trivial Stallings twist.

From L5,m, we will inductively begin to add additional components Ci for 6 ≤ i ≤ n such
that the union of the components defines our link Ln,m. In order to build Ln+1,m from a

given Ln,m, we add a new strand with closure Cn+1 in the following way. From the viewpoint

of following along Cn, we see that Cn has 2 crossings with Cn+1, followed by 2 crossings with

Cn−1, and then ending with 2 more crossings with Cn+1 where the signs of the crossings

are chosen such that the link Ln+1,m is always alternating. Each time we continue with this

procedure of inserting a new component, we introduce 4 more crossings to the link.

Lastly, for the special case when n = 4, we simply replace the 2m negative crossings

in L5,m with (2m − 1) negative crossings. The resultant link will be our 4-component link
denoted by L4,m.

We will now proceed with finding our elements in the mapping class group using the

2-parameter link family Ln,m.

Proposition 5.2.3. Suppose that either n = 4 and g ≥ 3, or g + 2 ≥ n ≥ 5, then there are
infinitely many pairwise independent pseudo-Anosov elements of M od(Σg,n) that satisfy the

AMU conjecture.

90

Proof. By construction, the 2-parameter link family Ln,m contains the Borromean rings as a

sublink for every n ≥ 4 and m ≥ 1. Now notice by Proposition 2.1.6, drilling out tori from
our 3-manifold does not decrease the exponential growth rate of our Turaev-Viro invariant.

In particular, if we start with any link where the Turaev-Viro invariant grows exponentially,

then any new link obtained by adding additional components will also have the Turaev-

Viro invariant growing exponentially. Since the Borromean rings have this property, this is

also the case for the 2-parameter link family Ln,m which contains the Borromean rings as a

sublink. Now by Theorem 2.2.16, any mapping torus of the link complement of Ln,m will

have its corresponding element in the mapping class group satisfy the AMU conjecture.

Additionally, our link family Ln,m will be hyperbolic by Menasco’s criterion [41] which

states that any prime, non-split, and alternating diagram of a link that is not the standard

diagram of the T (2, q) torus link is a hyperbolic link. By Theorem 2.2.8, this implies that

the element in the mapping class group is also pseudo-Anosov.

Now by construction, our 2-parameter link family Ln,m is the closure of a homogeneous

braid such that we can consider its Stallings fibration with fiber surface Σg,n for some g. As

noted previously, our corresponding braid contains the subword σ2

3σ−2

4

such that we have a

non-trivial Stallings twist. Now by Theorem 5.1.4, we can obtain infinitely many pairwise

independent pseudo-Anosov elements satisfying the AMU conjecture for each fiber surface

obtained from Ln,m using the Stallings fibration.

We will end the proof by calculating the genus for each of the fiber surfaces Σg,n of

the Stallings fibration of Ln,m. From Equation (5.1), we can calculate the genus of the

homogeneous braid as

g =

2 + #(Crossings) − n − s

2

where n is the number of components and s the index of the corresponding braid group.

We first consider the case when n = 4. By counting, we have (2m + 11) crossings with

91

braid group B5 such that s = 5. We can calculate that

g =

2 + #(Crossings) − n − s

2

=

2m + 4

2

= m + 2,

therefore, g = m + 2 for m ≥ 1.

Now we consider the case for n ≥ 5. Here, the number of components is equal to the
index of the braid group such that n = s. As mentioned earlier, each time we increase n by

one, we add four additional crossings. This implies that there are (2m + 4n − 8) crossings.
From this, we calculate that

g =

2 + #(Crossings) − n − s

2

=

2m + 2n − 6

2

= m + n − 3,

therefore, g = m + n − 3 for m ≥ 1. These two cases imply our final result that our fiber
surface Σg,n has either n = 4 and g ≥ 3, or g + 2 ≥ n ≥ 5.

As mentioned previously, Proposition 5.2.3 and its proof should be compared to [18,

Theorem 1.4]. Here, the authors similarly find infinitely many pairwise independent pseudo-

Anosov elements for the surface Σg,n where n = 2 and g ≥ 3, or g ≥ n ≥ 3. Although
the method is the same, by considering the Borromean rings as opposed to the figure-eight

knot as our sublink, we gain additional infinite families for M od(Σg,n) for certain g and n

not previously discussed. Noticeably, their examples always have genus larger than or equal

to the number of boundary components; however, with the Borromean rings, we can obtain

infinitely many pairwise independent pseudo-Anosov elements where the genus is strictly less

than the number of boundary components.

Remark 5.2.4. An important aspect of the proof of Proposition 5.2.3 is the use of Menasco’s

criterion. Because of this, we chose to use the Borromean rings from Table 5.1 as our example.

In this case, our 2-parameter family Ln,m will be alternating which will then imply that the

link is hyperbolic by Menasco’s criterion. Our previous construction would have worked with

92

any other homogeneous braid from Table 5.1; however, the resulting manifold may not be

hyperbolic. In these cases, the Turaev-Viro invariant will still grow exponentially; however,

the associated mapping class group element may not be pseudo-Anosov. Instead, the element

may only be reducible and contain pseudo-Anosov parts. In either of these two settings, the

conclusion of the AMU conjecture will still follow.

We will now end the subsection with a proof of Theorem 5.2.1.

Proof of Theorem 5.2.1. It is immediate that the combination of Corollary 5.2.2 and Propo-

sition 5.2.3 provides a pseudo-Anosov element of M od(Σg,4) for every g ≥ 1. In particular,
we see from Proposition 5.2.3 that for g ≥ 3, there are infinitely many pairwise independent
pseudo-Anosov elements satisfying the AMU conjecture. For the case when g = 0, Ander-

sen, Masbaum, and Ueno in [3] verified the conjecture for every pseudo-Anosov element in

M od(Σ0,4) as noted in Section 1.3.

5.2.2 Applications of Corollary 4.5.4

In this subsection, we will demonstrate how Corollary 4.5.4 may be utilized to construct

links fibering over S1. As we will see in Subsection 5.2.3, Subsection 5.2.2 will be used in

preparation of obtaining additional examples of pseudo-Anosov elements in the mapping

class group which satisfy the AMU conjecture. We also note that the results of Theorem

5.2.5 are obtained using the fundamental shadow links. Therefore, any of the corresponding

manifolds will also satisfy the Turaev-Viro invariant volume conjecture.

Our main result of the subsection states that given any complexity k, we can find a

fundamental shadow link which fibers over S1 with fiber surface a disk with punctures. In

order to write the statement, we will first define the family of braids {bk} as follows:

• For k = 1, define bk := σ−2

1 σ2

2 ∈ B3.

93

L(cid:48)

B

k

B

L(cid:48)

k

Figure 5.2: The complexity k fundamental shadow links realized in S3 obtained from the
braid family αk. On the left, for k even, there are (k + 4) components, and on the right, for
k odd, there are (k + 3) components. Here, our eventual braid axis component is labeled as
B, and the component that bounds a (k + 1)-punctured disk is labeled L(cid:48).

• For k = 2m, define bk ∈ Bk+3 by

m(cid:89)
(σ2i+1σ2i ··· σ3σ2σ2

i=1

1σ−1

bk :=

2 σ3σ4 ··· σ2iσ2i+1)(σ2i+2σ2i+1 ··· σ3σ2

2σ3 ··· σ2i+1σ2i+2).

• For k = 2m − 1, define bk ∈ Bk+3 by

m(cid:89)
(σ2iσ2i−1 ··· σ3σ2

i=2

bk := σ

2σ3 ··· σ2i−1σ2i)(σ2i+1σ2i ··· σ3σ2σ2

2 σ3σ4 ··· σ2iσ2i+1),

1σ−1

where σ = (σ1)(σ3σ2σ2

1σ−1

2 σ3).

Theorem 5.2.5. For k ≥ 1, let Mk := S3\IntNbd(bk) denote the complement of the braided
link bk. Then

1.) Mk fibers over S1 with fiber surface a disk with punctures,

2.) Mk is hyperbolic with volume 2kv8, and

94

1B

B

RB

σ1σ−1

1

Figure 5.3: On the leftmost diagram, we have the closure of the braid α2 = σ1σ−1
1 with the
region labeled RB corresponding to the B labeled component in the second diagram. From
the braid, we can apply Corollary 4.5.4 to obtain the link in the second diagram where B
will eventual be the braid axis component. Through an isotopy, we obtain the link in the
third diagram.

3.) Mk satisfies the Turaev-Viro invariant volume conjecture.

Proof. For k ≥ 2, we will begin by considering the unreduced braids αk = (σ1σ−1
1 ) k
even, and αk = σ1(σ−1

2 , for k odd. Now from the braids αk, we will apply Corollary

1 σ1) k−1

2 , for k

4.5.4 followed by an isotopy to obtain either the left link in Figure 5.2, for k even, or the

right link, for k odd. For clarity, an example is shown in Figure 5.3 for when k = 2. In

this case, the region corresponding to the component B is labeled as RB. Now since our

link is always constructed from the closure of a braid, we will always have an inside region

RB, and we can label the corresponding component in the link by B as shown in Figure 5.2.

Eventually, the component B will be our braid axis in our fibered link. We note that when

k is even, the sublink obtained from the braid closure of αk contains two components. In

the case when k is odd, the sublink obtained from the braid closure of αk contains only one

component. This accounts for the difference between the total components in the left and

right diagrams of Figure 5.2.

Now we let L be a link of Figure 5.2 with link component L(cid:48) as labeled. Notice that L(cid:48)

bounds a (k + 1)-punctured disk where one of the punctures comes from the component B.

We consider the self-homeomorphism obtained by performing a full-twist along the (k + 1)-
punctured disk that L(cid:48) bounds. Under this self-homeomorphism, we will obtain a new link

95

1L(cid:48)

L(cid:48)

L(cid:48)

B

B

L(cid:48)(cid:48)

B

Figure 5.4: The case of obtaining a fibration for k = 2. Performing a full-twist on
the 3-punctured disk shaded blue in the leftmost diagram results in the middle diagram.
Through an isotopy of the link, we obtain a link fibering over S1 with fiber surface D5
as represented by the last diagram. Here, the monodromy may be written explicitly as
(σ3σ2σ2

2 σ3)(σ4σ3σ2

2σ3σ4).

1σ−1

in S3 where we define the complement of the manifold as Mk. In this new link, B, as a

braid axis, now bounds a (k + 3)-punctured disk Dk+3. The components of our new link

in S3 may now be isotoped to our defined braided link bk. Since we initially began with a

fundamental shadow link of complexity k, our manifold Mk will be hyperbolic with volume

2kv8 and satisfy the Turaev-Viro invariant volume conjecture.

Now from the relationship between braided links and mapping tori, we will see that our

link may be viewed as a mapping torus M (Γ(bk)) where Γ is the surjective homomorphism

between Bk+3 and M CG(Dk+3) defined in Subsection 2.2.2. Additionally, the family bk in

Bk+3 will be seen explicitly as the elements

m(cid:89)
(σ2i+1σ2i ··· σ3σ2σ2

i=1

1σ−1

2 σ3σ4 ··· σ2iσ2i+1)(σ2i+2σ2i+1 ··· σ3σ2

2σ3 ··· σ2i+1σ2i+2),

when k = 2m, and

(σ1σ3σ2σ2

1σ−1

2 σ3)

m(cid:89)
(σ2iσ2i−1 ··· σ3σ2

i=2

2σ3 ··· σ2i−1σ2i)(σ2i+1σ2i ··· σ3σ2σ2

2 σ3σ4 ··· σ2iσ2i+1),

1σ−1

when k = 2m − 1 and k (cid:54)= 1.

96

1Figure 5.5: An isotopy of the Borromean twisted sister to a braided link where the braid
axis is colored black. The resulting manifold has fiber surface D3 with monodromy the braid
group element σ−2

2 ∈ B3 under the surjective homomorphism Γ.

1 σ2

Obtaining this fibration of the link in S3 with the given monodromy is straightforward,

and we will demonstrate an example of this process for k = 2 in Figure 5.4. We begin by

isotoping the component B so that locally we have a copy of Dk+3 × [−
, 
] for some 
. After
this isotopy, we may cut along the surface bounded by B such that we obtain the braid

element bk which corresponds to our initial definition of the elements of the family. After

this step, we have our desired fibration with monodromy Γ(bk).

Now lastly, we consider the case when k = 1. This will be done by showing an explicit

isotopy of the Borromean twisted sister. For reference, the Borromean twisted sister appears

as the rightmost diagram of Figure 4.3; however, it will be illustrated here again. As shown

in Corollary 4.4.5, the complement of the Borromean twisted sister is a complexity one

fundamental shadow link such that it satisfies the Turaev-Viro invariant volume conjecture

with hyperbolic volume 2v8. Now from following the isotopies in Figure 5.5, we can see that

one of the components is a braid axis that bounds a 3-punctured disk such that we have a

fibered link. By cutting along the 3-punctured disk, we can see that our monodromy is the
braid group element σ−2

1 σ2

2 ∈ B3 under the map Γ.

In conclusion, we can find a manifold with volume 2kv8, for any integer k ≥ 1, which
satisfies the Turaev-Viro invariant volume conjecture, fibers over S1 with fiber surface a disk

with punctures, and has explicit monodromy represented by bk.

Remark 5.2.6. Once we have a fibering link from Theorem 5.2.5, we may continue to

97

1construct more examples of links satisfying the Turaev-Viro invariant volume conjecture by

performing full-twists along the punctured disks the link components bound. Through this

process, the new link with a homeomorphic complement may have a different fiber surface and

monodromy. Additionally, there are self-homeomorphisms which may also remove the braid

axis such that the resulting link is the closure of a braid. For example, a link with complement

homeomorphic to the complement of a fundamental shadow link may be obtained as the braid

closure of the element

(σ4σ3σ2σ2

1σ2σ−1

3 σ4)(σ5σ4σ2

3σ4σ5)(σ1σ2σ3σ4σ2

5σ4σ3σ2σ1) ∈ B6.

This can be seen by performing a full-twist along the component labeled L(cid:48)(cid:48) in the rightmost

diagram of Figure 5.4.

Remark 5.2.7. In the proof of Theorem 5.2.5, we obtained our results by applying Corollary

4.5.4 to a particular family of 2-braids. In general, this process may be repeated for other n-

braids to obtain other links fibering over S1 which satisfy the Turaev-Viro invariant volume

conjecture; however, the monodromies may be more difficult to write down explicitly. In

Subsection 5.2.3, we will follow the same steps for a particular family of 3-stranded braids

such that we obtain fibered links satisfying similar properties. These links in combination

with the links in Theorem 5.2.5 will be used to create explicit pseudo-Anosov elements in

the mapping class group of a genus zero surface with n boundary components.

We will end this subsection with a result for obtaining additional fibered links which

satisfy the extended Turaev-Viro invariant volume conjecture.

Corollary 5.2.8. Let Mk be a manifold from Theorem 5.2.5 of volume 2kv8 that is realized

as the complement of the braided link bk where bk ∈ Bn for some n. Then for any p ≥ 2,
there exists a manifold M(cid:48)
k ∈ Bn+p−1

k obtained as the complement of a braided link b(cid:48)

k where b(cid:48)

98

such that

lim
r→∞

(cid:16) 2π

Here, r is odd, q = exp

2π
r
√−1

log |T Vr(M(cid:48)
(cid:17)

k, q)| = v4(cid:107)M(cid:48)

k(cid:107) = 2kv8.

, (cid:107)M(cid:48)

k(cid:107) denotes the simplicial volume of M(cid:48), and b(cid:48)

k is

r

obtained as an embedding of bk into Bn+p−1.

Proof. For p ≥ 2, we will first define Tp with distinguished boundary T to be the invertible
cabling space as stated in Definition 2.1.7. To recall, Tp is the complement of a solid torus

with p strands parallel to the core where the outer boundary torus of Tp is labeled T . For

k ≥ 1, the manifold Mk from Theorem 5.2.5 satisfies the Turaev-Viro invariant volume
conjecture, and more generally, Mk satisfies the extended Turaev-Viro invariant volume
conjecture. We will now define M(cid:48)

k to be the manifold obtained by gluing ∂(Tp)\T for the
manifold Tp to the boundary component of Mk which corresponds to the braid axis. Now
by Proposition 2.1.8, the manifold M(cid:48)

k satisfies the extended Turaev-Viro invariant volume

conjecture with the following equalities

lim
r→∞

2π
r

log |T Vr(M(cid:48)

k, q)| = lim
r→∞

2π
r

log |T Vr(Mk, q)| = v4(cid:107)Mk(cid:107) = 2kv8

(cid:16) 2π

(cid:17)

.

√−1

r

where r runs along the odd integers and q = exp

Lastly, the manifold M(cid:48)

k may be viewed explicitly as the complement of a braided link.

This is because both the manifolds Mk and Tp are the complement of a link in a solid torus.
By gluing the outer torus of Mk to one of the inner tori of Tp, our resultant manifold M(cid:48)

k

is still a complement of a link in a solid torus.

In other words, the manifold M(cid:48)

k is the

complement of a braided link obtained by drilling out (p − 1) solid tori from Mk.

Although Proposition 2.1.6 states that the exponential growth rate does not decrease by

drilling out tori, it is not necessary for the growth rate to be related to the simplicial volume

of the new manifold; however, Corollary 5.2.8 provides explicit examples where this holds.

99

5.2.3 Explicit Pure Braids Satisfying the AMU Conjecture

In this subsection, we will define explicit elements in the braid group which are pure braids.

Because they are pure braids, they also represent elements in the mapping class group for

genus zero surfaces as stated in Subsection 2.2.2. Additionally, since they will be constructed

from fundamental shadow links, the explicit elements in the mapping class group will also

satisfy the AMU conjecture. As a result, we provide the first known examples of pseudo-

Anosov elements satisfying the conjecture in the mapping class group of Σ0,n for odd n.

Under the identification of elements of the pure braid group P Bn with M od(Σ0,n+1), we

will now define the family of pure braids {ωm}, for m ≥ 4, where we refer to {bk} as the
family of braids obtained from Theorem 5.2.5:

• For m = 4, define ω4 := b1 ∈ M od(Σ0,4).
• For m ≥ 6 and even, define ωm := bm−4 ∈ M od(Σ0,m).
• For m = 5 and m = 7, define ωm := ω(cid:48)

m−1 is an
embedding of ωm−1 ∈ M od(Σ0,m−1) into M od(Σ0,m) sending each generator σj to σj.
• For m = 9, define ω9 ∈ M od(Σ0,9) by

m−1 ∈ M od(Σ0,m) such that ω(cid:48)

ω9 :=(σ4σ2

3σ4)(σ5σ4σ3σ2
2σ−1

(σ7σ6σ5σ4σ3σ2

3 σ4σ5σ6σ7).

2σ3σ4σ5)(σ6σ5σ4σ3σ2σ2

2 σ3σ4σ5σ6)

1σ−1

• For m = 7 + 2s and s ≥ 2, define ωm ∈ M od(Σ0,m) by

ωm :=ω(cid:48)

9

s(cid:89)
(σ4+2iσ3+2i ··· σ4σ2
2σ−1

i=2

(σ5+2iσ4+2i ··· σ3σ2

3σ4 ··· σ3+2iσ4+2i)

3 σ4σ5 ··· σ4+2iσ5+2i)

100

where ω(cid:48)

9 is an embedding of ω9 ∈ M od(Σ0,9) into M od(Σ0,m) sending each generator

σj to σj.

Now in terms of the pure braid family {ωk}, we state the following result.

Corollary 5.2.9. For n ≥ 4, the element ωn ∈ M od(Σ0,n) satisfies the AMU conjecture.
Additionally, if n (cid:54)∈ {5, 7}, then ωn is a pseudo-Anosov element.

Proof. We will first consider the case for genus zero surfaces with an even number of boundary
components. We note that for every m ≥ 3 and odd, there exists a link fibering over S1 that
has fiber surface Dm such that the complement satisfies the Turaev-Viro invariant volume

conjecture. The examples of such manifolds are M1 and Mk, for even k, obtained from

Theorem 5.2.5. For reference, we will list the associated braids as follows:

• For k = 1, we defined bk := σ−2

1 σ2

2 ∈ B3.

• For k = 2j, we defined bk ∈ Bk+3 as

j(cid:89)
(σ2i+1σ2i ··· σ3σ2σ2

i=1

bk :=

1σ−1

2 σ3σ4 ··· σ2iσ2i+1)(σ2i+2σ2i+1 ··· σ3σ2

2σ3 ··· σ2i+1σ2i+2).

Notice that in these cases, the monodromies are obtained from pure braids. Under the

identification of the pure braid group P Bm with M od(Σ0,m+1), this implies that we can view

our manifolds as the mapping tori M (bk) with monodromy bk ∈ M od(Σ0,m+1).

Now by Theorem 2.2.8, since our manifolds are hyperbolic, the element bk ∈ M od(Σ0,m+1)
is pseudo-Anosov. By Theorem 2.2.16, since our manifolds are mapping tori which satisfy

the Turaev-Viro invariant volume conjecture where the elements are pseudo-Anosov, then

bk satisfies the AMU conjecture. This shows that for any n ≥ 4 and even, we can find a
pseudo-Anosov element bk ∈ M od(Σ0,n) that satisfies the AMU conjecture.

We now consider the cases when our surface is Σ0,n for n ∈ {5, 7}. Our argument will
follow the same logic in constructing elements in the mapping class group such that the

101

s

=

2s

B

L(cid:48)

(s − 1)

Figure 5.6: On the left, we define the boxed diagram as the identity 3-braid with 2s additional
components. For a box labeled with 0, the diagram is the identity 3-braid without additional
components. On the right, we show the link in S3 obtained from applying Corollary 4.5.4 to
the element (σ1σ−1
2 ) ∈ B3. Here, the braid axis is labeled B, the link component that
bounds a (3 + 2s)-punctured disk is labeled L(cid:48), and the link contains (7 + 2s) components.

1 )s(σ2σ−1

corresponding mapping tori have exponential growth; however, we will choose to utilize

Corollary 5.2.8. We first begin with any pure braid in P Bm for m ≤ (n − 2) where the
corresponding mapping torus satisfies the Turaev-Viro invariant volume conjecture. Now

by Corollary 5.2.8, for p ≥ 2, we may add an additional (p − 1) strands to obtain a pure
braid in P Bm+p−1 such that the corresponding mapping torus also has Turaev-Viro invariant

with an exponential growth rate. This implies that the constructed element in the mapping

class group M od(Σ0,m+p) will satisfy the AMU conjecture. For n ∈ {5, 7}, we can choose to
use our previous pure braids from the case of a genus zero surface with an even number of

boundary components.

It is important to note the following. By utilizing the previous procedure, any number of

elements in different mapping class groups satisfying the AMU conjecture may be obtained.

Although in these cases, the elements will only contain pseudo-Anosov parts, and they will

not be pseudo-Anosov.

102

1Lastly, we will consider the case when n ≥ 9 and odd. Similar to the even boundary
case, we will construct a pure braid through a combination of Corollary 4.5.4 and self-

homeomorphisms of the manifold. For s ≥ 1, consider (σ1σ−1

2 ) ∈ B3 as the unre-
duced word of length (2s + 2). From Corollary 4.5.4, we can construct a hyperbolic manifold

1 )s(σ2σ−1

satisfying the Turaev-Viro invariant volume conjecture which is the complement of the link

shown in the right diagram of Figure 5.6. Here, the box labeled with (s− 1) is defined on the
left diagram. We also note that on the right diagram when (s− 1) = 0, the boxed diagram is
defined to be the identity 3-braid without any additional components. In the same manner
of the proof of Theorem 5.2.5, we can perform a full-twist along the component L(cid:48) to obtain

a braided link ω7+2s with braid axis B such that ω7+2s is a pure braid.

Now since L(cid:48) and B bound a (3 + 2s)- and 4-punctured disk, respectively, then after

the full-twist, our resultant link will have fiber surface D6+2s. From the full-twist, we can

describe the obtained elements in the pure braid group explicitly as follows:

• For s = 1, define ω7+2s ∈ B8 by

ω7+2s :=(σ4σ2

3σ4)(σ5σ4σ3σ2
2σ−1

(σ7σ6σ5σ4σ3σ2

3 σ4σ5σ6σ7).

2σ3σ4σ5)(σ6σ5σ4σ3σ2σ2

2 σ3σ4σ5σ6)

1σ−1

• For s ≥ 2, define ω7+2s ∈ B6+2s by
s(cid:89)
(σ4+2iσ3+2i ··· σ4σ2
2σ−1

ω7+2s :=ω(cid:48)

(σ5+2iσ4+2i ··· σ3σ2

i=2

9

3σ4 ··· σ3+2iσ4+2i)

3 σ4σ5 ··· σ4+2iσ5+2i)

where ω(cid:48)

9 is an embedding of ω9 ∈ B8 into B6+2s sending each generator σj to σj.

Since ω7+2s ∈ B6+2s is a pure braid and the complement of the braided link ω7+2s satisfies the
Turaev-Viro invariant volume conjecture, we can use the same argument as for the case of the

103

genus zero surface with an even number of boundary components. Under the identification

of the pure braid group P B6+2s with the mapping class group M od(Σ0,7+2s), we can find

a corresponding element in M od(Σ0,7+2s) which is pseudo-Anosov and satisfies the AMU

conjecture.

In conclusion, this shows that for every n ≥ 4, our pure braid family {ωn} satisfies the

AMU conjecture such that ωn is pseudo-Anosov for n (cid:54)∈ {5, 7}.

5.3 Cosets of the Pure Braid Group

In this section, we will construct additional examples of elements in the pure braid group

which satisfy the AMU conjecture. This will be done by considering the forgetful maps for

pure braid groups as well as utilizing the normal form of the pure braid group shown by

Artin in [4]. Note that this consideration for using the normal form was found in Remark

5.7 of [18].

Definition 5.3.1. For 1 ≤ i ≤ n, we define the i-th forgetful map on the pure braid
group Fi : P Bn → P Bn−1 to be the surjective homomorphism obtained by deleting the
i-th strand of the pure braid. For 1 ≤ l ≤ n − 1, a forgetful map on the pure braid group
F : P Bn → P Bn−l is a composition of i-th forgetful maps.

For a given forgetful map F : P Bm+l → P Bm and an element b ∈ P Bm, we can see that
F −1(b) = {βk} is a collection of pure braids βk ∈ P Bm+l. Now notice that the corresponding

complement of the braided link βk is obtained from the complement of b by drilling out tori.

If our initial braided link b has Turaev-Viro invariant with exponential growth rate, then βk

also has exponential growth rate by Proposition 2.1.6. Additionally, by Theorem 2.2.16, since

the complement of βk corresponds to the mapping torus of βk ∈ M od(Σ0,m+l+1), then the
element βk also satisfies the AMU conjecture. Note that this is a more general construction

for finding elements that satisfy the AMU conjecture than the one found in the proof of

104

Corollary 5.2.9 for the n ∈ {5, 7} case. Here, the main difference is that our elements in the
proof of Corollary 5.2.9 also satisfy the extended Turaev-Viro invariant volume conjecture

which is not necessarily true for the general elements obtained from the inverses of the

forgetful map. Using the forgetful maps, we see that we can obtain many elements that

satisfy the AMU conjecture.

To add more structure to the elements, we will consider the normal form [4] of the pure

braid group. This will allow us to view the elements which satisfy the AMU conjecture

as cosets of the pure braid groups.

It is well-known that for any n > k, the pure braid

group P Bn may be written as a semidirect product of Wn and P Bk where Wn is itself a

semidirect product of free subgroups of P Bn. Because of this, any braid b ∈ P Bn has a
unique representation as a product of β · ω for β ∈ Pk and ω ∈ Wn. Now as before, notice
that the braided link b = β · ω contains the braided link β as a sublink such that if the
complement of β has Turaev-Viro invariant with an exponential growth rate, then b also has

Turaev-Viro invariant with an exponential growth rate. Once again, by Theorem 2.2.16, the

element b ∈ M od(Σ0,n+1) will satisfy the AMU conjecture. This leads us into the following
corollary where we define the collection of pure braids P to contain the elements from Table
5.2 and the elements from Corollary 5.2.9. Here, the elements in Table 5.2 are the links up

to 11-crossings from Theorem 4.3.2 which can be obtained as the closure of a pure braid.

Corollary 5.3.2. For a fixed n, let β ∈ P Bs be an element of P with n > s. Now for
any b ∈ P Bn with normal form b = β · ω where ω ∈ Wn, the corresponding element b in
M od(Σ0,n+1) satisfies the AMU conjecture.

Proof. The proof follows from the previous discussion. The braided link of every element β

in the collection P has Turaev-Viro invariant with an exponential growth rate. Now notice
that the braided link of an element b = β · ω ∈ P Bn also contains β as a sublink. Therefore,
the complement of b which is the mapping torus of b ∈ M od(Σ0,n+1) also has Turaev-Viro
invariant with an exponential growth rate. By Theorem 2.2.16, the element b ∈ M od(Σ0,n+1)
satisfies the AMU conjecture.

105

LinkInfo Name Pure Braid Group P Bs

L6a4
L6a4
L8n5
L8n7
L8n7
L8n7
L8n7
L8n7
L8n7
L8n7
L10n87
L10n97
L10n97
L10n97
L10n105
L10n105
L10n105
L10n105
L10n105
L10n105
L10n105
L10n105
L10n108
L10n108
L10n108
L10n108
L10n108
L10n108

s = 3
s = 3
s = 3
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 3
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4
s = 4

Pure Braid Element

2

2

1

1

(σ1σ−1
2 )3
(σ−1
1 σ2)3
1 σ2)2σ−1
1 σ−3
(σ−1
3 σ2σ−2
σ1σ2σ−1
1 σ2σ3σ2σ1
1 σ2σ3σ−1
2 σ−1
3 σ2σ−2
2 σ−1
σ−1
1 σ−1
3 σ−2
2 σ−1
2σ−1
σ1σ−2
2 σ3σ−2
1 σ2
1 σ2σ3σ−1
3 σ2σ−2
2 σ−1
2 σ−1
1 σ−1
σ−1
2σ1σ3σ−2
σ1σ2
2 σ3
σ1σ−2
2 σ1σ−1
3 σ−2
2 σ−1
3
3 σ2σ−1
σ1σ2σ−2
1 σ2σ2
3σ2
1σ2
1σ2
σ3
2σ2
2σ1
2 σ−2
3 σ−3
1 σ−1
σ1σ2σ−1
2 σ3σ2σ1
σ−3
1 σ−2
2 σ−1
1 σ−1
3 σ−2
2 σ−1
3
2 σ−1
2 σ−2
3 σ−1
σ1σ−3
1 σ2σ2
3σ2
σ−1
1 σ2σ1σ−1
2 σ1σ−1
3 σ−1
2 σ3σ−1
2 σ−1
3
σ1σ2σ−1
1 σ2σ1σ−1
3 σ2σ−1
3 σ−1
2 σ3
σ1(σ2σ3)2σ2σ−1
1 σ−1
2 σ3σ2σ−1
3 σ2
σ1σ−1
2 σ3σ−1
2 σ−1
3 σ2σ−1
1 σ2σ3σ−1
2 σ3σ2
(σ−1
1 σ−1
2 )2σ−1
1 σ3σ−1
2 σ−1
3 σ2σ−1
σ1σ−1
2 σ1σ2σ−1
1 (σ3σ2)2σ3
σ1σ2σ−1
3 σ−1
2 σ3σ−1
2 σ−1
1 (σ−1
2 σ−1
3 )2σ−1
σ1σ−1
2 σ−1
3 σ2σ−1
3 σ−1
2 σ−1
1 σ2σ−1
3 σ2σ3σ−1
(σ−1
1 σ−1
2 )3σ3σ−2
2 σ3
2 σ−1
1 (σ−1
3 σ2σ−1
σ1σ2σ−2
3 )3
2 σ3σ−2
2 σ−1
1 σ−1
2 σ3σ2σ−1
σ1σ−1
2 σ1σ2σ−1
1 σ−2
3 σ2σ−2
σ1σ2σ−1
1 σ2σ1σ3σ−2
2 σ3σ−1
σ1σ2σ−2
3 σ2σ−1
1 σ−1
3 σ2σ3σ−1

3 σ2
3

σ1σ−2

2 σ3σ2

2

2

3

2

Table 5.2: The fundamental shadow links from Theorem 4.3.2 which can be written as the
closure of a pure braid.

Corollary 5.3.2 allows us to construct cosets of elements in the pure braid group which

satisfy the AMU conjecture; however, there is a key distinction compared to the elements in

Corollary 5.2.9. For Corollary 5.2.9, we were careful in ensuring that all but finitely many

of the elements are pseudo-Anosov.

In the case of Corollary 5.3.2, there is no guarantee

on whether the elements are pseudo-Anosov or whether they only contain pseudo-Anosov

pieces.

106

Remark 5.3.3. Although the braided links β formed from the braids β in Table 5.2 are

not necessarily fundamental shadow links, β as an element in the mapping class group of a

genus zero surface still satisfies the AMU conjecture. This is because the mapping torus of

the element β as an element in the mapping class group of a genus zero surface corresponds

to the link complement of a braided link which inherently contains the braid axis. Now the

braided link β contains the braid closure of β as a sublink such that the complement of β

is a fundamental shadow link realized in S3 with a drilled out torus from the braid axis.

Since drilling out a torus does not decrease the exponential growth rate of the Turaev-Viro

invariant, the complement of the braided link β also has Turaev-Viro invariants with an

exponential growth rate, and we can apply Theorem 2.2.16 to see that β as an element in

the mapping class group of a genus zero surface satisfies the AMU conjecture.

107

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