THE MURASUGI SUM AND SYMPLECTIC FLOER HOMOLOGY
By
Jared Able
A DISSERTATION
Submitted to
Michigan State University
in partial fulllment of the requirements
for the degree of
Mathematics Doctor of Philosophy
2022
ABSTRACT
This Ph.D. dissertation studies the operation of Murasugi sum on pairs of knots (or links) in
the three-sphere S 3. This operation, which produces a knot (or link) involves several choices
which usually change the isotopy class of the produced object. This could lead one to believe
the Murasugi sum has no hope of preserving properties of knots (or links). Indeed, without
restricting the involved choices, we show in Chapter 2 that any knot is the Murasugi sum
of any two knots. However, we also show that restricting the possible choices restricts the
possible Murasugi sums. The contents of Chapter 2 are joint with Mikami Hirasawa [1].
Historically, the usual restriction has been to consider knots which are Murasugi summed
along minimal genus Seifert surfaces. In this setting, knot genus is additive under Murasugi
sums [11], and the rank of the top group of knot Floer homology HFK
[ is multiplicative
under Murasugi sums [4, 38]. Continuing this trend in Chapter 4, we study the symplectic
Floer homology HF∗ of a family of knots which are closures of a particular type of 3-string
braid. These knots can be viewed as Murasugi sums performed along minimal genus Seifert
surfaces, and we show that a large range of choices in these Murasugi sums all yield the
same rank of HF∗ . We carry out these calculations via the Bestvina-Handel algorithm [2]
and the combinatorial formula for HF∗ of pseudo-Anosov maps due to Cotton-Clay [5]. We
hope that these calculations shed some light on the behavior of HF∗ under Murasugi sums,
as this group has garnered recent interest for its connection to the next-to-top group of
knot Floer homology [15, 40].
Copyright by
JARED ABLE
2022
ACKNOWLEDGEMENTS
I would like to thank my advisor, Matt Hedden, for his willingness to listen to my ideas, and
for his exceptional enthusiasm and breadth of knowledge. I would also like to thank Mikami
Hirasawa, whose patience, as well as his concrete knowledge of Seifert surfaces and Murasugi
sums, still astounds me. Finally, I would like to thank Emily, who continues to support and
challenge me in this journey called life.
iv
TABLE OF CONTENTS
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 THE MURASUGI GRAPH OF KNOTS . . . . . . . . . . . . . . . . 5
CHAPTER 3 PSEUDO-ANOSOV MAPS AND THE BESTVINA-HANDEL
ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
CHAPTER 4 SYMPLECTIC FLOER HOMOLOGY . . . . . . . . . . . . . . . . . 31
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
v
CHAPTER 1
INTRODUCTION
Figure 1.1. Two trefoil knots (left), and their connected sum (right).
A recurring theme in topology is to understand how manifolds behave under various
operations. For example, a classical way to add two knots K1 , K2 is the connected sum
K1 #K2 , see Figure 1.1. This operation is well-behaved with respect to many geometric
and algebraic invariants of knots such as the genus g(K), the signature σ(K), the Alexander
polynomial ∆K (t), and knot Floer homology HFK(K)
[ [30, 41]. For this reason, the connected
sum and its various generalizations (e.g. satellite operations and band sums) have proven
fruitful in constructing innite families of distinct knots which all satisfy a given property,
c.f. [6, 18, 19]. Historically, such constructions have motivated the search for increasingly
rened invariants of knots.
We are mainly interested in the operation known as the Murasugi sum, which is a gen-
eralization of the connected sum of knots. Given two knots K1 , K2 with Seifert surfaces
F1 , F2 satisfying ∂F1 = K1 , ∂F2 = K2 , the Murasugi sum K1 ⋆m K2 identies K1 , K2 along
an even-sided m-gon embedded in F1 , F2 . See Figure 2.1, and see Denition 2.2.1 for a pre-
cise denition. This operation was introduced by Murasugi in [34] and was later coined as
the Murasugi sum and popularized by Gabai in [11] via the slogan The Murasugi sum is
a natural geometric operation. Indeed, Gabai showed that geometric invariants of Seifert
surfaces such as beredness, incompressibility, and being of minimal genus are preserved
under Murasugi sums.
Whereas the connected sum between two oriented knots is unique, the Murasugi sum
of two knots is highly dependent on the choices of surfaces and embedded m-gons therein for
1
m ≥ 4. In this thesis, we consider placing various restrictions on the choices involved in the
Murasugi sum. In Chapter 2, we begin by studying the unrestricted behavior of Murasugi
sums. As one might expect, allowing for all such choices yields interesting behavior. Indeed,
we produce an ecient construction to justify the slogan Any knot is the Murasugi sum of
any two knots. We then study the behavior of Murasugi sums when restricting the size of
the m-gon. In particular, we obtain an obstruction to forming a knot as the Murasugi sum
of two knots. In Chapter 3, we lay the groundwork necessary to understand pseudo-Anosov
monodromies of bered knots in terms of invariant train tracks obtained via the Bestvina-
Handel algorithm [2]. This will allow us, in Chapter 4, to use work of Cotton-Clay [5] to
calculate the rank of the symplectic Floer homology HF∗ of the monodromies of a certain
innite family of bered knots which we denote K(j, |βB | − j). A knot in this family is
the closure of a particular homogeneous 3-string braid, and such a knot can be viewed as
a Murasugi sum of a pair of closures of positive 2-string braids. One consequence of our
calculation is that there are many ways to Murasugi sum such a pair of closures of 2-string
braids which all yield the same rank of HF∗ .
1.1 Knot theory background
Figure 1.2. A positive crossing (left), and a negative crossing (right).
The objects this thesis is concerned with are Seifert surfaces having knot boundaries. In
this section, we recall some classical constructions and facts from knot theory. A link L of |L|
components is an embedding of |L| disjoint oriented circles in the 3-sphere S3 such that the
image can be written as a nite union of line segments. A knot K is a link of one component.
In a given diagram of L, we allocate to each crossing either +1 (a positive crossing) or −1
(a negative crossing) according to the convention indicated in Figure 1.2. While the sum of
2
all of these signs in some diagram of L fails to be an invariant of L, we have the following
invariant of two-component links.
Denition 1.1.1. For a two-component link L with components L1 , L2 , the linking number
lk(L1 , L2 ) of L1 and L2 is half the sum of the signs, in a diagram for L, of the crossings at
which one strand is from L1 and the other is from L2 .
The linking number has led to a wealth of calculable algebraic knot invariants, in part
through its connection to Seifert surfaces. A Seifert surface F for a link L is an oriented,
connected, compact surface with ∂F = L. Such surfaces always exist by Seifert's algorithm
[30], see Figure 1.3, but they are never unique. Given a Seifert surface F for a knot K of
genus g = g(F ), choose a representative basis of closed curves α1 , . . . α2g for H1 (F ; Z) ∼= Z2g .
We form the Seifert matrix S as Si,j = lk(αi , αj+ ), where 1 ≤ i, j ≤ 2g and αj+ denotes a
small positive push-o of αj . Up to the notion of so-called S -equivalence, the Seifert matrix
is an invariant of K. A coarse knot invariant is given by the signature σ(K) of K, which is
the signature of (S + S T ), i.e. the sum of the signs along the diagonal of a diagonalization
of (S + S T ). An often more rened invariant is the (symmetrized) Alexander polynomial
∆K (t) of K, which is the determinant of t−g (S −tS T ) up to multiplication by t±n . Even more
rened is the knot Floer homology HFK(K)
[ of K, which categories ∆K (t). This means
that χ(HFK(K))
[ = ∆K (t) up to multiplication by t±n , where χ denotes the graded Euler
characteristic.
Figure 1.3. Forming a Seifert surface for a trefoil knot.
More geometrically, the genus g(K) of a knot K is the minimum of the genera of all Seifert
surfaces for K. While computing g(K) from its denition alone is essentially impossible, a
variety of invariants have been developed to calculate or bound g(K), one such being HFK
[.
3
A minimal genus Seifert surface for a link is not usually unique, but it is unique (up to
isotopy) if our link is bered [9, 25, 39].
Denition 1.1.2. A link is called bered if there is a bration π : S3 − L → S1 with ber
a Seifert surface F for L, the so-called ber surface of L.
For L bered with ber surface F, the complement of a neighborhood of L in S3 is
dieomorphic to the mapping torus Mϕ of a dieomorphism ϕ : F → F, i.e.
F × [0, 1]
S 3 \ ν(L) ∼
= Mϕ := .
(x, 0) ∼ (ϕ(x), 1)
The isotopy class represented by ϕ is called the monodromy of L, and the monodromy is
well-dened up to conjugation by a dieomorphism of F.
We are particularly interested in knots which are Murasugi sums of Hopf bands, as these
knots have monodromies which are usually easy to describe. We begin by noting that the
monodromy of the positive Hopf band is a right Dehn twist about its core curve, and the
monodromy of a negative Hopf band is a left Dehn twist about its core curve. See Figure
1.4. The behavior of monodromies under Murasugi sums is given as follows in [12].
Figure 1.4. The positive Hopf band (left), and the negative Hopf band (right).
Theorem 1.1.1. Suppose that F is a Murasugi sum of F1 and F2 , where ∂F = L, ∂Fi = Li
for i = 1, 2, and Li is a bered link with monodromy ϕi such that ϕi |Li = id. Then L is a
bered link with ber surface F and monodromy ϕ = ϕ′2 ◦ϕ′1 , where ϕ′i |Fi = ϕi and ϕ′i |F −Fi = id
for i = 1, 2.
The order of composition is determined by applying the monodromy of the upper surface
F1 rst, and then the monodromy of the lower surface F2 . See the discussion following
Denition 2.2.1.
4
CHAPTER 2
THE MURASUGI GRAPH OF KNOTS
2.1 Introduction
One development in knot theory is to dene and study the structure of the topological
space S composed of isotopy classes of knots. In [20], the Gordian Complex G of knots was
dened as follows: The vertex set of G consists of isotopy classes of knots, and a set of n+1
vertices K0 , . . . , K n spans an n-simplex if and only if any pair of knots in it can be changed
into each other by a single crossing change. Since then, many studies have been done by
replacing the crossing change with other local operations on knots and on virtual knots as
in [16, 21, 23, 36, 47].
In the following, we consider the operation of Murasugi sum along Seifert surfaces of links.
An oriented, embedded surface F without a closed component is called a Seifert surface for
an oriented link L if its boundary ∂F coincides with the oriented link L. An m-Murasugi
sum is an operation to glue two Seifert surfaces F1 and F2 along an m-gon with m even (for
the precise denition, see Denition 2.2.1).
We may regard a xed knot K as an operation on the space of knots as follows. A knot
K1 is changed to a knot K2 via K if a Seifert surface for K2 is obtained from a Seifert surface
F1 for K1 by Murasugi-summing a Seifert surface F for K to F1 . Thus for each xed K,
the space of knots S has the structure of directed graph, where an edge is an arrow or a
double-headed arrow.
Denition 2.1.1. For a knot K, the Murasugi sum graph of knots MSG(K) is a directed
graph such that (1) the vertex set consists of all isotopy classes of knots, (2) two vertices
K1 and K2 are connected by an edge with arrowhead on K2 if there exist Seifert surfaces
F, F1 , F2 for K, K1 , K2 such that F2 is a Murasugi sum of F and F1 . The restricted Murasugi
sum graph MSG(K, n) is considered by only allowing Murasugi sums along m-gons with
m ≤ n.
For n = 2, a 2-Murasugi sum is the connected sum operation. Hence for any knot K,
5
MSG(K, 2) has an obvious structure where the edges are arrows from each knot K′ to the
connected sum of K and K ′. For n = 4, a 4-Murasugi sum is called a plumbing. It was
shown in [11, 34, 42] that nice geometric properties of knots and surfaces (such as beredness
and genus-minimality) were preserved under Murasugi sums and decompositions of minimal
genus Seifert surfaces. On the other hand, Thompson [43] gave examples where the trefoil
is obtained as a plumbing of two unknots, and the unknot is obtained as a plumbing of two
gure-eight knots. Thus, expectations to generalize preservation results to Murasugi sums
of non-minimal Seifert surfaces were negated.
In this chapter, we generalize Thompson's examples to show that given any three knots,
we can produce one of them as a Murasugi sum of the other two.
Theorem A1. For any three knots K1 , K2 , K3 , there exist Seifert surfaces F1 , F2 , F3 for
them such that F3 is a Murasugi sum of F1 and F2 .
Therefore we have the following:
Corollary. For a knot K , any set of knots {K1 , K2 , . . . , Kn } in MSG(K) composes a com-
plete graph where all the edges are bi-directed.
We rene the result of Theorem A1 by giving an algorithm to nd a closed braid for
K3 which naturally splits into closed braids for K1 and K2 . See Figures 2.7, 2.8 following
Example 2.2.6.
Theorem A2. For any three knots K1 , K2 , K3 , there are braids b1 , b2 , b3 such that Ki is the
closure of bi (i = 1, 2, 3), satisfying the following:
If the braid b3 is expressed as a braid word W3 with generators σ1 , σ2 , . . . , σk , . . . , σn , then
W1 (resp. W2 ) is obtained from W3 by deleting the generators σ1 , . . . , σk (resp. σk+1 , . . . , σn ).
To further study the structure of MSG(K, n) where the size of Murasugi sums is limited,
we give, in Section 2.3, lower and upper bounds on the minimal m-gon required to form K3 as
a Murasugi sum of K1 and K2 . Our bounds are in terms of dcb (K, K ′ ) and dbt (K, K ′ ), which
are, respectively, the minimal number of coherent band surgeries (resp. band-twists) required
to transform K into K ′. For the precise denition of these operations, see Denitions 2.2.2
6
and 2.3.2.
Theorem B. If K3 is an m-Murasugi sum of K1 and K2 such that m is minimal, then
dcb (K1 #K2 , K3 ) + 2 ≤ m ≤ 2(dbt (K1 , K3 ) + dbt (K2 , O) + 1),
where the roles of K1 and K2 can be switched to improve the upper bound.
2.2 Any knot is a Murasugi sum of any two knots
The original construction of the Murasugi sum was rst introduced by Murasugi in [34]
and was later coined as the Murasugi sum by Gabai in [11]. For simplicity, we dene the
Murasugi sum in terms of Murasugi decomposition. See Figure 2.1.
Denition 2.2.1. Let F be a Seifert surface in S 3, let Σ be a 2-sphere such that S 3 \ Σ is a
union of two open 3-balls B1 , B2 and such that Σ∩F is an m-gon Ω with m even. Denote
F ∩ B1 = F1 and F ∩ B2 = F2 . Then we say that F decomposes into F1 and F2 along Ω.
Figure 2.1. A local picture of a 6-Murasugi sum.
If F decomposes into F1 and F2 along an m-gon, then F is said to be an m-Murasugi
sum of F1 and F2 , which we denote by F1 ⋆m F2 . Given two knots K1 , K2 , we write K1 ⋆m K2
to denote the boundary of F1 ⋆m F2 for some Seifert surfaces F1 , F2 for K1 , K2 . Note that
the summing disk Ω can initially appear stretched and twisted, but we can isotope the given
surfaces to see Ω as at. We can also isotope the surfaces so that Σ corresponds to the plane
z =0 in R3 , and in this situation, if F1 lies above (resp. below) z =0 and F2 lies below
(resp. above) z = 0, then we say that we Murasugi sum F1 onto the positive (resp. negative)
side of F2 .
7
There are several operations one can perform on knot diagrams to obtain a new knot.
One such operation is a crossing change, and more generally, an antiparallel full-twisting.
Denition 2.2.2. An antiparallel full-twisting on an oriented link is a local move where
we select a pair of locally antiparallel strings and apply some number of full twists
.
For convenience, we sometimes refer to antiparallel full-twisting as band twisting. We can
realize this twisting operation along an arc α, where α is a short, unknotted arc connecting
two antiparallel strings of a link L, which is contained within a small ball B such that L∩B
is a trivial 2-string tangle. In this setting, we can span a Seifert surface F for L such that
F ∩B is a rectangle b containing the arc α. Then the twisting operation is realized by
applying some full-twists to b.
Consider the two Seifert surfaces of the unknot in Figure 2.2. The following proposition
states that a crossing change within a knot can be realized by either plumbing or deplumbing
these surfaces. More generally, by increasing the number of full twists in R+ or R− , we can
realize any antiparallel full-twisting by either plumbing or deplumbing Seifert surfaces for
the unknot.
Figure 2.2. Two Seifert surfaces for the unknot.
Proposition 2.2.3. Let K1 , K2 be knots such that K2 is obtained by changing a positive
crossing in K1 . Then there exist Seifert surfaces F1 , F1′ for K1 , F2 , F2′ for K2 , and R+ , R−
for the unknot satisfying the following:
1. F1′ is a plumbing of F2 and R+ .
2. F2′ is a plumbing of F1 and R− .
Proof. We illustrate both statements in Figure 2.3.
8
Figure 2.3. Changing a positive crossing.
Furthermore, we can perform any number of crossing changes simultaneously via a single
Murasugi sum with a Seifert surface for the unknot, which allows us to prove the following
lemma.
Lemma 2.2.4. Any knot K has a Seifert surface F which is a Murasugi sum of two Seifert
surfaces F1 , F2 for the unknot.
Proof. For any diagram D of K, we can choose a subset C of crossings such that we obtain
the unknot by simultaneously changing the crossings in C. To see this, start at some point in
D and walk along the knot. Then can specify C to consist of those crossings which we enter
rst along an under-path and later along an over-path. Near each crossing in C, apply
a Reidemeister II move to introduce an antiparallel clasp , where undoing the clasp
results in changing that crossing as in . Thus we obtain a new diagram D′ for K,
and we obtain the unknot by simultaneously undoing all the clasps. Put a 3-ball B in the
complement of K and isotope K so that all the clasps are in B, and span a Seifert surface
F for K as in the left part of Figure 2.4. Then F decomposes into two Seifert surfaces F1
and F2 , where ∂F2 is the unknot and ∂F1 is the result of undoing all the clasps in K, and
hence is the unknot.
Conversely, any knot is obtained from an unknot by simultaneously removing antiparallel
clasps. Therefore, in the proof above, we may regard F and F2 as surfaces for the unknot
and F1 as a surface for K. This gives the following.
9
Figure 2.4. Producing ∂F = K as a Murasugi sum of unknots.
Corollary 2.2.5. Any knot K has a Seifert surface F1 which becomes a Seifert surface for
the unknot by Murasugi summing F1 with some Seifert surface for the unknot.
Proof of Theorem A1. By Corollary 2.2.5, there is a Seifert surface F1 (resp. F2 ) for K1
(resp. K2 ) that Murasugi sums with a Seifert surface F1′ (resp. F2′ ) of the unknot O to yield
a Seifert surface for O. By Lemma 2.2.4, there exist two Seifert surfaces F3 , F3′ for O that
Murasugi sum to a Seifert surface for K3 . The boundary connected sum of F1 , F2′ , F3 (resp.
F1′ , F2 , F3′ ) is a Seifert surface for K1 (resp. K2 ), and we Murasugi sum these surfaces as in
Figure 2.5 along the shaded n-gon.
Figure 2.5. K3 as a Murasugi sum of K1 and K2 .
Proof of Theorem A2. By Theorem A1, there are Seifert surfaces F1 , F2 , F3 for K1 , K2 , K3
such that F3 is a Murasugi sum of F1 and F2 and the summing disk Ω is at. Apply
a trivial twist at each band attached to Ω as depicted in Figure 2.6. Then we have a
diagram D′ such that the summing disk is spanned by a Seifert circle C. Note that the
canonical Seifert surface F3′ is a Murasugi sum of F1′ and F2′ along the summing disk Ω,
10
where ∂F1′ = ∂F1 , ∂F2′ = ∂F2 , ∂F3′ = ∂F3 . Apply Yamada's braiding algorithm [46] to D′ ,
independently inside and outside C. Then we have the desired braids.
Figure 2.6. Making a canonical surface for the braid decomposition.
∂ ∂
We use the notation F →− K to mean that ∂F = K , and F1 = F2 to mean that ∂F1 = ∂F2 .
Example 2.2.6. In Figure 2.7, we illustrate 75 as a Murasugi sum of two unknots, and in
Figure 2.8, we illustrate the unknot as a Murasugi sum of 52 and 75 . In these gures, we also
express the Murasugi sums in terms of braid decompositions, as described in Theorem A2.
Note that we simplied some procedures in the proofs of Theorems A1 and A2. In particular,
we already have a Seifert circle corresponding to the summing disk without twisting as in
Figure 6, we eliminated some Seifert circles with two bands, and we used long bands to
save us from depicting many generators.
Figure 2.7. 75 as a Murasugi sum of two unknots, and its braid decomposition.
11
Figure 2.8. The unknot as a Murasugi sum of 52 and 75 , and its braid decomposition.
2.3 Constraining the minimal m-gon
In Theorem A1, it was shown that given three knots K1 , K2 , K3 , we can form K3 as a
Murasugi sum of K1 and K2 by using a suciently large m-gon. We begin this section
with Denition 2.3.1 in order to give lower and upper bounds for the minimal such m-gon,
culminating in Theorem B from Section 2.1. In particular, if we restrict the size of our m-
gons, Theorem B obstructs forming a knot as a Murasugi sum of two given knots. Previously,
the only restriction on Murasugi sums was for plumbings of ber surfaces, such as in [33],
where Melvin and Morton showed that the Conway polynomials of bered knots of genus 2
take a restricted form when the ber surface is a plumbing of Hopf bands.
Denition 2.3.1. Given three oriented knots K1 , K2 , K3 , dene the minimal size of Mura-
12
sugi summation for K3 = K1 ⋆ K 2 to be
dM (K1 , K2 ; K3 ) = min{m | F3 = F1 ⋆m F2 , ∂Fi = Ki for i = 1, 2, 3},
where the minimum is taken over F1 , F2 .
Recall that m is even. From the denition, dM (K1 , K2 ; K3 ) = 2 if and only if K3 =
K1 #K2 . Also, if K3 ̸= K1 #K2 can be realized as a plumbing of K1 and K2 , then
dM (K1 , K2 ; K3 ) = 4. Now we recall the notion of a coherent band surgery. See Figure 2.9.
Figure 2.9. A band surgery along an unlink yielding 31 #31 .
Denition 2.3.2. Given a knot or link L and an embedded band b : I × I → S3 with
L ∩ b(I × I) = b(I × ∂I), we obtain a new link L′ = (L \ b(I × ∂I)) ∪ b(∂I × I), and we say
that L′ is obtained from L by a band surgery. For oriented L, L′ , a coherent band surgery is a
band surgery that respects the orientations of both links, that is, L\b(I ×∂I) = L′ \b(∂I ×I)
as oriented spaces.
Given two oriented links L, L′ , we denote by dcb (L, L′ ) the minimal number of coherent
band surgeries required to produce L′ from L. This number is known as the coherent band-
Gordian distance, and in the case of knots it is equal to twice the SH(3)-Gordian distance
[26]. In [27], dcb is calculated for most pairs of knots up to seven crossings. For knots K, K ′ ,
note that dcb (K, K ′ ) is necessarily even because a coherent band surgery changes the number
of components by one.
We have the following lower bounds for dM in terms of dcb and the signature σ.
Theorem 2.3.3. For knots K1 , K2 , K3 , we have
dcb (K1 #K2 , K3 ) + 2 ≤ dM (K1 , K2 ; K3 ).
Consequently,
13
dcb (K1 ⊔ K2 , K3 ) + 1 ≤ dM (K1 , K2 ; K3 ),
where K1 ⊔ K2 is a split link, and
|σ(K1 ) + σ(K2 ) − σ(K3 )| + 2 ≤ dM (K1 , K2 ; K3 ).
Proof. Suppose K3 is an m-Murasugi sum of K1 and K2 , where m = 2n is minimal. Then
there is a sequence of 2(n − 1) = m − 2 coherent band surgeries between K3 and K1 #K2 ,
where each band lies within a Seifert surface for K3 , so that
dcb (K1 #K2 , K3 ) + 2 ≤ m.
See Figure 2.10. With one more band surgery, we have m−2+1 coherent band surgeries
between K3 and the split link K 1 ⊔ K2 , so that
dcb (K1 ⊔ K2 , K3 ) + 1 ≤ m.
If an oriented link L′ is obtained from L by a coherent band surgery, an estimate was
given by Murasugi [35] on the dierence of the signatures as |σ(L) − σ(L′ )| ≤ 1. Since
σ(K1 #K2 ) = σ(K1 ) + σ(K2 ), we obtain the third inequality.
Figure 2.10. Performing (6 − 2) band surgeries to recover K1 #K2 from K3 .
Example 2.3.4. By Theorem 2.3.3, the signature obstructs forming 91 as a plumbing of
two copies of 31 . However, Figure 2.11 depicts how 91 desums into two copies of 31 along
the shaded 6-gon, which is obtained by rst merging two 4-gons into an 8-gon which is then
reduced to the 6-gon. This process is explained in the proof of Lemma 2.3.8.
14
Figure 2.11. 91 as a 6-Murasugi sum of 31 and 31 .
Remark 2.3.5. For pairs of knots where dcb has not been determined, we may apply lower
bounds for dcb from [26] in terms of the smooth four-ball genus g4 , hence in terms of σ by
[35], and in terms of the Nakanishi index e. Indeed, for knots K1 , K2 we have
|σ(K1 ) − σ(K2 )| ≤ 2g4 (K1 # − K2 ) ≤ dcb (K1 , K2 ),
where −K2 is the reverse mirror of K2 , and
|e(K1 ) − e(K2 )| ≤ dcb (K1 , K2 ).
We now move on to some upper bounds of dM . Using Lemma 2.3.7 below, we can easily
modify the proof of Theorem A1 to form K3 as a 4(u(K1 ) + u(K2 ) + u(K3 ))-Murasugi sum
of K1 and K2 , where u(K) is the unknotting number of K. This gives the upper bound
dM (K1 , K2 ; K) ≤ 4(u(K1 ) + u(K2 ) + u(K3 )).
In what follows, we improve this upper bound into Theorem 2.3.9. Recall that the
Gordian distance dG between two knots K, K ′ is the minimal number of crossing change
operations required to transform a diagram for K into a diagram for K ′ , where the minimum
is taken over all diagrams for K. Since K can be transformed into K′ by a sequence of
crossing changes that passes through the unknot, we have dG (K, K ′ ) ≤ u(K) + u(K ′ ). More
generally, we consider the band-twist distance dbt .
15
Denition 2.3.6. Dene the band-twist distance between two oriented links L, L′ , denoted
by dbt (L, L′ ), as the minimal n such that there exists a sequence of links L = L0 , L1 , L2 , . . . , Ln =
L′ , where Li+1 is obtained from Li by an antiparallel full-twisting (a band-twist) as in De-
nition 2.2.2.
Since any crossing change may be realized as a band-twist, we have dbt (K, K ′ ) ≤ dG (K, K ′ ).
Just as with crossing changes, we may perform any number of band-twist operations simul-
taneously.
Lemma 2.3.7. For two links L and L′ with dbt (L, L′ ) = n, there exists a Seifert surface F
for L′ with a set A of n mutually disjoint properly embedded arcs such that a Seifert surface
F for L is obtained by applying a band-twist operation along each arc in A.
Proof. Let L = L0 , L1 , L2 , . . . , Ln = L′ be a sequence of links related by the band-twisting
operation. After obtaining L1 from L0 by twisting along an arc α0 , instead of erasing α0 ,
isotope it so that it is disjoint from the arc α1 used to obtain L2 . Repeating this, we obtain
a set of arcs α0 , α1 , . . . , αn−1 attached to Ln = L′ . By an isotopy, we can arrange the arcs
to be short and contained in a ball B with L′ ∩ B being a trivial 2n-string tangle. Splice L′
along the arcs and push the resulting link L′′ slightly o B. Span a Seifert surface for L′′
disjoint from B. Then we obtain the desired Seifert surface for L′ by attaching bands to L′′
that pass through B.
If we wish to perform several Murasugi sums of several surfaces with a single surface,
we can often combine these sums into a single sum as indicated by the following lemma.
One implication of the below is that we may perform any number of band-twist operations
simultaneously via a single Murasugi sum.
Lemma 2.3.8. Suppose that a Seifert surface F ′ is obtained by Murasugi summing S1 , S2 , . . . , Sn
on the same side of a connected Seifert surface F along mutually disjoint summing disks
Ω1 , Ω2 , . . . , Ωn . Let Ωi be an ei -gon (i = 1, 2, . . . , n). Then F ′ is a Murasugi sum of F with
a boundary connected sum of S1 , S2 , . . . , Sn along an m-gon, where m = ni=1 ei . Moreover,
P
16
if F is a Seifert surface for a knot, then we can merge the sums into an m′ -gonal sum with
m′ = m − 2(n − 1).
Proof. We may assume that each summand Si is contained in a thin blister neighborhood
of the summing disk Ωi . Denote the edges in ∂Ωi as ai,1 , bi,1 , ai,2 , bi,2 , . . . for i = 1, 2, . . . , n,
where the ai,· 's are sub-arcs of ∂F and the bi,· 's are properly embedded arcs in F . Since F is
connected, we may nd an embedded arc γ in F whose endpoints are the midpoints of b1,j and
b2,k for some j and k as in Figure 2.12. We merge the two (dark shaded) summing disks Ω1 and
Ω2 into an (e1 +e2 )-gon Ω′ whose boundary consists of ((∂Ω1 ∪ ∂Ω2 ) \ (b1,j ∪ b2,k ))∪(γ1 ∪γ2 ),
where γ1 and γ2 are properly embedded in F and b1,j ∪ γ1 ∪ b2,k ∪ γ2 is a rectangle R = γ ×I
Sn
such that R ∩ int i=1 Ωi =∅. We see that the boundary connected sum of S1 and S2 is
contained in a thin blister neighborhood of the new summing disk Ω′ . By repeating this
merging operation, we eventually combine all the summing disks into one and obtain the
desired Murasugi sum.
For the last part of the assertion, the assumption that ∂F being connected ensures the
existence of two arcs a1,p , a2,q for some p, q such that one segment of ∂F between a1,p and a2,q
does not pass through other summing disks. Then we can apply the previously mentioned
merging of Ω1 and Ω2 so that γ1 or γ2 , say γ1 , can be isotoped to a sub-arc of ∂F in
Sn
F\ i=1 Ωi . Then the three consecutive edges in Ω′ , say, a1,k , γ1 , a2,j for some k, j, can be
regarded as one edge by merging the bands of S1 and S2 attached to a1,k and a2,j as in Figure
2.13. Then the new summing disk Ω′′ is a p-gon, where p = e1 + e2 − 2.
Figure 2.12. Merging disks Ω1 , Ω2 in F along γ.
Combining Lemma 2.3.7 and Lemma 2.3.8, we arrive at the following improvement on
our upper bound.
17
Figure 2.13. Merging two 6-gons into a (6 + 6 − 2)-gon.
Theorem 2.3.9. Let K1 , K2 , K3 be knots. Then
dM (K1 , K2 ; K3 ) ≤ 2(dbt (K1 , K3 ) + dbt (K2 , O) + 1),
where the roles of K1 and K2 may be switched to improve the upper bound.
Proof. Suppose that dbt (K1 , K3 ) = p and dbt (K2 , O) = q . As guaranteed by Lemma 2.3.7,
there is a Seifert surface F1 for K1 (resp. F2 for K2 ) with a collection A of arcs α1 , . . . , αp
(resp. B of arcs β1 , . . . , βq ) such that performing band-twist operations along the arcs yields a
Seifert surface F3 for K3 (resp. F0 for O). Prepare Seifert surfaces A1 , . . . , A p and B1 , . . . , Bq
such that plumbing Aj along αj (resp. Bk along βk ) results in applying the band-twist
operation along αj (resp. βk ). More precisely, each of the surfaces is a plumbing of a trivial
annulus and an unknotted annulus with various numbers of full-twists (recall Figure 2).
Construct F1′ from F1 such that ∂F1′ = K3 by plumbing A1 , . . . , Ap along A on the positive
side of F1 . Also, construct F0′ from F0 such that ∂F0′ = O by plumbing B1 , . . . , Bq along B
on the negative side of F0 .
We merge the plumbed surfaces A1 , . . . Ap so that F1′ is a Murasugi sum of F1 and A,
where A is a boundary connected sum of A1 , . . . , Ap . Note that ∂A is an unknot O1 . By
Lemma 2.3.8, we may regard the Murasugi sum as along a (4p − 2(p − 1))-gon and hence a
(2p + 2)-gon. Similarly, F0′ is regarded as a (2q + 2)-Murasugi sum of F0 and a Seifert surface
B for an unknot O2 . Denote by F the 2-Murasugi sum (i.e., a boundary connected sum) of
F1′ and F0′ , where F0′ is summed on the positive side of F1′ .
Then F is a (2p + 2 + 2q + 2)-Murasugi sum of two summands, where one summand
is the boundary connected sum of F1 and B, and the other summand is the boundary
18
connected sum of F0 and A. The boundary of the rst (resp. second) summand is K1 #O2
(resp. K2 #O1 ). By Lemma 2.3.8, we can reduce the summing (2p + 2q + 4)-gon into a
(2p + 2q + 2)-gon. Therefore, we have expressed K3 as a 2(p + q + 1)-Murasugi sum of K1
and K2 .
By combining Theorems 2.3.3 and 2.3.9, we arrive at Theorem B. As an application of
Theorem B, we determine dM (31 , 31 ; K) for knots up to ve crossings. Theorem B shows
that
dM (31 , 31 ; 31 ) = 4, dM (31 , 31 ; O) = dM (31 , 31 ; 41 ) = 6,
while it only gives the bounds
4 ≤ dM (31 , 31 ; 51 ), dM (31 , 31 ; 52 ) ≤ 6.
We can show that dM (31 , 31 ; 51 ) = dM (31 , 31 ; 52 ) = 4 in the following way.
For a1 , a2 , . . . , an ∈ 2Z, denote by S[a1 , a2 , . . . , an ] a linear plumbing of n unknotted
annuli, where the ith annulus has ai half-twists. Note that all 2-bridge links have such a linear
plumbing as a Seifert surface [17], which is of minimal genus if and only if a1 a2 · · · an ̸= 0.
Using the notation of Example 2.2.6, we have the following:
1. S[a1 , a2 , . . . , an ] ⋆4 S[b1 , b2 , . . . , bm ] = S[a1 , . . . , an , b1 , . . . , bm ],
∂
2. S[a1 , a2 , . . . , an ] = S[a1 , a2 , . . . , an , an+1 , 0] for any an+1 ,
∂
3. S[a1 , a2 , . . . , ai , 0, ai+2 , . . . , an ] = S[a1 , a2 , . . . , ai + ai+2 , . . . , an ].
Using the above, we have:
Example 2.3.10. 1. 31 ⋆4 31 ←
∂
− S[2, 2] ⋆4 S[2, 2] = S[2, 2, 2, 2] →
∂
− 51
∂ ∂ ∂ ∂
2. 31 ⋆4 31 ← − S[2, 2, −2, 0] ⋆4 S[2, 2] = S[2, 2, 0, 2] = S[2, 4] → − 52 .
Using this notation, we summarize what Thompson showed in [43] as follows:
∂ ∂ ∂
1. O ⋆4 O ← − S[2, 0] ⋆4 S[0, 2] = S[2, 0, 0, 2] = S[2 + 0, 2] = S[2, 2] → − 31
∂ ∂ ∂ ∂
2. 41 ⋆4 41 ← − S[2, −2, 2, 0] ⋆4 S[−2, 2] = S[2, −2, 0, 2] = S[2, 0] → − O.
19
CHAPTER 3
PSEUDO-ANOSOV MAPS AND THE BESTVINA-HANDEL
ALGORITHM
3.1 The classication of surface homeomorphisms
A celebrated result in low-dimensional topology is the classication of surfaces. This result
states that two orientable surfaces are dieomorphic if and only if they have the same number
of boundary components b, the same genus g, and the same number of punctures n. Denote
the dieomorphism type of such a surface by F = Fb,g,n . Then the Euler characteristic of F
is χ(Fb,g,n ) = 2 − 2g − b − n.
A natural object of study is the set of all self-dieomorphisms of F. Let Diffeo+ (F, ∂F )
denote the group of orientation-preserving dieomorphisms of F which restrict to the identity
on ∂F (if ∂F ̸= ∅). After endowing Diffeo+ (F, ∂F ) with the compact-open topology, we
dene the mapping class group of F as
MCG(F ) := π0 (Diffeo+ (F, ∂F )).
In other words, MCG(F ) is the group of smooth isotopy class of elements of Diffeo+ (F, ∂F ),
where isotopies are required to x the boundary pointwise. Such a dieomorphism can be
represented as a composition of Dehn twists about some collection of essential simple closed
surves in F [29].
Denition 3.1.1. A right Dehn twist about a simple closed curve A ⊂ F, denoted
R
DA ,
is dened as follows. Let X = S 1 × [0, 1] be the annulus with orientation induced by
R2 , and identify a regular neighborhood ν(A) of A with X via an orientation-preserving
dieomorphism ϕ : X → ν(A). After dening the twist map T :X→X as
T (θ, t) = (θ − 2πt, t),
R
we dene the homeomorphism DA :F →F to be the identity outside of ν(A), and to be
the composition ϕ ◦ T ◦ ϕ−1 on ν(A). A left Dehn twist about A, denoted
L
DA , is dened
similarly, where we instead consider the twist map T (θ, t) = (θ + 2πt, t). See Figure 3.1.
20
Figure 3.1. The action of a right Dehn twist about A on a line segment.
Elements of the mapping class group are classied according to the following theorem of
Thurston [45]. Though we state this theorem for compact surfaces, a similar classication
holds for surfaces with punctures [2].
Theorem 3.1.2. A dieomorphism ϕ of a compact oriented surface F is isotopic to a dif-
feomorphism ϕ′ such that either
1. ϕ′ is periodic, i.e. there exists some positive integer m such that (ϕ′ )m = id; or
2. ϕ′ is pseudo-Anosov, i.e. there is a number λ > 1 and a pair of transverse measured
foliations (Fs , µs ), (Fu , µu ) such that ϕ′ (Fs , µs ) = (Fs , λ1 µs ) and ϕ′ (Fu , µu ) = (Fu , λµu );
or
3. ϕ′ is reducible, i.e. there is a collection of disjoint simple closed curves C in the interior
of F that is xed by ϕ.
Thus, we can isotope the monodromy ϕ of the ber surface F of a bered knot K until
it is either periodic, pseudo-Anosov, or reducible. We are particularly interested in bered
knots which have pseudo-Anosov monodromy, so-called hyperbolic knots. This class of knots
consists of all knots which are not torus knots or satellite knots [44]. Conjecturally, most
prime knots are hyperbolic knots, in the sense that the ratio of hyperbolic knots to prime
knots approaches nearly 1 as the number of crossings goes to innity [31].
While we called pseudo-Anosov maps dieomorphisms in Theorem 3.1.2, it is more ac-
curate to say that they are dieomorphisms away from the nite set of singularities of the
associated pair of measured foliations Fs , Fu . Such singularities in the interior of F are
locally modelled on p-prongs, where p ≥ 3. At a singularity which is a puncture of F, the
21
same p-pronged model applies, but we now allow p ≥ 1. From a puncture, we can recover a
boundary component by blowing up the puncture, e.g. as in §2.1 of [24]. See Figure 3.2. On
a component of ∂F , singularities of Fs and Fu alternate, such that the boundary component
is a cycle of leaves, and ϕ is the identity on the boundary.
Figure 3.2. A 3-pronged interior singularity (left), a 2-pronged puncture (middle), and a
blow-up of a 2-pronged puncture into a boundary component (right).
3.2 The Bestvina-Handel algorithm
While it has been established that a surface dieomorphism can be isotoped into one of the
classes of Thurston, it can be dicult to practically identify the particular class to which
a given map belongs. To rectify this, Bestvina-Handel provide an algorithm to isotope a
surface dieomorphism into its Thurston class [2]. In what follows, we record only the
parts of the algorithm which pertain to representatives of pseudo-Anosov maps on surfaces
with one puncture. The algorithm as applied to such maps produces a so-called invariant
train-track which combinatorially encodes the dynamics of the canonical pseudo-Anosov
representative.
We begin with a spine of a punctured ber surface F having monodromy ϕ, where the
spine is viewed as a graph g with vertices and arbitrarily oriented edges. For an oriented
edge e of this graph, we denote by e the same edge with opposite orientation.
Denition 3.2.1. The collection of oriented edges emanating from a vertex v is called the
link of v.
Continuing, we represent ϕ as some composition of Dehn twists within a regular neigh-
borhood of the spine. After applying this composition to the edges of g , we realize the image
22
of each edge as a connected edge-path which travels through the edges and vertices of g. By
abuse of notation, we refer to the induced graph map as ϕ.
Denition 3.2.2. Viewing ϕ as a graph map, we dene the derivative D of ϕ at an edge e
to be the rst edge in the edge-path ϕ(e). The derivative D partitions the link of a vertex
v into equivalence classes called gates, where two edges e1 , e2 emanating from v are in the
same gate if Dm (e1 ) = Dm (e2 ) for some positive integer m.
Our initial edge-path calculation usually produces some redundancy which prevents the
initial representative of ϕ from being pseudo-Anosov. This redundancy results from iterates
of edge-paths which produce sub-paths of the form ee, which we call back-tracking. However,
the Bestvina-Handel algorithm says that we can perform a sequence of isotopies to remove
such back-tracking. More precisely, the algorithm produces an ecient graph map.
Denition 3.2.3. The graph map ϕ:g→g is called ecient if g has no vertices of valence
1 or 2. Additionally, every iterate of ϕ sends each edge to an edge-path which does not
back-track. The second condition is equivalent to saying that for every edge e, the edge-path
ϕ(e) contains no back-tracking, and ϕ(e) enters and exits any given vertex at dierent gates.
From an ecient graph map ϕ, we can construct an associated pair of transverse measured
foliations, thus realizing ϕ as pseudo-Anosov. The six isotopies required to make a graph
map ecient, called moves, are described combinatorially as follows. See Figure 3.3 for an
example of how certain moves aect a graph. By abuse of notation, we denote the result of
isotopies of ϕ as ϕ.
Figure 3.3. From left to right: starting with a graph, we perform a valence 1 isotopy, then
a valence 2 isotopy, and then a fold.
23
Move 1 (Pulling tight)
If we encounter an edge-path containing a back-tracking subpath ee, we remove this subpath.
This has no eect on the graph.
Move 2 (Collapsing an invariant forest)
Suppose we encounter an invariant forest of g , i.e. a subgraph h of g such that ϕ(h) ⊆ h and
each component of h is contractible. Then we collapse each component of h to a vertex, and
we remove all occurrences of edges in the invariant forest from all edge-paths.
Move 3 (Valence 1 isotopy)
Suppose we encounter a valence 1 vertex. Then we remove this vertex and the emanating
edge, as well as removing all occurrences of the emanating edge from all edge-paths.
Move 4 (Valence 2 isotopy)
Suppose we encounter a valence 2 vertex with emanating edges e1 , e2 . Then we remove
this vertex and amalgamate e1 , e2 into an edge labeled, say, e′1 . The edge-path ϕ(e′1 ) is the
concatenation ϕ(e2 ) · ϕ(e1 ). Furthermore, we delete e2 , and we replace e1 with e′1 in all
edge-paths.
Move 5 (Folding)
Suppose we encounter a pairwise adjacent collection of emanating edges e1 , . . . , e m in the
link of v having common starting image the edge-path P. Then we identify (fold) the
starting portion of this collection of edges to a single edge f with ϕ(f ) = P , and we relabel
e1 = e′1 , . . . , em = e′m . Furthermore, we replace all occurences of e′1 , . . . , e′m in all edge-paths
with f e′1 , . . . , f e′m . We call the operation of a folding which involves the start of e: folding
the end of e.
Move 6 (Subdivision)
Suppose we encounter an edge e with interior point v mapping to a vertex of g. Then we
can subdivide e = e1 e2 , where e1 , e2 emanate from the valence 2 vertex v. As a result, we
replace all occurences of e with e1 e2 , and we have ϕ(e) = ϕ(e1 ) · ϕ(e2 ).
24
The order and way in which we perform these six moves is sometimes governed by the
following matrix.
Denition 3.2.4. For a graph g with edges e1 , . . . , em , the transition matrix of a graph map
ϕ:g→g is the matrix M whose entry mij is the number of times that ej and ej appear
in the edge-path ϕ(ei ). In other words, M records how many times ei runs over ej under ϕ,
regardless of orientation.
For ϕ to be in a pseudo-Anosov class, it is necessary that M is Perron-Frobenius and
hence has a unique positive eigenvector. The ith entry of this eigenvector corresponds to
a width w assigned to the edge ei . Similarly, the transpose MT is Perron-Frobenius and
hence has a unique positive eigenvector. The ith entry of this eigenvector corresponds to a
length l assigned to the edge ei . Furthermore, the positive eigenvalue associated to either
such eigenvector equals the spectral radius of M (and of M T ).
3.2.1 Producing an ecient graph map
As previously mentioned, we begin with a spine of a punctured ber surface F having
monodromy ϕ, where the spine is viewed as a graph g with vertices and arbitrarily oriented
edges. We represent ϕ as some composition of Dehn twists within a regular neighborhood
of the spine and apply this composition to the edges of g. Viewing ϕ as a graph map, we
perform the following sequence of steps involving Moves 1-6.
Step 1
Pull ϕ tight wherever possible.
Step 2
If ϕ has a nontrivial invariant forest, collapse it. If this results in back-tracking, perform
Step 1 again.
Step 3
Remove all valence 1 vertices by performing valence 1 isotopies. If this results in back-
tracking, perform Step 1 again. If this results in a nontrivial invariant forest, perform Step
25
2 again.
Step 4
Remove all valence 2 vertices by performing valence 2 isotopies. When performing a given
valence 2 isotopy in this step, we delete the incident edge having larger width w as determined
by M. If both edges have the same width, we can delete either edge. If this results in back-
tracking, perform Step 1 again. If this results in a nontrivial invariant forest, perform Step
2 again.
Step 5
At this point, if ϕ is ecient, the algorithm stops. Otherwise, there exists some edge e and
some positive integer m such that ϕm (e) backtracks. Choose m to be minimal. If m = 2,
so that e 7→ e1 e2 7→ · · · f f · · · , we fold to e′ all edges in the link of {e1 , e2 } having common
starting image the edge-path P, where ϕ(e′ ) = P . This operations allows us to pull ϕ(e)
tight, thus removing the back-tracking resulting from the edge-path e1 e2 . After performing
this fold, we return to Step 1.
If m > 2, so that e 7→ e1 e2 7→ · · · 7→ h1 h2 7→ i1 i2 7→ j1 j2 7→ kk , we morally fold at the
link of {j1 , j2 }, then at the link of {i1 , i2 }, then at the link of {h1 , h2 }, and so on until we
fold at the link of {e1 , e2 }. However, this sequence of folds may prevent us from pulling ϕ(e)
tight. Let p be the point in the interior of e which maps to the point ϕ(p) that separates
the edge-path e1 e2 . If at any step in our folding sequence, a folded edge-path from the
folded edge e starts (or ends) at ϕ(p), we subdivide e (possibly by rst subdividing the edges
appearing in the iterates of ϕ(e)). This allows us to carry on our folding sequence without
folding the point p at which some iterate of ϕ fails to be locally injective, so that we can
ultimately pull ϕ(e) tight. After performing this sequence of folds, we return to Step 1. Note
that this algorithm terminates because the pulling-tight portion of Step 6 always reduces
the eigenvalue λ of M, and no other steps increase λ, until we have arrived at the dilatation
λmin of ϕ.
26
3.2.2 Producing an invariant train track from an ecient graph map
After applying the algorithm described in the previous section to the spine graph g of a
punctured ber surface F having monodromy ϕ, we have an ecient graph map ϕ : g → g.
We now describe the local behavior of this graph map near vertices. At a given vertex v,
replace v with a small circle C having points on its boundary corresponding to the gates
of the link of v. Outside of C, an edge e of g in the link of v meets a point on the circle
corresponding to the gate of e, such that e is orthogonal to C. These are the so-called
real edges of our train track. Within C, we connect two points p1 , p2 on C corresponding
to distinct gates with an innitesimal edge f if there is some real edge e of g and some
positive integer k such that ϕk (e) enters C via p1 and exits C via p2 . Additionally, we require
that f is orthogonal to C. See Figure 3.5. By inserting innitesimal edges at each vertex of
g, we obtain an invariant train track τ for ϕ.
3.2.3 Producing a foliation from an invariant train track
Given an invariant train track τ for ϕ, we form a Markov partition of a regular neighborhood
of τ as follows. Cover each edge e (both real and innitesimal) by a rectangle of width w(e)
and length l(e) as determined by the unique positive eigenvectors of, respectively, M and
MT . We do so in such a way that each rectangle meets each circle C in an arc of C, and
such that rectangles covering adjacent edges within the same gate intersect at a single point.
Rectangles have a canonical pair of transverse measured foliations consisting of vertical strips
and horizontal strips, so our Markov partition yields a foliation of a regular neighborhood of
τ.
We extend this foliation to the components of F −τ as follows. Given a k -sided component
X having cusp vertices, some iterate of ϕ maps each side of ∂X over itself. This yields a
xed point of some iterate of ϕ associated to each of the k sides of ∂X , and any vertex of
∂X is equidistant from the xed points on adjacent sides in terms of the length l. We can
now collapse all sides of ∂X onto a spine of X in a length-preserving fashion by identifying
all xed points to a singular point (which is removed if X is the component containing the
27
puncture). See Figure 3.4.
Figure 3.4. Extending a foliation to a face of F −τ by collapsing the face onto a spine.
Example 3.2.5. We now carry out the process described in this section to produce an
ecient graph map, an invariant train track, and a foliation for a simple pseudo-Anosov
map ϕ on the punctured torus. The initial graph and Dehn curves are pictured in Figure
3.5, where ϕ is a right Dehn twist about A, followed by a left Dehn twist about B. The
resulting graph map
DR
A DLB DR
A DLB
a 7−−→ a 7−−→ ba, b 7−−→ ba 7−−→ bba
is already ecient since we have
a 7→ ba 7→ · · · ab · · · 7→ · · · ab · · · 7→ · · ·
b 7→ bb · · · 7→ · · · ab · · · 7→ · · · ab · · · 7→ · · ·
We now construct the invariant train track τ. Replacing the vertex with a small circle C,
we partition the real edges a, a, b, b into their gates at points on C. To calculate the gates,
we iterate D on the link {a, a, b, b} until the image becomes periodic. Once the image of D
on this link becomes periodic, we classify two edges as being in the same gate if their image
under this periodic iterate of D is the same, and as being in distinct gates otherwise. We
calculate
D
{a, a, b, b} 7− → {b, a, b, a} 7→ {b, a, b, a}
From this, we conclude that a, b are in the same gate, and a, b are in the same gate. Each
of these two gates corresponds to a cusp. Now we insert the innitesimal edges within C.
28
Figure 3.5. Our starting spine and Dehn curves (left), and adding the innitesimal edge
(right).
The edge-path a 7→ ba connects the distinct gates {b, a}, so we insert an innitesimal edge
e connecting these two gates. Now our graph map is written as a 7→ bea, b 7→ bebea, e 7→ e.
The result is an invariant train track where the component containing the puncture is a disk
with two cusps.
We now construct the pair of transverse foliations. The transition matrix is
1 1 0
M = 1 2 0
1 2 1
√ √ √ √
3− 5 5−1
which has unique positive eigenvector
2
, 2
, 1 , so that w(a) = 3−2 5 , w(b) = 5−1 2
,
√
5−1
w(e) = 1. The transpose MT has unique positive eigenvector
2
, 1, 0 , so that l(a) =
√
5−1
2
, l(b) = 1, l(e) = 0. We cover each edge with a rectangle having length and width
corresponding to that edge's l, w weight. See Figure 3.6 for the Markov partition and its
image under ϕ.
To obtain the associated unstable foliation, we foliate the rectangles of the Markov parti-
tion, and then we collapse the 2-gon containing the puncture onto a spine of the 2-gon. See
Figure 3.7.
29
Figure 3.6. The initial Markov partition (left), and its image under ϕ (right).
Figure 3.7. The unstable foliation of a regular neighborhood of τ (left), and the extension
of this foliation to F (right).
30
CHAPTER 4
SYMPLECTIC FLOER HOMOLOGY
4.1 Cotton-Clay's calculation of symplectic Floer homology
In [5], Cotton-Clay calculated the symplectic Floer homology HF∗ of mapping classes of
pseudo-Anosov maps of compact surfaces. For such a mapping class ϕ, the rank of HF∗
is simply a count of the (nondegenerate) xed points of a symplectic representative of ϕ.
Consequently, Cotton-Clay was able to give a combinatorial formula for the rank of HF∗
from an invariant train track for a singular representative of ϕ. However, this formula is
stated for closed surfaces, while our present application requires such a formula for surfaces
with one boundary component. At the end of this section, we give such a formula based on
Cotton-Clay's work.
We now outline the basic construction of HF∗ , which is also known as xed point Floer
homology. For more details, see [5, 7, 10, 32]. Throughout the rest of this section, we consider
a compact symplectic surface (F, ω) of negative Euler characteristic and an area-preserving
symplectomorphism ϕ:F →F with nondegenerate xed points.
Denition 4.1.1. A xed point x is nondegenerate if the linear dierential dϕx does not
have 1 as an eigenvalue. A map ϕ is nondegenerate if all of its xed points are nondegenerate.
In particular, such a ϕ has a nite xed point set, denoted Fix(ϕ), consisting of isolated
xed points. After enforcing certain technical conditions on ϕ and ω, we can take the
homology of the chain complex
CF∗ (ϕ) := Z2 ⟨x | x ∈ Fix(ϕ)⟩,
where the dierential ∂ counts mod 2 the ow lines from x to y in a suitable moduli space.
More precisely, let J = {Jt }t∈R be a smooth path of ω -compatible almost-complex structures
on F such that Jt+1 = ϕ∗ Jt . We dene the moduli space M1 (x, y, J) as the set of maps
u : R2 → F such that
1. u(s, t) = ϕ(u(s, t + 1)),
2. lims→+∞ u(s, ·) = x, lims→−∞ u(s, ·) = y,
31
3. ∂s u + Jt (u)∂t u = 0,
4. The Maslov index satises µ(u) = 1.
We call such a u an index one ow line from x to y. After perturbing J to be generic, we
realize M1 (x, y, J) as a smooth 1-dimensional manifold. The quotient of M1 (x, y, J) by the
free R-action of translation in the s-variable yields a compact 0-dimensional manifold, so we
can dene
X M1 (x, y, J)
∂x := mod 2 y.
R
y∈Fix(ϕ)
In Proposition 3.1.1 of [5], it is shown that HF∗ (ϕ) is well-dened for a nondegenerate
symplectomorphism ϕ in a pseudo-Anosov mapping class h. In particular, HF∗ (ϕ) = HF∗ (h)
is independent of choice of ω, J , and ϕ.
There are two necessary conditions for a ow line to exist between two xed points x, y .
The rst condition is that x, y belong to the same Nielsen class.
Denition 4.1.2. Two xed points x, y of ϕ are in the same Nielsen class if there exists a
path γ : [0, 1] → F with γ(0) = x, γ(1) = y such that ϕ(γ) is homotopic (rel. endpoints) to
γ.
Indeed, given a ow line u(s, t) from x to y, let γt : [0, 1] → F be a reparametrization of
u(·, t) (for each t). Then γ1 is the desired Nielsen equivalence of x and y as described in [5].
The second condition is that the Lefschetz signs of x, y are dierent.
Denition 4.1.3. The Lefschetz sign of a nondegenerate xed point x is the quantity dened
by (−1)ε(x) = sign(det(id − dϕx )) ∈ {±1}.
Indeed, this condition follows since we require that the Maslov index satises µ(u) = 1,
and we also have µ(u) = ε(x) − ε(y) mod 2 as in [10, 22]. Since the linear map dϕx is
symplectic, its two eigenvalues are either real with product 1, or imaginary with product 1.
Thus, we can partition Fix(ϕ) into the following three classes.
Denition 4.1.4. A nondegenerate xed point x is called elliptic if the eigenvalues of dϕx
lie on the unit circle. A nondegenerate xed point x is called positive hyperbolic if the
32
eigenvalues of dϕx are real and positive. A nondegenerate xed point x is called negative
hyperbolic if the eigenvalues of dϕx are real and negative.
Note that the Lefschetz sign of elliptic and negative hyperbolic xed points is 1, and
the Lefschetz sign of positive hyperbolic xed points is −1. In particular, there can only
be nontrivial ow lines between elliptic xed points and positive hyperbolic xed points, or
between positive and negative hyperbolic xed points.
4.2 Combinatorial formulae for HF∗
A (singular) pseudo-Anosov map ϕ : F → F is smooth away from the singularities of its
associated foliations. Furthermore, with respect to an appropriate area form on F, ϕ is
symplectic away from these singularities. In Sections 3.2 and 4.2 of [5], Cotton-Clay uses
Hamiltonian vector elds to give a symplectic smoothing of ϕ near these singularities and
near ∂F (if ∂F ̸= ∅). The result is a symplectic representative ϕsm of ϕ having nondegenerate
xed points.
In obtaining ϕsm , Cotton-Clay also determines the number and type of xed points that
result near each singularity and each puncture (i.e. each boundary component). At a rotated
interior singularity which is a p-prong, we have one elliptic xed point corresponding to the
singularity, a so-called Type IIIc xed point. Near a xed, unrotated interior singularity
which is a p-prong (p ≥ 3), we have (p − 1) positive hyperbolic xed points, which are
so-called Type IIIb-p xed points. Near a rotated puncture, we have no xed points. Near
a xed, unrotated puncture which is a p-prong (p ≥ 1), we have p positive hyperbolic xed
points, which are so-called Type IIId-p xed points.
In [3], it was shown that no two xed points of a singular pseudo-Anosov map ϕ on a
closed surface are in the same Nielsen class. Furthermore, Cotton-Clay shows that xed
points of ϕsm associated to a given interior singularity are in their own Nielsen class, and
xed points of ϕsm associated to a given puncture are in their own Nielsen class. Since
the xed points of ϕsm associated to a given interior singularity or puncture are all of the
same Lefschetz sign by the preceding discussion, we conclude that all dierentials vanish in
33
HF∗ (ϕsm ). Hence, the rank of HF∗ (ϕsm ) is given by Fix(ϕsm ), which equals the minimum
number of xed points of a nondegenerate symplectic representative of ϕ. In particular,
Fix(ϕsm ) is well-dened.
So we can calculate the rank of HF∗ (ϕsm ) by counting the number of normal xed points
of ϕ and then by accounting for how Fix(ϕ) is aected when we replace ϕ with ϕsm . An
invariant train track τ for ϕ yields the count of normal xed points. Indeed, as described in
Section 3.2.3, the transition matrix M of an ecient graph map ϕ yields a Markov partition of
a regular neighborhood of τ consisting of rectangles. Under ϕ, the length of these rectangles
is stretched by the dilatation λ, and the width of these rectangles is shrunk by λ. By the
contraction mapping theorem, ϕ has a xed point each time a rectangle maps over itself.
The total number of times that rectangles map over themselves is given by tr(M ), so that
tr(M ) is the xed point count of ϕ, up to some correction terms.
There are over-counted xed points resulting from a given edge bi that, without loss of
generality, has edge-path image ϕ(bi ) = bi · · · . This would mean that there is another edge bj
in the link of bi that, without loss of generality, has edge-path image ϕ(bj ) = bj · · · . Cotton-
Clay contends that such a xed point contributes 2 to tr(M ), so that we have double-counted
this xed point.
However, there appears to be a small error in Cotton-Clay's description of these over-
counted xed points (which ultimately does not aect our calculation in Section 4.3). Indeed,
P
Cotton-Clay captures such over-counting as si , where si equals one-half the number of
times that ϕ(bi ) starts or ends with bi which has the correct orientation. However, it is
P
possible that si is half-integer valued for once-punctured surfaces when the punctured
region of F −τ is unrotated and has two cusps or has one cusp. For example, using the
spine and monodromy from Example 3.2.3, we can isotope the curve A slightly upwards, so
that the image of ϕ is given by a 7→ ba, b 7→ bab. This also species an invariant train track,
P
and this would give si = 3/2. When this phenomenon occurs for a punctured region with
two cusps, two xed points are over-counted in tr(M ) as 3. When this phenomenon occurs
34
for a punctured region with one cusp, one xed point is over-counted in tr(M ) as 3. So we
P
modify Cotton-Clay's si count to be
P P
si , if si ∈ Z,
s˜i := ⌊ si ⌋, 1
P P
if si ∈ Z + 2
and the punctured region has two cusps,
1
P P
⌊ si ⌋ + 1,
if si ∈ Z + and the punctured region has one cusp.
2
This denition of s˜i would not accurately describe over-counting for all multi-punctured
surfaces, so we emphasize that we are only considering surfaces with at most one puncture.
Finally, there is an under-counting of xed points resulting from two distinct edges bi , bj
that, without loss of generality, share an initial switch and have edge-path images ϕ(bi ) =
bj · · · and ϕ(bj ) = bi · · · . Such a xed point does not contribute to tr(M ). This is called a
ip ; let f be the total number of ips.
From the preceding discussion, the number of normal xed points of ϕ is
Fix(ϕ) = tr(M ) − s˜i + f.
Assume, for now, that F is a closed surface. We describe the eect on the xed points of
ϕ when we replace ϕ with ϕsm . As mentioned above, the symplectic map ϕsm agrees with
ϕ away from the singularities. A xed interior p-pronged singular point (p ≥ 3) that is
nontrivially rotated has one elliptic xed point which is not accounted for in tr(M ). Denote
by r the number of rotated xed interior singular points. A xed interior p-pronged singular
point (p ≥ 3) that is unrotated has (p − 1) positive hyperbolic xed points, which are over-
counted in tr(M ) as p, corresponding to the p sides of the associated component of F −τ
that are collapsed to the singularity. Denote by u the number of unrotated xed interior
singular points. This yields the following formula of Cotton-Clay.
Theorem 4.2.1. For a closed surface F and a pseudo-Anosov map ϕ : F → F ,
X
rk HF∗ (ϕ) = tr(M ) − si + f + r − u.
i
35
Now we assume that F has one boundary component. The only dierence from the closed
case is that one component of F −τ contains a puncture. The contribution of xed points
of the faces which do not contain the puncture remains the same, so we must now consider
how the face containing the puncture aects our calculation. Say this face is a p-gon, p ≥ 1.
If this face is rotated, it contributes no normal xed points to ϕ (and nothing to tr(M )).
After replacing ϕ with ϕsm , we obtain no additional xed points. If this face is unrotated, it
contributes a normal xed point for each side of the p-gon, namely p xed points, and these
xed points are already accounted for in the trace. If this face has one cusp or two cusps,
there may be some over-counting as described above, but this is accounted for in s˜i . After
replacing ϕ with ϕsm , these p positive hyperbolic xed points are retained and are in the
same Nielsen class. Consequently, we have the following theorem.
Theorem 4.2.2. For a surface F with one boundary component and a pseudo-Anosov map
ϕ : F → F , the rank of HF∗ (ϕ) is
rk HF∗ (ϕ) = tr(M ) − s˜i + f + r − u.
4.3 Invariant train tracks for K(j, |βB | − j)
In this section, we record invariant train tracks for the monodromies of a particular family
of 3-string braids. Let LA be the closure of the braid βA = σ1 σ1 , and let LB be the closure
of the braid βB = σ1−1 · · · σ1−1 , where |βB | ≥ 2 is even. We consider those knots which can
be written as K = LA ⋆4 LB such that K is the closure of the braid
σ1 σ2−1 · · · σ2−1 σ1 σ2−1 · · · σ2−1 .
| {z } | {z }
j |βB |−j
Denote such a knot as K(j, |βB | − j), where j is odd.
Before we present our train tracks for K(j, |βB | − j), we make some remarks. Since the
images of innitesimal edges can be reconstructed from the images of real edges, we only
label the real edges in our graphs and calculate the images of the real edges. Furthermore,
since we are only interested in the behavior of our train tracks at switches, i.e. where edges
36
meet other edges, we illustrate the behavior of our train tracks only near switches. Finally,
by noting that K(j, |βB | − j) is isotopic to K(|βB | − j, j), one can verify that the ve train
tracks that follow are sucient to give a train track for the monodromy of any knot of the
form K(j, |βB | − j). These calculations are explicitly performed in the Appendix.
An invariant train track for the monodromy of the closure of K(1, 1) = 41 is given by
Figure 4.1. An invariant train track for the monodromy of the closure of K(3, 3) = 85 is given
by Figure 4.2. An invariant train track for the monodromy of K(|βB | − 1, 1), where |βB | ≥ 3,
is given by Figure 4.3. An invariant train track for the monodromy of K(j, 3), where j ≥ 5,
is given by Figure 4.4. An invariant train track for the monodromy of K(j, |βB | − j), where
j≥5 and |βB | − j ≥ 5 is given by Figure 4.5.
a 7→ ba
b 7→ bba
Figure 4.1. An invariant train track for the monodromy of K(1, 1) = 41 .
a 7→ c3
b6 c 3 , l=1
b ,
l−1 2≤l≤4
bl 7→
d 3 b 4 ,
l=5
b5 , l=6
b3 c1 ad1 d3 c1 b6 ,
l=1
cl 7→ c1 , l=2
ad1 , l=3
c 2 ,
l=1
dl 7→ d3 d1 , l=2
d2 , l=3
Figure 4.2. An invariant train track for the monodromy of K(3, 3) = 85 .
37
a 7→ b|βB |
cb|βB | ,
l=1
bl 7→ bl−1 , 2 ≤ l ≤ |βB | − 1
b|βB |−1 b|βB | , l = |βB |
c 7→ b|βB |−1 cac
Figure 4.3. An invariant train track for the monodromy of K(|βB | − 1, 1), where |βB | ≥ 3.
a 7→ c3
bj+3 c3 , l=1
b ,
l−1 2≤l ≤j+1
bl 7→
dj+2 dj bj+1 , l =j+2
bj+2 , l =j+3
bj c1 ad1 dj+2 dj c1 bj+3 ,
l=1
cl 7→ c1 , l=2
ad1 , l=3
c2 ,
l=1
dl 7→ dj+2 d1 , l=2
dl−1 3≤l ≤j+2
Figure 4.4. An invariant train track for the monodromy of K(j, 3), where j ≥ 5.
38
a 7→ c|βB |+2−j
b|βB | c|βB |−j c|βB |+2−j , l=1
b ,
l−1 2≤l ≤j+1
bl 7→
dj+2 dj bj+1 , l =j+2
j + 3 ≤ l ≤ |βB |
bl−1 ,
bj c1 c|βB |+1−j ad1 dj+2 dj c1 b|βB | , l=1
c ,
1 l=2
cl 7→
ad1 ,
l=3
4 ≤ l ≤ |βB | + 2 − j
cl−1
c 2 ,
l=1
dl 7→ dj+2 d1 , l=2
dl−1 3≤l ≤j+2
Figure 4.5. An invariant train track for the monodromy of K(j, |βB | − j), where j≥5 and
|βB | − j ≥ 5.
39
4.4 Calculation of HF∗(ϕK ) for K = K(j, |βB | − j)
As described in Section 4.3, we can calculate the rank of HF∗ (ϕ) from an invariant train
track for ϕ. In particular, we apply Theorem 4.2.2 to each of the invariant train tracks
just presented. As described in the Appendix, in each such train track, every interior p-
pronged singular point (p ≥ 3) is permuted with other singular points, so the contribution
P
of p (rp − up ) is 0. In each train track, there are also no ips, so the contribution of f is 0.
Indeed, one can check that in each train track, there are no edges e1 , e2 which emanate from
the same vertex such that ϕ(e1 ) = e2 · · · and ϕ(e2 ) = e1 · · · . We have an over-counting of
xed points in the train track for K(1, 1) = 41 corresponding to the pair a 7→ · · · a, b 7→ b · · · .
The trace of the transition matrix of this train track is 3. Similarly, we have an over-counting
of xed points in the train track for K(|βB | − 1, 1), where |βB | ≥ 3, corresponding to the
pair b|βB | 7→ · · · b|βB | , c 7→ · · · c. The trace of the transition matrix of this train track is 3.
All of the other train tracks have no over-counting and have transition matrices with trace
equal to 2. So the rank of HF∗ is 2 in every case.
Theorem 4.4.1. For K = K(j, |βB | − j) as dened above,
rk HF∗ (ϕK ) = 2.
Such calculations of HF∗ (ϕ), where ϕ is some composition of Dehn twists along a nicely
chosen collection of simple closed curves, have been performed via other methods [8]. How-
ever, to the author's knowledge, the maps ϕ considered here are not present in the literature.
It should be noted that an invariant train track for the pseudo-Anosov monodromy of an it-
erated plumbing of Hopf bands is given in [13], but this is done without the Bestvina-Handel
algorithm and without graphs.
One reason that such calculations are relevant is the recently established correspondence
for a bered knot K between the next-to-top group of HFK(K)
[ and HF∗ (ϕK ) [15]. Indeed,
while the top group of HFK
[ yields the genus and bered status of a knot [14, 37, 39], the
next-to-top group of HFK
[ of a nontrivial bered knot K yields the number of nondegenerate
40
xed points (plus one) of a symplectic representative of the monodromy of ϕK . Consequently,
we have the following corollary.
Corollary 4.4.1. For K = K(j, |βB | − j) as dened above,
rk HFK(K,
[ g(K) − 1) = 3.
Furthermore, the Alexander polynomial takes the form
∆K (t) = t−g(K) − 3t1−g(K) · · · − 3tg(K)−1 + tg(K)
4.5 Further research
In Chapter 2, we have shown that given a knot K and any two knots K1 , K2 , we can form K
as a Murasugi sum of K1 and K2 , and that this situation can be illustrated in a closed braid
form. So anything can happen when we use Murasugi sums of general complexity, but as we
have also shown, Murasugi sums are more well-behaved when the size of Murasugi sums is
restricted. For further study, we may also put other restrictions on the Murasugi sums, for
example by restricting the genera of the Seifert surfaces involved in the Murasugi sums. So
we can ask the following decomposition question about the result of the Murasugi sum:
Question. For a nontrivial knot K with genus g(K), what is the minimal genus of a Seifert
surface F for K which is a Murasugi sum of two unknots?
In this situation, if K has an alternating diagram which can be unknotted with u(K)
crossing changes, then Seifert's algorithm yields a minimal genus Seifert surface, so our
constructions give that 1 ≤ g(F ) − g(K) ≤ u(K). Note that this inequality is also true for
knots with u(K) = 1, because by [28], there exists a minimal genus Seifert surface for K on
which the unknotting crossing change can be done by twisting a band. Similarly, we can ask
the following composition question about the summands of the Murasugi sum:
Question. For two nontrivial knots K1 , K2 , what are the minimal genera of Seifert surfaces
F1 , F2 of K1 , K2 that Murasugi sum to the unknot?
Answering these questions in general seems to be more involved than what we treat in
this thesis, where we typically form some Seifert surfaces, manipulate them, then dissolve
41
them to obtain their knot boundaries. In particular, one would need to be more explicit with
how the original Seifert surfaces are formed. Studying these questions via band surgery and
band twisting (or perhaps other moves) should yield insight into a collection of subgraphs of
MSG(K) similar to MSG(K, n).
In Chapter 4, we calculated the rank of HF∗ of the monodromies of a family of knots
which are the closures of particular homogeneous 3-string braids. It is the hope of the author
that similar techniques can be used to calculate HF∗ of the knot closure of any homogeneous
braid with pseudo-Anosov monodromy. Such knots can be viewed as iterated Murasugi sums
of closures of positive braids. Noting that such a statement for closures of positive braids
(which include torus knots having periodic monodromy) might be proven by using the Skein
exact sequence of HFK
[, we present the following conjecture.
Conjecture 4.5.1. Suppose β is a homogeneous braid in the generators σ1 , . . . , σn such
that each σi appears at least twice, and suppose that the closure of β is a knot K . For
1 ≤ i ≤ n − 1, consider a subword of β which begins with σi±1 , ends with σi+1 ∓1
, and contains
no other σi±1 or σi+1∓1
. Similarly, consider a subword of β which begins with σi+1 ∓1
, ends with
σi±1 , and contains no other σi±1 or σi+1 ∓1
. Let ai be the number of times that a subword of one
of these two types appears in β . Then
n−1 l m
X ai
rk HFK(K,
[ g(K) − 1) = 1 + .
i=1
2
For example, the knot 12n22 is the closure of the homogenous braid
σ1−1 σ2 σ3 σ1−1 σ3 σ4−1 σ2 σ4−1 σ3 σ4−1 σ3 σ4−1 .
Here, a1 = 3, a2 = 0, a3 = 5, so our conjecture (accurately) predicts that
rk HFK(12n 22 , g(12n22 ) − 1) = 6.
[
More generally, it would be interesting to describe the behavior of the next-to-top group of
HFK
[ of all bered knots under Murasugi sum along minimal genus Seifert surfaces. While
the author has not succeeded in proving a statement of this type using Heegaard diagrams,
we nevertheless state the following generalization of Conjecture 4.5.1.
42
Conjecture 4.5.2. For bered knots K, K1 , K2 , suppose that K = K1 ⋆m K2 , where the
Murasugi sum is performed along minimal genus Seifert surfaces. Then
m
rk HFK(K,
[ g(K) − 1) − rk HFK(K
[ 1 , g(K1 ) − 1) − rk HFK(K
[ 2 , g(K2 ) − 1) ≤ − 1.
2
Clearly, this conjecture is true for m = 2, as this agrees with the Künneth formula for
HFK
[ of the connected sum of knots due to Ozsváth-Szabó [41]. In proving this conjecture,
one would give a deeper understanding of how symplectic xed points behave under Murasugi
sums, and one would also give another obstruction to forming Murasugi sums.
43
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8(1):603608, 2008.
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46
APPENDIX: CALCULATION OF INVARIANT TRAIN TRACKS FOR
K(j, |βB | − j)
Figure 4.6. A spine of our surface, and the core curves A, Bj .
Here we explicitly calculate invariant train tracks of the knots K(j, |βB | − j) dened in
Section 4.3. We begin with a spine of our surface, which consists of two vertices and the
oriented edges a, b1 , . . . , b|βB | . See Figure 4.6. We choose the canonical homology basis A
for the Hopf band which is the closure of βA = σ1 σ1 , so that A ≃ a, and the canonical
homology basis B1 , . . . , B|βB |−1 for the closure of βB = σ1−1 · · · σ1−1 , so that Bl ≃ bl bl+1
for 1 ≤ l ≤ |βB | − 1. We isotope these curves so that the intersection between A and
Bl is empty except for some xed l = j. We further isotope A, B1 , . . . , B|βB |−1 so that A
intersects b1 , . . . , b j each geometrically once, Bl intersects bl , bl+1 each geometrically once for
1 ≤ l ≤ |βB |, Bj intersects a geometrically once, and there are no other intersections.
As described in Section 1.1, the monodromy of K is given as the composition of Dehn
L L R R L
twists ϕ = DB |β
◦ · · · ◦ DB 1
◦ DA , where we rst apply DA , then DB 1
, and so on. In this
B |−1
L
setup, we calculate the nontrivial action of each Dehn twist as: DB j
(a) = bj+1 bj a, and
L L
For 1 ≤ l ≤ |βB | − 1 : DB l
(bl ) = bl+1 , DB l
(bl+1 ) = bl+1 bl bl+1
R
For 1 ≤ l ≤ j : DA (bl ) = bl a
We construct our invariant train tracks for ve dierent cases. In each case, we assume
that |βB |, j + 1 are even so that K(j, |βB | − j) is a knot. For Case 1, we additionally assume
that |βB | − j = j = 1. For Case 2, we instead assume that |βB | = j + 1 and j ≥ 3. For Case
47
3, which involves the most moves out of all of the cases, we instead assume that |βB | ≥ j + 5
and j ≥ 5. From the invariant train track for Case 3, we recover invariant tracks for Case
4, where we instead assume that j≥5 and |βB | = j + 3, and for Case 5, where we instead
assume that |βB | = j +3 and j = 3. Throughout our calculations, we only draw local pictures
of our graphs around the vertices.
Case 1
Assume that |βB | − j = j = 1. The Bestvina-Handel algorithm is illustrated in Figure 4.7,
where we begin on the left with an initial spine graph. The initial image of the edges under
ϕ is given by, after we pull tight,
a 7→ b2 b1 a, b1 7→ b1 a, b2 7→ b2 b1 b2 .
Figure 4.7. A sequence of moves to produce an invariant train track for Case 1.
At this point, we have a valence 2 vertex with link {b1 , b2 }, so we perform a valence 2
isotopy to remove b2 . After dropping the subscript on b1 , we obtain a 7→ ba, b 7→ bba. We
claim that this species an invariant train track. To verify this claim, we must calculate
the gates occurring at the vertex specied by {a, a, b, b}. To do so, we must iterate the
graph derivative D at this vertex until the image becomes periodic. The image of an edge
emanating from a vertex under D is the rst edge in the edge-path of the emanating edge.
Once the image of D at a vertex becomes periodic, we classify two edges as being in the
same gate if their image under this periodic iterate of D is the same, and as being in distinct
gates otherwise. We calculate
D
{a, a, b, b} 7−
→ {b, a, b, a} 7→ {b, a, b, a}
48
From this, we conclude that a, b are in the same gate, and a, b are in the same gate. Each
of these two gates corresponds to a cusp. Now we insert the innitesimal edges within our
vertex. The edge-path a 7→ ba connects the distinct gates {b, a}, so we insert an innitesimal
edge connecting these two gates. See the right of Figure 4.7 for the result of distinguishing
the gates and adding the innitesimal edge (in red). The result is an invariant train track
where the component containing the puncture is a disk with two cusps.
Case 2
Assume that |βB | = j + 1 and j ≥ 3. The Bestvina-Handel algorithm is illustrated in Figure
4.8, where we begin on the left with an initial spine graph. The initial image of the edges
under ϕ is given by, after we pull tight,
a 7→ b|βB | b|βB |−1 a
b
|βB |−1 a, l=1
bl 7→ b|βB | bl−1 b|βB |−1 a, 2 ≤ l ≤ |βB | − 1
b|βB | b|βB |−1 b|βB | ,
l = |βB |
Figure 4.8. A sequence of moves to produce an invariant train track for Case 2.
Move 1
Fold the start of b2 , . . . , b|βB | to c1 7→ b|βB | , so that the new edges b′2 , . . . , b′|βB | are specied by
b2 = c1 b′2 , . . . , b|βB | = c1 b′|βB | .
49
This folding leaves a valence 2 vertex with link {b1 , c1 }, so we also perform a valence 2 isotopy
to remove c1 . This move does not change the appearance of the graph. After pulling tight
and dropping primes, we obtain
a 7→ b|βB | b|βB |−1 a
b|βB | b|βB |−1 a, l=1
bl 7→ bl−1 b|βB |−1 a, 2 ≤ l ≤ |βB | − 1
b|βB |−1 b|βB | ,
l = |βB |
Move 2
We now perform a sequence of folds to eliminate the backtracking
b|βB | 7→ b|βB |−1 b|βB | 7→ · · · b|βB |−2 b|βB |−1 · · · 7→ · · · aa · · · .
First, we fold the end of a, b1 , . . . , b|βB |−1 to c1 7→ b|βB |−1 a, so that the new edges are specied
by
a = a′ c1 , b1 = b′1 c1 , . . . , b|βB |−1 = b′|βB |−1 c1 .
After pulling tight and dropping primes, we obtain
a 7→ b|βB |
b|βB | , l=1
bl 7→ c1 bl−1 , 2 ≤ l ≤ |βB | − 1
c1 b|βB |−1 b|βB | ,
l = |βB |
c1 7→ b|βB |−1 c1 ac1
Move 2.1
As a result of Move 2, one more move is needed to remove the given backtracking since
b|βB | 7→ b|βB |−1 b|βB | 7→ · · · c1 c1 · · · .
50
Next, we fold the start of b2 , . . . , b|βB | to c2 7→ c1 , so that the new edges are specied by
b2 = c2 b′2 , . . . , b|βB | = c2 b′|βB | .
This folding leaves a valence 2 vertex with link {b1 , c2 }, so we also perform a valence 2 isotopy
to remove c2 . This move does not change the appearance of the graph. After pulling tight,
dropping primes, and dropping the subscript on c1 , we obtain
a 7→ b|βB |
cb|βB | , l=1
bl 7→ bl−1 , 2 ≤ l ≤ |βB | − 1
b|βB |−1 b|βB | ,
l = |βB |
c 7→ b|βB |−1 cac
We claim that this species an invariant train track. To verify this claim, we must calculate
the gates occurring at the three vertexs {c, a, b|βB | }, {a, b1 , . . . , b|βB |−1 }, {b1 , . . . , b|βB | }. We
carry out this calculation for the vertex {c, a, b|βB | }:
D
{c, a, b|βB | } 7− → {c, b|βB | , b|βB | } 7→ {c, b|βB | , b|βB | }
From this, we conclude that a, b|βB | are in the same gate, which corresponds to a cusp, while
c is in its own gate. Similarly, the emanating vertices at the other vertexs are all in distinct
gates except for the gate {a, b1 }, which corresponds to another cusp.
Now we insert the innitesimal edges within each vertex between some number of distinct
gates. For instance, the edge-path b1 7→ cb|βB | connects the distinct gates {c, b|βB | }, so we
insert an innitesimal edge e connecting these two gates. Furthermore, we calculate the
image of e under iterates of ϕ and insert innitesimal edges between the two gates which
ϕk (e) connects for some k ∈ Z≥0 . This is done similarly to how we calculated the gates,
where we now iterate the graph derivative D on the gate {c, b|βB | } until it becomes periodic.
51
We calculate
D
{c, b|βB | } 7−
→ {c, b|βB | }
We conclude that no additional innitesimal edges are introduced from e. By performing a
similar calculation for all other edge-paths, we conclude that within a vertex, each pair of
adjacent gates is connected by an innitesimal edge. See Figure 4.8 for the result of distin-
guishing the gates and adding innitesimal edges (in red). The result is an invariant train
track where the component containing the puncture is a disk with two cusps. Furthermore,
there are two |βB |-prongs which are permuted.
Case 3
Assume that j≥5 and |βB | ≥ j + 5. Our initial spine is pictured in Figure 4.9. The image
of the edges under ϕ is given by, after we pull tight,
a 7→ b|βB | bj a
b a,
j l=1
bl 7→ b|βB | bl−1 bj a, 2≤l≤j
b|βB | bl−1 b|βB | ,
j + 1 ≤ l ≤ |βB |
Move 1
Fold the start of b2 , . . . , b|βB | to c1 7→ b|βB | . This leaves a valence 2 vertex with link {b1 , c1 },
so we also perform a valence 2 isotopy to remove c1 . This does not change the appearance
of the graph. After pulling tight and dropping primes, we obtain
a 7→ b|βB | bj a
b|βB | bj a, l=1
bl 7→ bl−1 bj a, 2≤l≤j
bl−1 b|βB | ,
j + 1 ≤ l ≤ |βB |
52
Figure 4.9. The initial spine graph of Case 3, and the results of Moves 1-7.
Move 2
At this point, we would like to eliminate the backtracking
bj 7→ bj−1 bj · · · 7→ · · · bj−2 bj−1 · · · 7→ · · · aa · · ·
However, this would involve a folding which increases the valence of the point p within bj
where ϕ3 (p) fails to be locally injective. To correct this, we subdivide bj = b1j b2j , where
b1j 7→ bj−1 , b2j 7→ b1j b2j a.
53
Now we can fold the end of a, b1 , . . . , bj−1 , b2j to c1 7→ b2j a, so that the new edges are specied
by
a = a′ c1 , b1 = b′1 c1 , . . . , bj−1 = b′j−1 c1 , b2j = (b2j )′ c1 .
After pulling tight and dropping primes, we obtain
a 7→ b|βB | b1j
b|βB | b1j ,
l=1
c1 bl−1 b1j ,
2≤l ≤j−1
bl 7→
c1 b2j b1j b|βB | l =j+1
bl−1 b|βB | ,
j + 2 ≤ l ≤ |βB |
b1j 7→ c1 bj−1
b2j 7→ b1j
c1 7→ b2j c1 ac1
Move 2.1
As a result of Move 2, one more move is needed to remove the given backtracking since
bj 7→ bj−1 b1j 7→ c1 c1
We fold the start of b2 , . . . , bj−1 , b1j , bj+1 to c2 7→ c1 , so that the new edges are specied by
b2 = c2 b′2 , . . . , bj−1 = c2 b′j−1 , b1j = c2 (b1j )′ , bj+1 = c2 b′j+1 .
After pulling tight and dropping primes, we obtain
54
a 7→ b|βB | c2 b1j
b|βB | c2 b1j ,
l=1
b1 c2 b1j ,
l=2
bl−1 b1j ,
3≤l ≤j−1
bl 7→
b2j b1j c2 b|βB | l =j+1
bj+1 c2 b|βB | l =j+2
bl−1 b|βB | ,
j + 3 ≤ l ≤ |βB |
b1j 7→ bj−1 c2
b2j 7→ c2 b1j
c1 7→ b2j c1 ac1
c2 7→ c1
Move 3
At this point, we have a valence 2 vertex with link {b1j , b2j }, so we perform a valence 2 isotopy
to remove b1j . After pulling tight, dropping primes, and dropping the superscript on b2j , we
have
55
a 7→ b|βB | c2
b|βB | c2 , l=1
b 1 c2 , l=2
bl−1 ,
3≤l≤j
bl 7→
bj c2 b|βB | l =j+1
bj+1 c2 b|βB | l =j+2
bl−1 b|βB | ,
j + 3 ≤ l ≤ |βB |
c1 7→ bj c1 ac1
c2 7→ c1
Move 4
At this point, we would like to eliminate the backtracking
b|βB | 7→ b|βB |−1 b|βB | 7→ · · · b|βB |−2 b|βB |−1 · · · 7→ · · · b|βB | b|βB | · · ·
However, this would involve a folding which increases the valence of the point p within b|βB |
where ϕ3 (p) fails to be locally injective. To correct this, we subdivide b|βB | = b1|βB | b2|βB | , where
b1|βB | 7→ b|βB |−1 , b2|βB | 7→ b1|βB | b2|βB | .
Now we can fold the start of a and the end of bj+1 , . . . , b|βB |−1 , b2|βB | to c3 7→ b2|βB | , so that the
new edges are specied by
a = c3 a′ , bj+1 = b′j+1 c3 , . . . , b|βB |−1 = b′|βB |−1 c3 , b2|βB | = (b2|βB | )′ c3 .
Together, Moves 4, 4.1, 5 do not change the appearance of the graph. After pulling tight
and dropping primes, we obtain
56
a 7→ b1|βB | c2
c3 b2|βB | b1|βB | c2 , l=1
b1 c 2 , l=2
bl−1 ,
3≤l≤j
bl 7→
bj c2 b1|βB |
l =j+1
c3 bj+1 c2 b1|βB |
l =j+2
c3 bl−1 b1|β | ,
B
j + 3 ≤ l ≤ |βB | − 1
b1|βB | 7→ c3 b|βB |−1
b2|βB | 7→ b1|βB |
c1 7→ bj c1 c3 ac1
c2 7→ c1
c3 7→ c3 b2|βB |
Move 4.1
As a result of Move 4, one more move is needed to remove the given backtracking since
b|βB | 7→ b|βB |−1 b1|βB | 7→ c3 c3
Now we can fold the start of b1 , bj+2 , . . . , b|βB |−1 , b1|βB | to c4 7→ c3 so that the new edges are
specied by
b1 = c4 b′1 , bj+2 = c4 b′j+2 , . . . , b|βB |−1 = c4 b′|βB |−1 , b1|βB | = c4 (b1|βB | )′
After pulling tight and dropping primes, we obtain
57
a 7→ b1|βB | c4 c2
b2|βB | b1|βB | c4 c2 , l=1
b1 c 4 c 2 , l=2
bl−1 ,
3≤l≤j
bl 7→
bj c2 c4 b1|βB |
l =j+1
bj+1 c2 c4 b1|βB |
l =j+2
bl−1 b1|β | ,
B
j + 3 ≤ l ≤ |βB | − 1
b1|βB | 7→ b|βB |−1 c4
b2|βB | 7→ c4 b1|βB |
c1 7→ bj c1 c3 ac1
c2 7→ c1
c3 7→ c3 b2|βB |
c4 7→ c3
Move 5
At this point, we have three valence 2 vertices with respective links {b1|βB | , b2|βB | }, {c1 , c3 },
{c2 , c4 }. So we perform three valence 2 isotopies to remove b1|βB | , c3 , c4 . After pulling tight
and dropping the superscript on b2|βB | , we obtain
58
a 7→ c2
b|βB | c2 , l=1
b1 c 2 , l=2
bl−1 ,
3≤l≤j
bl 7→
bj c 2 l =j+1
bj+1 c2 l =j+2
bl−1 ,
j + 3 ≤ l ≤ |βB |
c1 7→ bj c1 ac1 b|βB |
c2 7→ c1
Move 6
We would like to remove the backtracking
c1 7→ · · · c1 b|βB | 7→ · · · b|βB | b|βB |−1 · · ·
ϕ|βB |−j−3
7−−−−−−→ · · · bj+3 bj+2 · · · 7→ · · · bj+2 bj+1 · · · 7→ · · · c2 c2 · · ·
through a sequence of pairs of folds. We begin by sliding the end of bj+1 , bj+2 over a, and
then we fold the start of bj+2 , bj+3 to c3 7→ a. Then for even k satisfying j + 3 ≤ k ≤ |βB | − 2,
perform the following pairs of folds in increasing k -order. First, fold the end of bk , bk+1 to
ck+1−j 7→ ck−j , then fold the start of bk+1 , bk+2 to ck+2−j 7→ ck+1−j . After performing this
sequence of folds, we then fold the end of c1 , b|βB | to c|βB |+1−j 7→ c|βB |−j . From all of this
folding, we obtain
59
a 7→ c2
c|βB |+1−j b|βB | c|βB |−j c2 , l=1
bl 7→ b1 c2 , l=2
bl−1 ,
3 ≤ l ≤ |βB |
bj c1 c|βB |+1−j ac1 b|βB | , l=1
c|βB |+1−j c1 ,
l=2
cl 7→
a,
l=3
cl−1
4 ≤ l ≤ |βB | + 1 − j
Move 7
We now have the backtracking
b2 7→ b1 c2 7→ · · · c|βB |+1−j c|βB |+1−j · · · ,
so we fold the start of b1 , c2 to c|βB |+2−j 7→ c|βB |+1−j . After pulling tight, we have
a 7→ c|βB |+2−j c2
b|βB | c|βB |−j c|βB |+2−j c2 , l=1
bl 7→ b1 c2 , l=2
bl−1 ,
3 ≤ l ≤ |βB |
bj c1 c|βB |+1−j ac1 b|βB | , l=1
c 1 ,
l=2
cl 7→
a, l=3
cl−1
4 ≤ l ≤ |βB | + 2 − j
60
Move 8
We now have the backtracking
ϕj−3
c1 7→ bj c1 · · · 7→ · · · bj−1 bj · · · 7−−→ · · · b2 b3 · · · 7→ · · · b1 b2 · · · 7→ · · · c2 c2 · · · .
In a way similar to Move 6, we must perform a sequence of pairs of folds. The rst such
pair of folds is to fold the end of a, b1 , b2 to d1 7→ c2 , and then to fold the start of b2 , b3
to d2 7→ d1 . Then for odd k satisfying 3 ≤ k ≤ j − 2, perform the following pairs of folds
in increasing k -order. First, fold the end of bk , bk+1 to dk 7→ dk−1 , then fold the start of
bk+1 , bk+2 to dk+1 7→ dk . After performing this sequence of folds, we then fold the start of c1
and the end of bj , bj+1 to dj 7→ dj−1 . From all of this folding, we obtain
a 7→ c|βB |+2−j
b|βB | c|βB |−j c|βB |+2−j ,
l=1
bl−1 ,
2≤l≤j
bl 7→
dj bl−1 , j+1≤l ≤j+2
bl−1 ,
j + 3 ≤ l ≤ |βB |
bj c1 c|βB |+1−j ad1 dj c1 b|βB | , l=1
c1 dj ,
l=2
cl 7→
ad1 ,
l=3
cl−1
4 ≤ l ≤ |βB | + 2 − j
c2 ,
l=1
dl 7→
dl−1
2≤l≤j
61
Figure 4.10. The Case 3 graph after Moves 8 and 9.
62
Move 9
We now have the backtracking
b1 7→ b|βB | c|βB |−j · · ·
ϕ|βB |−j−3
7−−−−−−→ · · · bj+3 c3 · · · 7→ · · · bj+2 d1 · · · 7→ · · · bj+1 c2 · · · 7→ · · · dj dj · · · .
Again, we must perform a sequence of pairs of folds. The rst such pair of folds is to fold
the start of bj+1 and the end of c2 to dj+1 7→ dj , and then to fold the end of bj+2 , d1 to
dj+2 7→ dj+1 , and then to fold the end of c3 and start of bj+3 to dj+3 7→ dj+2 . Then for even
k satisfying 4 ≤ k ≤ |βB | − j − 1, perform the following pairs of folds in increasing k -order.
First, fold the start of ck and end of bj+k to dj+k 7→ dj+k−1 , then fold the start of bj+k+1
and end of ck+1 to dj+k+1 7→ dj+k . After performing this sequence of folds, we then have
|βB | − j − 2 valence 2 vertices, so we perform valence 2 isotopies to remove dj+3 , . . . , d|βB | .
As a result, we obtain
63
a 7→ c|βB |+2−j
b|βB | c|βB |−j c|βB |+2−j , l=1
bl−1 ,
2≤l ≤j+1
bl 7→
dj+2 dj bj+1 , l =j+2
bl−1 ,
j + 3 ≤ l ≤ |βB |
bj c1 c|βB |+1−j ad1 dj+2 dj c1 b|βB | , l=1
c 1 ,
l=2
cl 7→
ad1 , l=3
cl−1
4 ≤ l ≤ |βB | + 2 − j
c2 , l=1
dl 7→ dj+2 d1 , l=2
dl−1
3≤l ≤j+2
We claim that this species an invariant train track. To verify this claim, we must
calculate the gates. We give a sample calculation for the gates corresponding to the vertex
with emanating edges {a, b1 , b2 , d1 }. In this example,
D
{a, b1 , b2 , d1 } 7−
→ {c|βB |+2−j , c|βB |+2−j , b1 , c2 }.
Since D(a) = D(b1 ), the edges a, b1 emanate from the same gate, and this corresponds to a
cusp in the train track. Continuing to iterate D, we obtain:
64
{c|βB |+2−j , c|βB |+2−j , b1 , c2 } 7−→ {c|βB |+1−j , c|βB |+1−j , b|βB | , c1 }
7−→ {c|βB |−j , c|βB |−j , b|βB |−1 , b|βB | }
D|βB |−j−3
7−−−−−−→ {c3 , c3 , bj+2 , bj+3 }
7−→ {d1 , d1 , dj+2 , bj+2 }
7−→ {c2 , c2 , dj+1 , bj+1 }
7−→ {c1 , c1 , dj , bj }
7−→ {bj , bj , dj−1 , bj−1 }
Dj−3
7−−−→ {b3 , b3 , d2 , b2 }
7−→ {b2 , b2 , d1 , b1 }
7−→ {b1 , b1 , c2 , c|βB |+2−j }
By repeating this iteration of D three more times, we return to {c|βB |+2−j , c|βB |+2−j , b1 , c2 }, so
the edges a, b1 lie in the same gate, and the edges b2 , d1 lie in distinct gates. By performing
a similar calculation at all vertexs, we conclude that all other edges are in distinct gates
except for the edges {c1 , bj+1 }, which correspond to another cusp.
Finally, we must calculate the innitesimal edges. We produce a sample calculation
corresponding to the innitesimal edge which is added from the image of c1 7→ bj c1 . This
edge-path connects the distinct gates {bj , c1 }, and its image under iterations of D is specied
by
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{bj , c1 } 7−→ {bj−1 , bj }
7−→ {bj−2 , bj−1 }
Dj−3
7−−−→ {b1 , b2 }
7−→ {c|βB |+2−j , b1 }
7−→ {c|βB |+1−j , b|βB | }
D|βB |−j−2
7−−−−−−→ {c3 , bj+2 }
7−→ {d1 , dj+2 }
7−→ {c2 , dj+1 }
7−→ {c1 , dj+1 }
The nal gates in this calculation correspond to the same vertex and a dierent innites-
imal edge from our starting gates {bj , c1 }. By repeating this iteration of D two more
times, we return to {bj , c1 }, so we conclude that all vertexs with three gates have adja-
cent gates connected by an innitesimal edge. Similarly, the adjacent gates within each of
{a, c4 , . . . , c|βB |−1−j , c|βB |+1−j }, {c3 , . . . , c|βB |−j , c|βB |+2−j }, {d3 , . . . , dj−2 , dj , dj+2 },
{d2 , . . . , dj−1 , dj+1 } are connected by innitesimal edges. See Figure 4.11 for the result of
distinguishing the gates and adding innitesimal edges. The result is an invariant train track
where the component containing the puncture is a disk with two cusps. Furthermore, there
|βB |+1−j j+1
are two -prongs which are permuted (colored blue), there are two -prongs which
2 2
are permuted (colored gray), and there are (|βB | + 2) 3-prongs which are permuted (colored
red).
Case 4
Now we assume that j ≥ 5, |βB | = j + 3. Our initial spine graph is pictured in the top left
of Figure 4.9, where we relabel |βB | = j + 3. The moves we perform are essentially the same
as those performed in Case 3, except for several key dierences. After Move 4, there are
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Figure 4.11. Adding innitesimal edges at vertices to obtain the train track of Case 3.
images of bl for j + 3 ≤ l ≤ |βB | − 1, which would be empty in our current case. One can
check that this does not aect how the images are written after Move 5.
In Move 6, our sequences of folds for j + 3 ≤ k ≤ |βB | − 2 is now empty. Consequently,
we delete the following labeled edges from the bottom of Figure 4.9. From the upper graph,
we delete the edges labeled bj+3 , . . . , b|βB |−1 , c4 , . . . , c|βB |−1−j , and from the lower graph, we
delete the edges labeled bj+4 , . . . , b|βB | , c5 , . . . , c|βB |−j . Then, we relabel |βB | = j1 + 3. This
results in two valence 2 vertices with respective links {a, c4 }, {c3 , c5 }. The image of our
current case's edges after Move 7 is obtained from the previous case's image by relabeling
|βB | = j + 3, and by performing two valence 2 isotopies to remove c4 , c5 .
In Move 9, our sequence of folds for 4 ≤ k ≤ |βB | − j − 1 is now empty, but in the same
move, we removed all such folds by valence 2 isotopy. Consequently, this has no eect on our
current case's graph or the image of edges. See Figure 4.12 for the nal graph. The image
of our edges is given by
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a 7→ c3
bj+3 c3 , l=1
bl−1 ,
2≤l ≤j+1
bl 7→
dj+2 dj bj+1 , l =j+2
bj+2 ,
l =j+3
bj c1 ad1 dj+2 dj c1 bj+3 , l=1
cl 7→ c1 , l=2
ad1 ,
l=3
c2 , l=1
dl 7→ dj+2 d1 , l=2
dl−1
3≤l ≤j+2
Figure 4.12. The nal Case 4 graph.
We claim that this species an invariant train track. By calculating the gates in a
way similar to the previous case, we conclude that at each vertex, every emanating edge
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is in a distinct gate except for the gates {a, b1 }, {bj+1 , c1 } which correspond to cusps. By
another innitesimal edge calculation, we conclude that within each vertex, adjacent gates
are connected by an innitesimal edge. See Figure 4.4. The result is an invariant train track
where the component containing the puncture is a disk with two cusps. Furthermore, there
j+1
are two
2
-prongs which are permuted, and there are (j + 5) 3-prongs which are permuted.
Case 5
Now we assume that j = 3, |βB | = j + 3. Similarly to Case 4, we trace through the moves
of Case 3 and alter our graph and the image of our edges as required. See Figure 4.13. As
a result, we obtain
a 7→ c3
b6 c 3 , l=1
bl−1 ,
2≤l≤4
bl 7→
d 3 b4 , l=5
b 5 ,
l=6
b3 c1 ad1 d3 c1 b6 , l=1
cl 7→ c1 , l=2
ad1 ,
l=3
c2 , l=1
dl 7→ d3 d1 , l=2
d 2 ,
l=3
We claim that this species an invariant train track where the component containing
the puncture is a disk with two cusps. Furthermore, there are eight 3-prongs which are
permuted. See Figure 4.2.
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Figure 4.13. The nal Case 5 graph.
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