INVARIANTS OF TOPOLOGICAL AND LEGENDRIAN LINKS IN LENS SPACES
WITH A UNIVERSALLY TIGHT CONTACT STRUCTURE
By
Christopher R. Cornwell

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics
2011

ABSTRACT
INVARIANTS OF TOPOLOGICAL AND LEGENDRIAN LINKS IN LENS
SPACES WITH A UNIVERSALLY TIGHT CONTACT STRUCTURE
By
Christopher R. Cornwell
In this thesis a HOMFLY polynomial is found for knots and links in a lens space L(p, q).
Further study of this polynomial invariant finds a relationship with the classical invariants
of Legendrian and transverse links, when L(p, q) is endowed with a universally tight contact
structure. In fact certain criteria are found which, if satisfied by any numerical invariant of
links in L(p, q), guarantee that the invariant fits into a Bennequin type inequality. A linear
function of the degree of the HOMFLY polynomial is then shown to satisfy these criteria. A
corollary is that certain “simple” Legendrian and transverse realizations of knots admitting
grid number one diagrams maximize the classical invariants in their knot type.
In order to obtain the above results, formulae are found for computing the classical
invariants of Legendrian and transverse links from a toroidal front projection. Having these
formulae, and known results about fibered links that support a given contact structure, it is
found whether the duals of some families of Berge knots support the universally tight contact
structure.

ACKNOWLEDGMENT

I first express my gratitude to my dissertation advisor, Effie Kalfagianni, for her part in my
progress through the mathematics PhD program at Michigan State University. Her guidance
and support have been an invaluable resource to me in my studies and my development as
a mathematician.
I’d like to thank Matt Hedden for many enlightening conversations on many topics, and
I want to thank Ken Baker for an “illustrated conversation” which related to some of the
results in this thesis. Also, I thank Lenny Ng and Dan Rutherford for conversations about
my thesis and related problems; in particular, Dan pointed out how the proof of Theorem
1.1.3 implied Corollary 1.1.5.
I’d also like to thank my fellow graduate students at MSU. I’ve learned a great deal from
many of you. I appreciate the atmosphere of mathematical learning, as well the wonderful
friendship, that we’ve been able to enjoy during my years here. Finally, I want to thank my
wife, Whitney. Your unfailing love, encouragement, and support have not only helped me
succeed, but have made me happy.

iii

TABLE OF CONTENTS

List of Figures

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v

1 Introduction
1.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2

2 Definitions and Background
2.1 Skein Polynomials . . . . . . . . . . . . . . . . . . . . .
2.2 Contact structures on 3-dimensional manifolds . . . . .
2.3 Review of known results . . . . . . . . . . . . . . . . .
2.3.1 Power series invariants in Q-homology 3-spheres
2.3.2 Bennequin type inequalities . . . . . . . . . . .
2.3.3 Links and grid diagrams in L(p, q) . . . . . . . .
3 Homotopies and the HOMFLY polynomial
3.1 Homology classes . . . . . . . . . . . . . . . . .
3.2 Skein Theory . . . . . . . . . . . . . . . . . . .
3.3 The HOMFLY polynomial in L(p, q) . . . . . .
3.3.1 Trivial Links in L(p, q). . . . . . . . . . .
3.3.2 Every link is homotopic to a trivial link .
3.3.3 The polynomial invariant . . . . . . . . .

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29
29
30
37
37
43
47

4 Classical invariants of Q-nullhomologous Legendrian and transverse links
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Formulae for tbQ (K), rotQ (K), and slQ (K) from a grid projection in L(p, q)
4.3 Bennequin-type bounds in L(p, q) . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Examples and Computations . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
55
59
64
71

5 Fibered knots and the universally tight contact structure
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Surgeries on fibered knots, contact structures and Berge knots . . . . . . . .
5.3 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76
77
80
83

iv

LIST OF FIGURES

2.1

Projections in a skein relation . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

The front projection of a Legendrian figure 8 knot . . . . . . . . . . . . . . .

13

2.3

Positive and negative stabilization of K . . . . . . . . . . . . . . . . . . . . .

15

2.4

(A) shows a grid diagram (with grid number 2) in L(7, 2) on a fundamental
domain of T . In (B) we alter the fundamental domain. (C) is the “straightening” of (B). For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation. . .

23

2.5

A grid diagram for L(5, 3) with corresponding grid projection. . . . . . . . .

25

2.6

Stabilization of type X: NW (top) and of type O: SW (bottom) . . . . . . .

26

2.7

A non-interleaving commutation in L(7, 2). . . . . . . . . . . . . . . . . . . .

27

2.8

A move which is neither an interleaving nor non-interleaving commutation. .

28

2.9

D: the lift to S 3 of the grid diagram D in L(5, 1). . . . . . . . . . . . . . . .

28

3.1

A crossing change on a grid projection. . . . . . . . . . . . . . . . . . . . . .

30

3.2

1 → 2: crossing change; 2 → 3: row commutation; 3 → 4: column commutation; 4 → 5: two row commutations; 5 → 6: destabilization; 6 → 7: many
column commutations; 7 → 8: destabilization . . . . . . . . . . . . . . . . .

32

3.3

Positive and negative skein crossings. . . . . . . . . . . . . . . . . . . . . . .

33

3.4

Resolution at a positive skein crossing . . . . . . . . . . . . . . . . . . . . . .

34

3.5

Resolution at a negative skein crossing . . . . . . . . . . . . . . . . . . . . .

35

3.6

The interleaving regions of p, a point in the image of H : C (D) → R2 . . . .

37

v

3.7

The grid number 1 knot K3 in L(7, 2) . . . . . . . . . . . . . . . . . . . . . .

38

3.8

The trivial link diagram D(I) in L(5, 2) with I = (0, 1, 2, 0, 3). . . . . . . . .

40

3.9

Decreasing Grid Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.10 Decreasing Grid Number: Case 2a . . . . . . . . . . . . . . . . . . . . . . . .

46

3.11 Decreasing Grid Number: Case 2b . . . . . . . . . . . . . . . . . . . . . . . .

46

3.12 The interleaving of unaltered columns in D . . . . . . . . . . . . . . . . . . .

52

3.13 The interleaving of altered columns in D with a common good choice of α0 .

53

3.14 The interleaving of altered columns in D without a common good choice of α0 54
4.1

Constructing a rational Seifert surface . . . . . . . . . . . . . . . . . . . . .

56

4.2

How tb changes with a disk slide . . . . . . . . . . . . . . . . . . . . . . . .

61

4.3

On an X : N E destabilization cd (P ) increases. . . . . . . . . . . . . . . . . .

66

4.4

A skein crossing change and resolution at a positive skein crossing. . . . . . .

67

4.5

A skein crossing change and resolution at a negative skein crossing . . . . . .

68

4.6

Example: A link B in L(5, 1) and its skein tree. . . . . . . . . . . . . . . . .

72

4.7

Example: A link L in L(5, 1) and its skein tree. . . . . . . . . . . . . . . . .

73

4.8

Grid diagram associated to the link Ln .

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

74

5.1

Berge knots on a Seifert surface of the trefoil . . . . . . . . . . . . . . . . . .

83

5.2

Berge knots on a Seifert surface of the figure 8 knot . . . . . . . . . . . . . .

87

5.3

In surface (A) the embedded curve obtained from the weighted edges as indicated is already a negative braid. In surface (B), this is not the case. By
isotopy of the surface, we get the top-left surface in Figure 5.4, where it shows
how to get a negative braid closure. . . . . . . . . . . . . . . . . . . . . . . .

90

vi

5.4

The isotopy of Σ above starts at the top-left figure and ends at the bottomright figure. The isotopy from the top-right figure to the bottom-left figure is a
“Reidemeister I move” of one of the bands in the closure, which move cancels
the twisting. From the bottom-left figure to the bottom-middle figure, one
performs a finger move through the full twist. . . . . . . . . . . . . . . . . .

vii

91

Chapter 1
Introduction
In this thesis we study knots and links in lens spaces through the tool of toroidal grid
diagrams. Grid diagrams of knots and links in S 3 have garnered recent attention, due
to applications in Legendrian knot theory and knot Floer homology. Their counterparts
that describe links in a lens space are of similar interest in this setting. In [5] toroidal
grid diagrams were considered in order to construct combinatorial knot Floer homology in
lens spaces. Later, in [4], the correspondence between Legendrian links in universally tight
contact lens spaces and toroidal grid diagrams was developed.
The construction in [5] is of particular relevance to an open problem known as the Berge
Conjecture, which posits a complete description of knots in S 3 that admit Dehn surgery
resulting in a lens space. In fact, the conjecture is only still open for integer surgeries, and
thus can be reformulated in terms of which knots in a given lens space yield an integer
surgery that gives S 3 . In this setting, the Berge Conjecture states that such a knot must
admit a toroidal grid diagram with grid number 1.
In the work of Baker and Grigsby [4], that indicates the correspondence between Legen1

drian links in universally tight contact lens spaces and toroidal grid diagrams, a relationship
is found between the Thurston–Bennequin number of a knot, that of its mirror, and the
grid number of the knot. This relationship suggests that finding bounds for the Thurston–
Bennequin number of a knot might be useful in controlling the grid number.
Our work here uses skein polynomials (specifically the HOMFLY polynomial) to find
such a bound on the Thurston–Bennequin number. In fact, previous to our work, the
existence of a HOMFLY polynomial for links in lens spaces was an open problem. In the
work below we resolve this problem, and toroidal grid diagrams are fundamental to the
proof of the existence of the HOMFLY polynomial. In addition, we develop a skein theory
of toroidal grid diagrams. We then give sufficient criteria for an invariant of links in lens
spaces to bound from above the self-linking number of a transverse link in the universally
tight contact structure. Equivalently, this means that the invariant bounds the sum of the
Thurston-Bennequin number and absolute value of the rotation number, of a Legendrian
representative of the link. We then derive an invariant from the HOMFLY polynomial that
satisfies these criteria.

1.1

Summary of results

Throughout what follows we let p, q be coprime integers with 0 ≤ |q| < p. The lens space
p
which results as − q surgery on the unknot in S 3 is denoted by L(p, q) (note that this excludes
the case S 1 × S 2 ). For each L(p, q) we construct a collection of links called trivial links.
There is exactly one trivial link in each homotopy class of links. We prove the following.
Theorem 1.1.1. Let L be the set of isotopy classes of oriented links in L(p, q) and let
T L ⊂ L denote the set of isotopy classes of trivial links. Define T L ∗ ⊂ T L to be
2

those trivial links with no null-homotopic components. Let U be the isotopy class of the
standard unknot, a local knot in L(p, q) that bounds an embedded disk. Suppose we are
given a value Jp,q (τ ) ∈ Z[a±1 , z ±1 ] for every τ ∈ T L ∗ . Then there is a unique map
Jp,q : L → Z[a±1 , z ±1 ] such that
(i) Jp,q satisfies the skein relation
a−p Jp,q (L+ ) − ap Jp,q (L− ) = zJp,q (L0 ).
(ii) Jp,q (U ) = a−p+1 .
(iii) Jp,q (U

−p −ap
L) = a z
Jp,q (L).

As usual, the links L+ , L− , and L0 differ only in a small neighborhood. The exact
construction of these links in L(p, q) is made clear in the subsequent text.
Remark 1.1.2. In a large class of Q-homology spheres, Kalfagianni has found a power series
valued invariant of framed links that satisfies the Kauffman skein relation [32]. The ideas in
the proof of Theorem 1.1.1 should also be capable of showing that this invariant provides a
Kauffman polynomial for links in L(p, q).
We also give criteria for a Q-valued invariant of links in L(p, q) to bound the classical
invariants in (L(p, q), ξU T ), where ξU T is a universally tight contact structure on L(p, q)
defined by the pushforward of the standard contact structure. Our result is a lens space
analogue of a theorem of Lenny Ng [41]. To a given trivial link τ let T (τ ) be a particular
transverse representative of τ defined below, in Remark 4.1.5.
Theorem 1.1.3. Let i be a Q-valued invariant of oriented links in L(p, q) such that

i(L+ ) + 1 ≤ max i(L− ) − 1, i(L0 )

and
3

i(L− ) − 1 ≤ max i(L+ ) + 1, i(L0 ) ,

where L+ , L− , and L0 are oriented links that differ as in the skein relation. If slQ (T (τ )) ≤
−i(τ ) for every trivial link τ in L(p, q), then for every link L in L(p, q),

slQ (Lt ) ≤ −i(L),

where Lt is any transverse representative in (L(p, q), ξU T ) of L. Moreover, if Ll is a Legendrian representative of L then

tbQ (Ll ) + rotQ (Ll ) ≤ −i(L).

In order to prove Theorem 1.1.3 we provide explicit formulae to calculate the invariants
tbQ , rotQ , and slQ from a projection of the link to a Heegaard torus. These formulas are in
the spirit of those used to compute the classical invariants in (S 3 , ξst ) from a front projection
(see [16]). Moreover, combining the formula for tbQ with a result of Baker and Grigsby [4],
we find a very short proof of a result of Fintushel and Stern [18], that if integral surgery on
a knot in L(p, q) yields S 3 , then ±q is a quadratic residue mod p.
Given a link K in L(p, q), the polynomial Jp,q (K) is a two-variable polynomial in variables a and z, and its definition depends on a normalization on the set of trivial links. The
following is a corollary of Theorem 1.1.3.

Corollary 1.1.4. Let Jp,q denote the HOMFLY polynomial invariant in L(p, q), normalized so that if τ is a trivial link with no nullhomotopic components, or is the unknot, then
Jp,q (τ ) = a

p·slQ (T (τ ))+1

. Given an oriented link L in L(p, q), set e(L) to be the lowest
4

degree in a of Jp,q (L). If Lt is a transverse representative of L in (L(p, q), ξU T ), then

slQ (Lt ) ≤

e(L) − 1
.
p

The Franks-Williams-Morton (FWM) inequality is a celebrated result that relates the
braid index and algebraic crossing number of a braid to the HOMFLY polynomial of its
closure (which is a link in S 3 ). We point out that Corollary 1.1.4 is a generalization of this
result to lens spaces. For more details, see the discussion of the FWM inequality below.
Finally, we pinpoint the maximum self-linking and maximum Thurston-Bennequin numbers
of trivial knots as a corollary of Corollary 1.1.4.
Corollary 1.1.5. If τ is a trivial link in L(p, q), then T (τ ) has maximal self-linking number
among all transverse representatives of τ . If τ is a trivial knot, then the Legendrian knot
associated to its grid number one diagram has maximal Thurston-Bennequin number.
The thesis is organized as follows: In Chapter 2 we give definitions and background that
are fundamental to the study of polynomial invariants, and of Legendrian and transverse
links. We also review the work of Kalfagianni and Lin that paves the way for Theorem 1.1.1.
We end the chapter with a discussion of toroidal grid diagrams and their correspondence
to Legendrian links. In Chapter 3 we consider homotopies of links in L(p, q) and develop
a skein theory of toroidal grid diagrams. Using these tools, we prove the existence of the
HOMFLY-PT link polynomial for links in L(p, q). In Chapter 4 we review constructions of
[7],[4] that extend classical invariants of Legendrian and transverse knots to the rationally
null-homologous setting. We then give formulae for the Thurston-Bennequin, rotation, and
5

self-linking numbers from data given by a projection adapted to a grid diagram. In that
same chapter we prove Theorem 1.1.3 and Corollaries 1.1.4 and 1.1.5. We end the chapter
with a computation, giving a sequence of Legendrian knots and links in (L(5, 1), ξU T ) on
which the FWM inequality is sharp and arbitrarily negative. Finally, in Chapter 5 we discuss
fibered knots in certain lens spaces L(p, q) that are the cores of surgery on two families of
Berge knots. In particular, we show that many of these knots do not support the universally
tight contact structure on L(p, q).

6

Chapter 2
Definitions and Background

2.1

Skein Polynomials

For our study of oriented knots and links in a lens space it is requisite that we consider
n-singular links. Let M be an oriented 3-dimensional manifold, and let S be a disjoint
union of oriented circles. A piecewise-linear (or alternatively, smooth) map K : S → M is
an n-singular link if it has exactly n transverse double points, and is an embedding away
from these points. We often write K for the image K(S). Two n-singular links K1 , K2
are equivalent if there is an ambient isotopy M × I → M , taking K1 to K2 , such that
the double points remain transverse throughout the isotopy. We sometimes say that K1
and K2 are “smoothly” isotopic, to distinguish such isotopy classes from more restrictive
notions of equivalence, such as Legendrian and transverse isotopy (defined below). Note that
a 0-singular link is a link in M , and is a knot if the domain S is connected.
Much of classical knot theory (the study of knots and links in S 3 ) benefits from the
ability to discretize isotopies. That is, two links K1 , K2 in S 3 are isotopic if and only if any
7

regular projection of K1 to the equatorial plane can be taken to a regular projection of K2
via a finite sequence of three types of local changes to the projection, known as the three
Reidemeister moves.
Given any n-singular link K, give S the structure of a simplicial complex so that, for
each double point x in K, there are distinct 1-simplexes σ1 , σ2 ⊂ S with the interior of σi
containing one of the points in K −1 (x). We may assume that K(σ1 ) ∪ K(σ2 ) is contained
in an embedded disk D ⊂ M with K(∂σ1 ) ∪ K(∂σ2 ) ⊂ ∂D. Moreover, since σi inherits an
orientation from that of S, we may refer to the initial and terminal points of σi .
Let x ∈ M be a double point of K. Subdividing the simplicial structure on S if necessary,
there is a small ball neighborhood B ⊂ M of x such that D is properly embedded in B and
K ∩ B = K(σ1 ) ∪ K(σ2 ). Define a1 and a2 to be two simple arcs in distinct components of
∂B \ ∂D, going from the initial point to the terminal point of K(σ1 ). Define b1 and b2 to
be simple arcs on ∂D with b1 going from the initial point of K(σ1 ) to the terminal point of
K(σ2 ) and b2 going from the initial point of K(σ2 ) to the terminal point of K(σ1 ).
We define three (n − 1)-singular links from K in the following manner:

K+ = K(S \ σ1 ) ∪ a1 ;
K− = K(S \ σ1 ) ∪ a2 ;

(2.1.1)

K0 = K(S \ (σ1 ∪ σ2 )) ∪ (b1 ∪ b2 ).

Remark 2.1.1. Note that a different choice of a1 and a2 would interchange K+ and K− .
If M is oriented, this ambiguity can be dealt with. Since x is a transverse double point,
consider the ordered pair of tangent vectors K (σ1 ), K (σ2 ) at x. There is a unique vector
normal to D at x that completes this ordered pair to an oriented frame that agrees with the
8

orientation on B at x. Such a vector points into one of the components of B \ D. Take a1
to be in the same component.
The first motivation for discussing n-singular links is to define a skein triple of links.
Definition 2.1.2. Let K1 , K2 , K3 be any three links in M that are isotopic to K+ , K− ,
and K0 respectively, for some 1-singular link K. Then (K1 , K2 , K3 ) is called an oriented
skein triple in M . Given a 3-manifold M , let L denote the set of isotopy classes of (oriented)
links in M . A (Laurent) polynomial invariant of oriented links in the variables x1 , . . . , xm
±1
is a well-defined map J : L → Z[x1 , . . . , x±1 ]. A polynomial invariant of oriented links
m
J is called a skein polynomial if there is a homogeneous degree 1 polynomial F (X, Y, Z)
with coefficients in Z[x±1 , . . . , x±1 ] such that for any skein triple (K+ , K− , K0 ), we have
m
1
F (J(K+ ), J(K− ), J(K0 )) = 0.
Skein polynomials in the setting of S 3 include the Alexander, Jones, and HOMFLY-PT
polynomials. We note that the Alexander and Jones polynomials are one-variable specializations of the two-variable HOMFLY-PT polynomial. Furthermore, all mentioned skein
polynomials can be computed from regular projections.
The linear polynomial F (X, Y, Z) in the definition of a skein polynomial is often called the
skein relation for that polynomial invariant. As our focus is on the HOMFLY-PT polynomial,
we will use this skein relation as an example. Note that the HOMFLY polynomial was
introduced in [20] (see also [50]), and is determined by its skein relation, along with a choice
of normalization on the unknot. As observed by Ocneanu, the HOMFLY polynomial can
also be expressed using a trace of representations into Hecke algebras (see also [28]).
Let K+ , K− , K0 be three links in S 3 which admit regular projections that are identical
outside a neighborhood of some crossing, and that differ as in Figure 2.1 near the crossing.
9

L+

L−

L0

Figure 2.1: Projections in a skein relation

Then (K+ , K− , K0 ) is a skein triple. The HOMFLY-PT oriented link polynomial J :
L → Z[v ±1 , z ±1 ] satisfies the skein relation

v −1 J(K+ ) − vJ(K− ) = zJ(K0 ),

(2.1.2)

(that is, F (X, Y, Z) = v −1 X − vY − zZ) and is usually normalized so that J(unknot) = 1.
In fact, using this normalization on the unknot, we can use induction to see that any
oriented invariant of links satisfying (2.1.2) must be Laurent polynomial-valued. Indeed,
consider a regular projection D of a link K in S 3 and let c(D) be the set of crossings of D.
There is a subset of c(D) such that if we change which strand is over-crossing at each crossing
in this subset, we obtain a projection of the standard unlink. Let u(D) be the number of
crossings in the smallest such subset. Suppose D has a positive crossing in this subset, as on
the left in Figure 2.1 (the case where the subset has only negative crossings being similar).
Then we may consider K as K+ in a skein triple (K+ , K− , K0 ). Further, c(D0 ) < c(D+ )
and u(D− ) < u(D+ ), where D = D+ and D− , D0 are diagrams of K− , K0 as shown in
Figure 2.1. Thus we may inductively assume that J(K− ), J(K0 ) are Laurent polynomials,
noting that (2.1.2) implies

J

L

=
10

v −1 − v
J (L) .
z

Our proof that there exists a HOMFLY-PT polynomial for links in a lens space uses a
complexity function argument, as we did in the previous paragraph. We define a complexity
function that behaves nicely with respect to skein triples. Then we deduce that an invariant
of links JM found by Kalfagiannin and Lin [33], which satisfies the HOMFLY-PT skein
relation, takes Laurent polynomial values in the setting of a lens space. By construction,
JM takes values in a ring of power series, but it was not known to converge to a polynomial
invariant. It is still an open problem whether JM is Laurent polynomial-valued when M is
not a lens space (see [31]).
The construction of JM , which we will return to in Section 2.3, uses Vassiliev (or finitetype) link invariants. Note that, given any invariant of (n − 1)-singular links (in particular,
beginning with an honest link invariant), we can define an invariant of n-singular links by
setting F (L× ) = F (L+ ) − F (L− ). Thus any invariant of n-singular links gives an invariant
of m-singular links for m > n. A link invariant is called a Vassiliev invariant of order ≤ n
if the derived invariant vanishes on any n + 1-singular link, but does not on some n-singular
link.

2.2

Contact structures on 3-dimensional manifolds

Contact structures are a natural geometric structure that may be studied on any odddimensional manifold. We focus on the setting of a 3-dimensional manifold Y . In this
case, a contact structure ξ on Y is a rank 2 subbundle of T Y that does not agree locally (as
subbundles) with the tangent bundle of any embedded surface in Y . If ξ is oriented then
there is a global 1-form α satisfying α ∧ dα > 0 such that ξ = ker α. The pair (Y, ξ) is
called a contact 3-manifold. Two contact 3-manifolds (Y1 , ξ1 ), (Y2 , ξ2 ) are contactomorphic
11

if there is a diffeomorphism Y1 → Y2 which carries ξ1 to ξ2 .
An embedded disk in (Y, ξ) is overtwisted if its tangent bundle restricted to its boundary
agrees with ξ. A contact structure ξ is overtwisted if (Y, ξ) contains an overtwisted disk
and ξ is tight otherwise. The property of ξ being tight or overtwisted is clearly invariant
˜
under contactomorphism. Finally given any map of 3-manifolds, ϕ : Y → Y , and a contact
structure ξ on Y , one can take the pullback bundle ϕ∗ ξ, and this will be a contact structure
˜
˜
on Y . If Y is the universal cover of Y and ϕ∗ ξ is tight, then ξ is called universally tight.
If K ⊂ Y is a link and, for some contact structure ξ on Y , K is tangent to ξ at each
point, then K is called a Legendrian link in (Y, ξ). Two Legendrian links are Legendrian
isotopic if there is an isotopy from one to the other, through Legendrian links. A link K
in Y is called transverse if T K is transverse to ξ at each point of the image of K, and two
such links are transversely isotopic if there is an isotopy through transverse embeddings from
one to the other. Note that (assuming ξ is oriented) a transverse link comes with a natural
orientation.
A fundamental example of a contact 3-manifold is (R3 , ξst ), where R3 has coordinates
(x, y, z) and ξst = ker(dz − ydx). Projection to the xz-plane in (R3 , ξst ) is called front
projection, and the front projection of a Legendrian link has a particularly nice form since
the y-coordinate can be recovered from the slope in the xz-plane of the front projection. As
a consequence, a diagram of a link in the xz-plane is the front projection of some Legendrian
link in (R3 , ξst ) if and only if the only singular points of the projection are semi-cubical
cusps and transverse double points, the projection has no vertical tangencies, and at a
double point the strand with more negative slope is the over-crossing strand. An example is
given in Figure 2.2.
12

Figure 2.2: The front projection of a Legendrian figure 8 knot

In addition to the underlying smooth isotopy class of a Legendrian link K, there are
two invariants of Legendrian links called the classical invariants of K. These invariants are
traditionally defined when K is nullhomologous in Y . The first is the Thurston-Bennequin
number tb(K), which measures the framing induced from the contact planes, relative to the
framing given by a Seifert surface for K (a connected oriented surface, properly embedded
in the link complement, with boundary equal to K). That is, consider the simple closed
curve K on the boundary of a regular neighborhood of K which is determined by a nonzero section in the (real) line bundle ξK ∩ ν, where ν is the normal bundle of K. Choose a
Seifert surface Σ for K. Then tb(K) is the algebraic intersection Σ · K . The second classical
invariant is the rotation number rot(K), which is defined as the winding number of T K after
a trivialization of ξΣ , given a Seifert surface Σ for K. We note that rot(K) actually depends
on the homology class of Σ, but not on the chosen trivialization. However, as this thesis will
only work in the setting of lens spaces, rot(K) will only depend on K.
In (R3 , ξst ), one can compute the Thurston-Bennequin number and rotation number of
an oriented Legendrian link from its front projection in the following way. Let P be a front
projection of a Legendrian link K, let w(P ) be the writhe of P , and c(P ) the number of
cusps in the projection. Then, using the fact that the vector field ∂/∂z is transverse to
ξst everywhere, one can show that tb(K) = w(P ) − 1 c(P ). Also, given an orientation,
2
13

let cd (P ) (resp. cu (P )) be the downward (resp. upward) oriented cusps, then rot(K) =
1
2 (cd (P ) − cu (P )). If K is the Legendrian knot with front projection shown in Figure 2.2,
then tb(K) = −3 and, given either orientation, rot(K) = 0.
Transverse links have one classical invariant other than the underlying smooth link type.
If K is a transverse link, this invariant is called the self-linking number of K and written
sl(K). The self-linking number is defined as follows: given a Seifert surface Σ for K, since
one can trivialize ξΣ , we can choose a non-vanishing vector field v in ξΣ . Normalizing v
determines a simple closed curve K on the boundary of a regular neighborhood of K, since
K is transverse to ξ. Define sl(K) to be the linking of K with K , i.e. Σ · K .
Our study is that of links in a lens space, which of course need not be nullhomologous. However, the definitions given above can be extended to the setting of a rationally
nullhomologous link with the help of rational Seifert surfaces (defined below), which are a
generalization of a Seifert surface and are embedded in the exterior of the link.
There is an operation one can perform on a Legendrian knot K in any contact 3-manifold
called positive (resp. negative) stabilization on K that gives a different Legendrian knot
denoted S + (K) (resp. S − (K)). To describe the operation, we point out that a theorem of
Darboux states that given a point p in a contact 3-manifold (Y, ξ), there exists a neighborhood
U of p, such that there is a contactomorphism ϕ : (U, ξ|U , p) → (R3 , ξst , 0). If K is
Legendrian in (Y, ξ) then suppose p is a point on K. Suppose that Figure 2.3(a) is the front
projection of ϕ(U ∩ K). Then S + (K) is defined by replacing U ∩ K with the image of Figure
2.3(b) under ϕ−1 , and S − (K) is defined by replacing U ∩ K with the image of Figure 2.3(c)
under ϕ−1 . The Legendrian isotopy type of S ± (K) does not depend on the choice of point
p or contactomorphism ϕ.
14

(a)

(b)

(c)

Figure 2.3: Positive and negative stabilization of K

2.3

Review of known results

In this section we review what is known about the problem of finding skein polynomials in
manifolds other than S 3 , in addition to results that relate invariants of smooth isotopy of
knots to the classical invariants of their Legendrian and transverse realizations. We will also
introduce one of the main tools for our study of links in L(p, q), toroidal grid diagrams, and
their correspondence to Legendrian links in L(p, q) with a universally tight contact structure.

2.3.1

Power series invariants in Q-homology 3-spheres

Original constructions of the HOMFLY polynomial have been difficult to reproduce in manifolds other than S 3 , as they rely on planar link projections. However, it was shown in
[12] that the HOMFLY and Kauffman polynomials are generating functions for particular
sequences of Vassiliev (or finite-type) link invariants (see also [8],[9], and [34]). In [30] and
[33], Kalfagianni and Lin considered Vassiliev invariants of links in (certain) Q-homology
spheres in their effort to find a HOMFLY polynomial. They succeeded in finding an invariant that takes values in a ring of power series.
Kalfagianni and Lin begin by considering homotopies of links in Q-homology spheres. In
particular, they show that any free homotopy of a link in M can be perturbed slightly to a
particularly nice form called almost general position. Among other properties, a homotopy in
almost general position only fails to be isotopy at finitely many moments, at which times one
15

has a 1-singular link, and near these times one passes from one resolution of the 1-singular
link to another (e.g. L+ to L− ).
We described above how singular link invariants may be derived from a link invariant.
Kalfagianni and Lin give an “integrability condition” in [33], determining when a given
1-singular link invariant is derived from a link invariant. Their theorem placed certain
restrictions on the ambient manifold. Recently Kalfagianni [32] was able to remove some of
these restrictions, and we cite her theorem here, adapted to the setting of unframed links.
Recall that an atoroidal 3-manifold M is one such that π2 (M ) = 0 and M has no essential
tori. Let L (1) denote the isotopy classes of 1-singular links in M .
Theorem 2.3.1 ([32]). Suppose that M is a Q-homology sphere with π2 (M ) = 0, such that
if H1 (M ) = 0 then M is atoroidal. Also assume that R is a ring which is torsion free as an
abelian group and let f : L (1) → R be an invariant of 1-singular links. Then there exists a
link invariant F : L → R so that

f (L× ) = F (L+ ) − F (L− )

for any 1-singular link L× if and only if

f (∝) = 0
f (L×+ ) − f (L×− ) = f (L+× ) − f (L−× ),

where L×± and L±× denote the four 1-singular resolutions of any 2-singular link, and ∝
denotes a 1-singular link with a “small” loop near some double point.
Almost general position of homotopies was an essential ingredient in proving Theorem
16

2.3.1. One easily checks that some f derived from a link invariant F will satisfy the two
conditions of the theorem. Given a singular link invariant satisfying these conditions, one
takes a homotopy in almost general position from a trivial link in T L to the given link and
sums the values of the singular invariant at the finite singular moments in the homotopy.
To see this is well-defined one must show that the definition is independent of the homotopy
used, or that a “global integrability” condition holds.
It is known [29] that the 2-variable HOMFLY polynomial for links in S 3 is equivalent to a
sequence of 1-variable Laurent polynomials {Jn (t)}, and that replacing t by ex gives a power
series invariant Jn (x) with coefficients that are Vassiliev invariants [8, 12]. Kalfagianni and
Lin use these ideas in [33] to arrive at Theorem 2.3.2 below. They define their invariant JM
by defining inductively a sequence of Vassiliev invariants whose generating function is a power
series satisfying the HOMFLY-PT skein relation. The ith coefficient of this power series is
defined from a singular link invariant, which they show “integrates” to a link invariant by
showing it satisfies the conditions of Theorem 2.3.1.
The following theorem is then obtained. Let U be the isotopy class of a knot in M that
is the standard unknot in a small ball neighborhood of a point of M . Fix T L , a collection
of links in M such that there is exactly one representative in T L for each homotopy class
of links in M , and furthermore, if T L ∈ T L has k components that are homotopically
trivial then T L = L

k U for some link L that has no homotopically trivial components.

Define R := C[[x, y]] to be the ring of formal power series in the variables x and y over
y3
y2
C. Define v ∈ R by v := ey = 1 + y + 2 + 6 + · · · and let z ∈ R be defined by
3
5
z := ex − e−x = 2x + x + x + · · · .
3
60

Theorem 2.3.2 ([33]). Let M be a Q-homology 3-sphere with π2 (M ) = 0, such that if
17

H1 (M ) = 0 then M is atoroidal. Let L be the set of isotopy classes of links in M . Then,
−1
given values JM (T ) for each T ∈ T L such that JM (T U ) = v z −v JM (T ), there is a
unique map JM : L → R such that

v −1 JM (L+ ) − vJM (L− ) = zJM (L0 ).

Given M as above, it is unknown whether JM can be made to take values that are
Laurent polynomials. In this consideration, Kalfagianni asked the following question [31]
which encapsulates the primary difficulty of the problem, and which we answer affirmatively
when M is a lens space:
Question 1. Is there a choice of T L ∗ such that every link L ⊂ M can be reduced to
disjoint unions of unlinks and elements in T L ∗ by a series of finitely many skein moves?

2.3.2

Bennequin type inequalities

Much effort has gone into finding upper bounds for the classical invariants in (S 3 , ξst ), where
ξst is the standard tight contact structure on the 3-sphere, e.g. [10], [19, 37], [51], [52], [48],
[49], [55], [40], [58]. Much of this work benefits from the fact that the classical invariants of
Legendrian/transverse knots in (S 3 , ξst ) can be computed easily from a front projection.
Less is known about these invariants of Legendrian knots in other contact manifolds. A
theorem of Eliashberg [15] generalizes the Bennequin inequality for null-homologous Legendrian knots in any 3-manifold with a tight contact structure:
Theorem 2.3.3 (Eliashberg-Bennequin inequality). Let ξ be a tight contact structure on a
18

3-manifold, Y . If K is a null-homologous knot in Y and F is a Seifert surface for K, then

tb(Kl ) + rotF (Kl ) ≤ 2g(F ) − 1

(2.3.1)

for any Kl , a Legendrian representative of K.

This bound can be improved in some settings. Lisca and Matic improved the bound in the
case that the contact structure is Stein fillable [36], and this improvement was extended to
the setting of a tight contact structure with non-vanishing Seiberg-Witten contact invariant
by Mrowka and Rollin [39]. An analogous theorem was proved by Wu [57] for the Ozsv´tha
Szab´ contact invariant.
o
These improvements involved replacing the Seifert genus in the Eliashberg-Bennequin
inequality with the genus of a surface which is properly embedded in a 4-manifold bounded
by Y . As such bounds involve the negative Euler characteristic of a surface with boundary
K, they must be no less than -1. In [25], Hedden introduced an integer τξ (K) that is defined
via the filtration on knot Floer homology associated to (Y, [F ], K), where [F ] is the homology
class of a Seifert surface for K. He showed that in the case that the Ozsv´th-Szab´ contact
a
o
invariant is non-zero, the right side of (2.3.1) can be replaced by 2τξ (K) − 1. With such a
bound he showed that for any contact manifold with non-zero contact invariant, there exist
prime Legendrian knots with arbitrarily negative classical invariants.
In another direction, one could consider rationally null-homologous knots in a contact
manifold (Y, ξ). In such a setting there is a notion of rational Seifert surface and corresponding classical invariants tbQ , rotQ , and slQ (see Definition 4.1.2 below). Baker and Etnyre
[7] extend the Eliashberg-Bennequin inequality to this setting:
19

Theorem 2.3.4. Let (Y, ξ) be a contact 3-manifold with ξ a tight contact structure. Let K
be a knot in Y with order r > 0 in homology and let Σ be a rational Seifert surface for K.
Then for Kt , a transverse representative of K,
1
slQ (Kt ) ≤ − χ(Σ).
r

Moreover, if Kl is a Legendrian representative of K then
1
tbQ (Kl ) + rotQ (Kl ) ≤ − χ(Σ).
r

There is an inequality found by Franks and Williams [19], and independently by Morton
[37], that relates the index and algebraic crossing number of a braid to a degree of the HOMFLY polynomial of its closure. Later, using the Bennequin’s work, Fuchs and Tabachnikov
reinterpreted the result in terms of the self-linking number of a transverse knot in (S 3 , ξst )
[21]. This inequality has come to be known as the Franks-Williams-Morton (FWM) inequality.
More precisely, we describe the FWM inequality as follows. The HOMFLY polynomial
J(K) is a polynomial invariant of links in the variables v, z such that if U ⊂ S 3 is the unknot
then J(U ) = 1, and J satisfies

v −1 J(K+ ) − vJ(K− ) = zJ(K0 ),

where K+ , K− , and K0 form a skein triple.

20

(2.3.2)

Theorem 2.3.5 (Franks-Williams-Morton (FWM) inequality). Let e(K) denote the minimum degree of v in J(K). Then for any transverse representative Kt of K,

sl(Kt ) ≤ e(K) − 1.

Moreover, if Kl is a Legendrian representative of K then

tb(Kl ) + rot(Kl ) ≤ e(K) − 1.

As discussed in the introduction, Theorem 1.1.1 gives a HOMFLY polynomial for links
in lens spaces, and Corollary 1.1.4 extends the FWM inequality to universally tight contact
lens spaces.

2.3.3

Links and grid diagrams in L(p, q)

In this section we review toroidal grid diagrams in L(p, q), which are central to our discussion
and view of links in lens spaces, and were fully studied in [4] (see also [5]). We will also
(briefly) review the correspondence between Legendrian links in L(p, q) with a universally
tight contact structure and toroidal grid diagrams. Let us begin with the definition of that
contact structure.
p
The manifold L(p, q), which is the result of − q surgery on the unknot, is also obtained as a
2πi
q
quotient of S 3 ⊂ C2 by the equivalence relation (u1 , u2 ) ∼ (ωp u1 , ωp u2 ), where ωp = e p .
Let π : S 3 → L(p, q) be the quotient map.
Represent points (u1 , u2 ) of S 3 in polar coordinates, letting ui = (ri , θi ). The kernel ξst
21

2
2
of the 1-form α = r1 dθ1 + r2 dθ2 is the unique (up to orientation) tight contact structure on
S 3 [22]. The 1-form α is constant along any torus in S 3 determined by a fixed r1 . Since such
q
a torus is fixed (not pointwise) under the action (u1 , u2 ) → (ωp u1 , ωp u2 ), the pushforward
ξU T = π∗ (ξst ) is a well-defined, and is clearly a contact structure on L(p, q). Since ξU T
pulls back to the standard contact structure on S 3 , which is tight, ξU T is universally tight.
The points of L(p, q) can be identified with points in a fundamental domain of the cyclic
action on S 3 . Thus, since r2 is determined by r1 in S 3 , we can describe L(p, q) by

L(p, q) =

2π
(r1 , θ1 , θ2 ) | r1 ∈ [0, 1], θ1 ∈ [0, 2π) , θ2 ∈ 0,
p

.

Analogous to the correspondence between planar grid diagrams and Legendrian links in
(S 3 , ξst ) ([42],[53]), we can define toroidal grid diagrams in L(p, q) to get a correspondence
between grid diagrams and Legendrian links in (L(p, q), ξU T ). To be precise we define grid
diagrams in L(p, q) as follows (cf. [4]).

Definition 2.3.6. A grid diagram D with grid number n in L(p, q) is a set of data (T, α, β, O, X),
where:
• T is the oriented torus obtained via the quotient of R2 by the Z2 lattice generated by
(1, 0) and (0, 1).
i
• α = α0 , . . . , αn−1 , with αi the image of the line y = n in T . Call the n annular
components of T − α the rows of the grid diagram.
p
i
• β = β0 , . . . , βn−1 , with βi the image of the line y = − q (x − pn ) in T . Call the n
annular components of T − β the columns of the grid diagram.
22

• O = O0 , . . . , On−1 is a set of n points in T − (α ∪ β) such that no two Oi ’s lie in
the same row or column.
• X = X0 , . . . , Xn−1 is a set of n points in T − (α ∪ β) such that no two Xi ’s lie in
the same row or column.
The components of T − α − β are called the fundamental parallelograms of D and the
points O∪ X are called the markings of D. Two grid diagrams with corresponding tori T1 , T2
are considered equivalent if there exists an orientation-preserving diffeomorphism T1 → T2
respecting the markings (up to cyclic permutation of their labels).
Such a grid diagram has “slanted” β curves. For considerations of both convenience and
aesthetics, we alter the fundamental domain of T and “straighten” our pictures so that the
β curves are vertical. Figure 2.4 shows how this “straightening” is accomplished.
α0

fundamental parallelogram
z

z
α1

o

x

o

(A)

x

(B)
x

o

x

z

o

z
β0

β1

z
o

(C)

x

x

o

z
Figure 2.4: (A) shows a grid diagram (with grid number 2) in L(7, 2) on a fundamental
domain of T . In (B) we alter the fundamental domain. (C) is the “straightening” of (B). For
interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this dissertation.
A link K ⊂ L(p, q) is associated to a grid diagram D in L(p, q) in the following manner.
23

√
Let Σ be the torus in L(p, q) of constant radius r1 = 1/ 2 which splits L(p, q) into two solid
tori V α and V β . Identify T with −Σ such that the α-curves of D are negatively-oriented
meridians of V α and the β-curves are meridians of V β . Next connect each X to the O in its
row by an “horizontal” oriented arc (from X to O) that is embedded in T and disjoint from
α. Likewise, connect each O to the X in its column by a “vertical” oriented arc embedded
in T and disjoint from β. The union of the 2n arcs makes a multicurve γ. Remove selfintersections of γ by pushing the interiors of horizontal arcs up into V α and the interiors of
vertical arcs down into V β .
We note that in the association of a link to a grid diagram in L(p, q) the author used
the opposite convention as that adopted in other places in the literature [4, 5, 45, 47]. The
convention used in other places in the literature was adopted to fit conventions coming from
knot Floer homology. However, for the purposes of this thesis, it is more clear to use the
approach presented below (as in [13]) as there is no reference to Floer homology theories.
Definition 2.3.7. Let K be a link associated to a grid diagram D in L(p, q) with grid
number n. For some 0 < m < n, suppose D is a subcollection of m rows and m columns
of D such that the 2m markings contained in the rows of D are exactly the 2m markings
contained in the columns of D . Then D is a grid diagram for some sublink of K. If this
sublink has one component then D is called a component of D.
Remark 2.3.8. No part of Definition 2.3.6 prohibits a marking in X and a marking in O
from being in the same fundamental parallelogram. To a grid diagram that has grid number
one (and so, only one marking in X and one marking in O) and its two markings in the same
fundamental parallelogram, we associate a knot in L(p, q) that is contained in a small ball
neighborhood and bounds an embedded disk.
24

Remark 2.3.9. Except for the case described in Remark 2.3.8, we assume that each marking
of D is the center point of the fundamental parallelogram that contains it. Let the straightened fundamental domain of T have normalized coordinates {(θ1 , θ2 ) | θ1 ∈ [0, p), θ2 ∈ [0, 1)} ,
so that each O and X sharing the same column have the same θ1 -coordinate mod 1.
Under the requirements of Remark 2.3.9, the projection of K to −Σ (with vertical arcs
crossing under horizontal arcs) is called a grid projection associated to DK (the authors of
[4] call this a rectilinear projection). Note that K has an orientation given by construction
and so the grid projection is also oriented. Figure 2.5 shows an example of a grid diagram
with a corresponding grid projection.
Given a grid diagram in a lens space L(p, q) (with the identification of T to −Σ), the basis
of vectors given by parallel translates of tangent vectors to {α0 , β0 } is coherently oriented
with the global frame {d/dθ1 , d/dθ2 }.
The analogue to front projections in this setting, called toroidal front projection, projects
radially through the tori V α and V β to Σ. A slight perturbation of a grid projection gives
a toroidal front projection, which determines a Legendrian link in (L(p, q), ξU T ) since the
slope dθ2 /dθ1 on the projection determines the radial coordinate r1 . The cusps of the front
projection correspond to lower-left and upper-right corners of the grid projection, and we
call these corners the cusps of a grid projection.
2

3
o

4

0

0

2
x

x

o

x
x
1

1

x

o
2

x

o

o

x
3

4

o

o
0

Figure 2.5: A grid diagram for L(5, 3) with corresponding grid projection.
25

If D has grid number n then there are 22n different grid projections as there are two
choices of vertical arc for each column, and two choices of horizontal arc for each row. In a
given row (resp. column), the difference in choice of horizontal (resp. vertical) arc corresponds
to a Legendrian isotopy across a meridional disk of V α (resp. V β ) (see [4]). So the Legendrian
isotopy class of the link is independent of the choice of grid projection.
In view of the correspondence above, Legendrian links in (L(p, q), ξU T ) can be discussed
via grid diagrams. There is a set of grid moves such that two grid diagrams correspond to
the same Legendrian link if and only if there is a sequence of such grid moves taking one
grid diagram to the other [4]. These moves come in two flavors: grid (de)stabilizations and
commutations.
X

O

x
xo

o

X

O

x

ox
o

Figure 2.6: Stabilization of type X: NW (top) and of type O: SW (bottom)

Grid Stabilizations and Destabilizations: Grid stabilizations increase the grid number by one and should be thought of as adding a local kink to the knot. They are named
with an X or O, depending on the type of marking at which stabilization occurs, and with
NW, NE, SW, or SE, depending on the positioning of the new markings. Figure 2.6 shows
an X:NW stabilization and an O:SW stabilization. Destabilizations are the inverse of a
stabilization. Any (de)stabilization is a grid move that preserves the isotopy type.
However, the correspondence between our grid diagrams and toroidal front projections is
such that cusps correspond to upper-right and lower-left corners of a grid projection. Only
26

(de)stabilizations of types NW and SE preserve Legendrian isotopy type.
5

6
x

0

0

1

2

x

o
x
1

o
2

3

4

o

5

x

4
x

5

6

0

6

5

0

3
o

0

1

2

x

o
x
1

o
2

3

4

3
o
o

4

5

x
5

6

0

Figure 2.7: A non-interleaving commutation in L(7, 2).
Commutations: A commutation interchanges two adjacent columns (or rows) of the
grid diagram. Let A be the annulus consisting of the two adjacent columns c1 , c2 (resp. rows
r1 , r2 ) involved in the commutation. This annulus is sectioned into pn segments of the n
rows (resp. columns) of the grid diagram. Let s1 , s1 be the two segments in A containing
the markings of c1 (resp. r1 ). If the markings of c2 (resp. r2 ) are contained in separate
components of A − s1 − s1 , the commutation is called interleaving. If they are in the
same component of A − s1 − s1 the commutation is called non-interleaving. We note that
in the literature a commutation typically refers only to what we call a non-interleaving
commutation. We have extended the terminology to include the interleaving case. A noninterleaving commutation of columns (resp. rows) is a grid move that preserves Legendrian
isotopy type [4]. An interleaving commutation corresponds to a crossing change (see Lemma
3.2.1). An example of non-interleaving commutation is shown in Figure 2.7.
Note that a commutation (interleaving or non-interleaving) does not include a column
exchange of the type illustrated in Figure 2.8, where there is a row containing markings of
both c1 and c2 .
We end the section with a construction that plays an important role in what follows.
27

x o

o x

Figure 2.8: A move which is neither an interleaving nor non-interleaving commutation.
Given a grid diagram D in L(p, q) define a grid diagram D of a link in S 3 as follows: cut T
along α0 to get an annulus A. The boundary of A is a disjoint union of two copies of α0 .
+
Let one of these copies be α0 and the other be
−
+
and glue α0 on Ai to α0 on Ai+1(mod p) for

−
α0 . Take p copies of A, say A0 , . . . , Ap−1
i = 0, . . . , p − 1 by the identity map. Note

that the torus constructed from A0 , . . . , Ap−1 covers T under the cover π : S 3 → L(p, q)
and so the link associated to D covers the link associated to D. Call D the lift of D to S 3 .
(An example is shown in Figure 2.9, where the grid diagram on the right is the lift of the
grid diagram in L(5, 1) on the left. The link corresponding to the lift is a Hopf link.)
0

1

2

3 4 0
x o
o
x o
o x
o
o x

x
4 0 1
o
o x
0 1 2

2
3

3 4
x
4

x
x

0

o
o

x

o

x

o x
0 1 2

D

3

4

0

D

Figure 2.9: D: the lift to S 3 of the grid diagram D in L(5, 1).

28

Chapter 3
Homotopies and the HOMFLY
polynomial

3.1

Homology classes

Definition 3.1.1. For an oriented link K in L(p, q), define µ(K) to be the homology class
of K in H1 (L(p, q)).
Remark 3.1.2. Let K, K be oriented links. We note that µ(K) = µ(K ) if and only if K
and K are freely homotopic, since the free homotopy class of an oriented knot in L(p, q) is
determined by its homology class.
Let C be the core of the handlebody Vα in L(p, q) (dual to a meridional disk of Vα ).
Then H1 (L(p, q)) ∼ Z/p is cyclically generated by γ = [C] ∈ H1 (L(p, q)). We identify µ(K)
=
with the multiple of γ it represents.
Suppose DK = (T, α, β, O, X) is a grid diagram with associated link K in L(p, q). Orient
the α and β curves so that the algebraic intersection αi · βj is positive on T , for all pairs
29

i, j. Let RK be a grid projection for DK with its given orientation. We can compute µ(K)
from the grid diagram as follows.
Lemma 3.1.3. Given DK and RK as above, choose an α-curve αi in the grid diagram.
Then µ(K) is equal (mod p) to αi · RK , the algebraic intersection of αi with RK .
Proof. Push the interiors of horizontal arcs on RK slightly into Vα to get a knot K contained
in Vα . The winding number of K in Vα is clearly counted by αi · RK and K is isotopic to
K, so µ(K ) = µ(K).

W

W

o

O

X

x

O

o
o x
x

o

x

X

Figure 3.1: A crossing change on a grid projection.

3.2

Skein Theory

We now develop a skein theory of grid diagrams in L(p, q). Note that grid diagrams have
been discussed in the setting of singular links in S 3 , and used to extend link Floer homology
to singular links [2]. What we describe here as grid diagrams that differ by a “skein crossing
30

change” are exactly the grid diagrams in [2] that correspond to the two resolutions of the
double point of a 1-singular link (see, for example, Figure 10 in [2]).
If two links L+ and L− differ as described in (2.1.1), they are isotopic to links with
associated grid diagrams differing by the grid move in Figure 3.1. The next lemma refines
this. In a figure of a grid diagram, slanted gray bars indicate some number of columns (and
their markings) which may be between the columns of interest.
Lemma 3.2.1. Let L, L be links corresponding to grid diagrams that differ by an interleaving
commutation. Then L, L differ by a crossing change.

Proof. We prove the statement for row commutation. The case with columns is similar.
The proof is given by the sequence of grid moves detailed in Figure 3.2, where arrows
indicate commutations to be performed, and a triple of markings that is to be destabilized
is circled. Referring to Figure 3.2, there are 8 steps. From step 1 to 2 we perform a crossing
change. From 2 to 5 we perform a number of non-interleaving commutations. Going from 5 to
6 is destabilization. From 6 to 7 involves several commutations. Each of these commutations
is non-interleaving since the markings of the column being moved to the right are in adjacent
rows. Finally we destabilize from 7 to 8.
Note that all grid moves above correspond to isotopy of the link, except the first which,
as in Figure 3.1, corresponds to interchanging the two resolutions of a singular link defined
by (2.1.1).

Lemma 3.2.1 shows that if two grid diagrams differ only by the interleaving commutation
of two adjacent columns (or rows), then the corresponding links are some pair L+ , L− .
Motivated by this result, we make the following definition.
31

1

2
X

O
X

o

X
x

O

3

o
o x
x

o

4
X
x

o

o

X

o
o x
x

x

o
ox

o

5

x

o

O
X

O

6
X
x

o
ox

o
x

X
o
X

destabilize

7
X
X

O

8

O
XO

X
X

O

O
O

Figure 3.2: 1 → 2: crossing change; 2 → 3: row commutation; 3 → 4: column commutation;
4 → 5: two row commutations; 5 → 6: destabilization; 6 → 7: many column commutations;
7 → 8: destabilization
Definition 3.2.2. A pair of adjacent columns in a grid diagram for L(p, q) is called a skein
crossing if their commutation is an interleaving commutation. Define a skein crossing change
to be the commutation of a skein crossing pair. A skein crossing is called positive if it appears
as in the left side of Figure 3.3 under some cyclic permutation of the rows and columns. It
is negative if (after some cyclic permutation) it appears as in the right side of Figure 3.3.
Given two grid diagrams that differ only by a skein crossing change, we call the diagram
with the positive skein crossing D+ and the diagram with the negative skein crossing D− .
There is a function on grid diagrams that is related to the idea of skein crossings. We
32

x

x

x

x
o

o

o
D+

o
D−

Figure 3.3: Positive and negative skein crossings.
will use this function later, both to define trivial links and to produce a complexity function
on grid diagrams that will prove Theorem 1.1.1.
Definition 3.2.3. Consider a (not necessarily adjacent) pair of columns c1 , c2 in a grid
diagram of L(1, 0) = S 3 . Let Ri (o) be the row in the diagram that contains the O marking
of ci and let Ri (x) be the row containing the X marking of ci . Call the columns c1 , c2
interleaving if R2 (o) and R2 (x) are in different annular components of T − (R1 (o) ∪ R1 (x)).
Note that if c1 , c2 are adjacent and interleaving then they comprise a skein crossing.
Given a grid diagram D for a link in L(p, q), define scr(D) to be the number of interleaving
pairs of columns in D, the lift of D to S 3 .
Note that, despite the name of the function, scr(D) does not count the number of skein
crossings of D. Instead, it counts all pairs of columns of D which, if they were adjacent,
would make a skein crossing. Let us point out a few important properties of the function
scr.
Proposition 3.2.4. Let D(L(p, q)) denote the set of all grid diagrams in L(p, q). The
function scr : D(L(p, q)) → Z satisfies the following:
1. (Orientation invariance) Given a grid diagram D, let rD denote the grid diagram
obtained by exchanging every O marking for an X marking and vice versa. Then
scr(D) = scr(rD).
33

2. (Column commutation invariance) Let D, D be two grid diagrams that only differ by
a column commutation. Then scr(D) = scr(D ). Note that this is not true for row
commutation.

Proof. The fact that orientation invariance holds is obvious.
To prove the second property, first note that an adjacent pair of columns in D will
correspond to p adjacent pairs of columns in D. Thus, if D and D differ by a column
commutation, then D and D differ by p column commutations. Given a column c of D, if
R(o) and R(x) are the rows that its markings occupy, then its markings are still in rows R(o)
and R(x) after column commutation. It is clear then from Definition 3.2.3 that the number
of columns in D that are interleaving with c is the same before and after a commutation
with a column adjacent to c.
We now consider how resolutions fit into the skein theory of grid diagrams. To begin, let
us define what we mean by such a resolution.
D+

D0

(D+ )0
x

x
o

o
R

c1

c2

c1

x
o

xo
c2

c1

c2

Figure 3.4: Resolution at a positive skein crossing
Given a grid diagram D+ and a positive skein crossing of D+ , choose one of the O
markings in the skein crossing. There exists some choice of grid projection that, near this
marking, looks like the leftmost projection in Figure 3.4 (or a 180◦ rotation of this picture).
Let L× be the 1-singular link that has one double point at the crossing depicted in this
34

projection, and is identical elsewhere to the link associated to D+ . We take the resolution
L0 of L× defined in (2.1.1).
This operation is local and we see that the link associated to the grid diagram (D+ )0 ,
pictured in the middle of Figure 3.4, is isotopic to L0 . Since columns c1 and c2 are adjacent in the diagram (D+ )0 any commutation of row R will be non-interleaving, and so an
isotopy. Therefore, we may commute this row until we have a triple of markings that can be
destabilized. After this destabilization we get the grid diagram D0 , as shown in Figure 3.4,
with an associated link that is isotopic to L0 . Note that the grid diagram D0 is obtained
from D+ by exchanging the θ1 coordinates of the O markings in the two columns.
Likewise, given a grid diagram D− , a negative skein crossing of D− , and an O marking
in the skein crossing, there is a grid projection of D− that looks like the leftmost projection
in Figure 3.5. We then define the resolution of this skein crossing similarly, getting the grid
diagram D0 depicted in Figure 3.5. In this case, we get the diagram D0 by interchanging
the X markings in the columns. Based on our observations, we make the following definition.
D−

(D− )0
x

x

o

ox

D0
x

o

o

R
c1

c2

c1

c2

c1

c2

Figure 3.5: Resolution at a negative skein crossing

Definition 3.2.5. Given a grid diagram D and a positive (resp. negative) skein crossing
of D, define D0 to be the grid diagram that differs from D only in the columns of the
skein crossing, where it differs only by interchanging the θ1 coordinates of the O (resp. X)
35

markings.
Remark 3.2.6. In Figure 3.4 (resp. 3.5), the grid diagram D0 is obtained by interchanging
the θ1 -coordinates of the O (resp. X) markings in the skein crossing. Note that if we change
the θ1 -coordinates of the X (resp. O) markings instead, the resulting diagram differs from
D0 by a non-interleaving commutation of columns, and so its associated link is also isotopic
to L0 . Finally, note that if D+ and D− differ only as shown, by a skein crossing change,
then the grid diagram D0 is obtained as a resolution of each, so D+ , D− , and D0 form a
skein triple.
In the course of the next section we will prove Theorem 1.1.1, and in the proof we will
need the function scr to decrease under a resolution D±

D0 . To see that scr behaves

this way, we use a map H from the columns of a grid diagram to the coordinate plane. The
map H is defined below and we point out some of its properties.
Let C (D) be the set of columns of a grid diagram D in L(1, 0) = S 3 and choose an
α-curve α0 . Define a map Hα0 : C (D) → R2 in the following way. Using the orientation
on β we can define a height function θ2 on the annulus T − α0 . For c ∈ C (D), let m
(resp. n) be the smaller (resp. larger) of the θ2 coordinates of the markings of c. We define
Hα0 (c) = (m, n). For convenience of notation, we will drop the subscript of Hα0 unless we
wish to emphasize the dependence of the map on the choice of α0 .
By our definition, if D has grid number n, then H(C (D)) is a set of n points in the plane
between the y-axis and the line y = x. Further the map only depends on the positions of the
markings in a column, so if rD is the grid diagram obtained from D by reversing orientation
(interchanging the X and O markings), then the image set H(rD) equals the image H(D).
Finally, we point out that if c is a column of D and p = H(c), it can be easily checked
36

that another column c is interleaving with c if and only if H(c ) is in one of the shaded
regions of the plane shown in Figure 3.6. Call these regions the interleaving regions of p.

p

Figure 3.6: The interleaving regions of p, a point in the image of H : C (D) → R2

Remark 3.2.7. Observe that the property of two columns being interleaving in a grid
diagram for a link in S 3 is completely independent of a preferred α-curve (Definition 3.2.3).
This implies that, although the map H depends on the choice of α0 , the property of H(c )
being in one of the interleaving regions of H(c) does not depend on such a choice.

3.3
3.3.1

The HOMFLY polynomial in L(p, q)
Trivial Links in L(p, q).

Fix a lens space L(p, q) with Heegaard torus Σ as described in Section 2.3. As mentioned in Remark 2.3.9, after a choice of α0 and β0 all grid diagrams are given coordinates
{(θ1 , θ2 ) | θ1 ∈ [0, p), θ2 ∈ [0, 1)}.
Given a grid diagram D, let GN (D) denote the grid number of D. Given a link K, set
37

GN (K) to be

GN (K) = min {GN (D) | D is a grid diagram for a link isotopic to K} .

5

6
O

0

0

1

2

3

4

5

2

3

4

5

6

0

X
1

Figure 3.7: The grid number 1 knot K3 in L(7, 2)

Definition 3.3.1. A knot K in L(p, q) is called a trivial knot if GN (K) = 1. If µ(K) = i,
we write K = Ki . Under this notation the unknot described in Remark 2.3.8 is a trivial
knot and written K0 .

Remark 3.3.2. Note that µ is only defined up to congruence class mod p. Let DK denote
i
the grid diagram with grid number one of the trivial knot Ki . Since p, q are coprime, there
exists some Ki for each 0 ≤ i ≤ p − 1. See Figure 3.7 for the diagram and a projection of
K3 in L(7, 2).

Having defined trivial knots we now define a collection of trivial links so that one component trivial links are trivial knots. Let I = (m0 , m1 , . . . , mp−1 ) be a p-tuple of non-negative
integers. We construct a grid diagram D(I) with grid number n =
38

mi as follows. Place

the markings O = {O1 , O2 , . . . , On } so that 0 < θ1 (Oi ) < 1 for each 1 ≤ i ≤ n, and so that
1
θ1 (Oi ) + = θ1 (Oi+1 )
n
1
θ2 (Oi ) − = θ2 (Oi+1 ).
n

and

for each 1 ≤ i ≤ n − 1. Note that these conditions force O1 to be in the left-uppermost
fundamental parallelogram; i.e. O1 has coordinates

1
1
2n , 1 − 2n .

In order to place the markings X = {X1 , X2 , . . . , Xn }, we define a bijection

σ : {0, 1, . . . , p − 1} → {0, 1, . . . , p − 1}

by the congruence σ(i) · q ≡ i mod p. Now, for each i such that

l−1 m
+1 ≤ i ≤
k=0 σ(k)

l
k=0 mσ(k) , with 0 ≤ l ≤ p − 1, let vi be the directed vertical arc (in the column of Oi )
that satisfies α0 · vi = σ(l) and has one endpoint at Oi and the other at the center of a
fundamental parallelogram that is in the same row as Oi . Place Xi at the other endpoint of
vi . We call the resulting grid diagram D(I). An example is shown in Figure 3.8 below.
Recall the definition of the components of a grid diagram (Definition 2.3.7). We first
note that every component of D(I) has grid number one, since for each 1 ≤ i ≤ n, Oi and
Xi are in the same row and the same column. Secondly, let vi be the vertical arc from Oi
to Xi as in the construction of D(I). Denote the homology class of the component of D(I)
corresponding to Oi by µ(Oi ) = α0 · vi . Then our construction (by the definition of σ)
guarantees that if
µ(Oi )q (mod p) < µ(Oj )q (mod p)

(3.3.1)

then i < j. Further, the coordinates of the O markings place these markings on a diagonal
39

3
o

4

0

o

2

3

3

4

0

x
o

x
o

x
o

x
o

0

1
x

1

x
2

Figure 3.8: The trivial link diagram D(I) in L(5, 2) with I = (0, 1, 2, 0, 3).
of a parallelogram in T − α0 − β0 .
Definition 3.3.3. For any p-tuple I of non-negative integers, define K(I) to be the link
associated to D(I). We call a link K in L(p, q) a trivial link if K is isotopic to K(I) for
some I. The p-tuple I is called the index set associated to K and D(I) is called a trivial
link diagram. Note that each grid number one knot Ki corresponds to an index set with 1
in the ith position and 0 otherwise.
In the course of proving Theorem 1.1.1 we will need that each trivial link K admits a
grid diagram that minimizes the function scr in that free homotopy class of link, among
diagrams with minimal grid number (just as n component unlinks minimize crossing number
in their homotopy class). It may be that the diagram D(I) for K does not achieve this,
however in the following lemma we prove the existence of a grid diagram that does.
Let I = (m0 , m1 , . . . , mp−1 ), with mi ≥ 0 for each i. Consider all grid diagrams
D in L(p, q) such that the components of D have grid number one (they are in the set
}), and so that D has mi components of type DK for each 0 ≤
{DK , DK , . . . , DK
0
1
p−1
i
i ≤ p − 1. Call this finite collection of grid diagrams D(I). Let n =

mi . Then the grid

number of every diagram in D(I) is n.
Lemma 3.3.4. Consider the link K(I) in L(p, q) for an index set I. There is a grid diagram
40

ˆ
ˆ
D(I) which has associated link isotopic to K(I), such that scr(D(I)) is the minimum of the
image scr(D(I)) ⊂ Z≥0 .
Proof. Definition 3.2.3 gives a map scrI : D(I) → Z≥0 by restriction. If z is the minimum
−1
of Im scrI , let M (I) = scrI (z). We must show that there is a diagram in M (I) whose
associated link is isotopic to K(I). To do so, we consider M (I) more thoroughly.
Recall from Proposition 3.2.4 that scr is invariant under column commutation, so if
D ∈ M (I) and we get D from D by a column commutation, then D ∈ M (I). Begin with
any grid diagram D ∈ M (I) and perform column commutations until the θ1 coordinate of
every O marking is between 0 and 1 (this is possible since each component has grid number
one). The resulting grid diagram (which by abuse of notation we call D) is still in M (I).
Note that the O markings of D are in one-to-one correspondence with the components
and with the columns. Therefore, perhaps after more commutations of columns, we may
guarantee that if O, O are two O markings and if

µ(O)q (mod p) < µ(O )q (mod p)

(3.3.2)

then θ1 (O) < θ1 (O ) (here µ(O) is the homology class of the component of D corresponding
to O, as in (3.3.1)). As a result, all O markings with a fixed homology type are grouped
into consecutive columns. Finally, among the O markings with homology µ(O) = i, perform
commutations of columns until the θ2 coordinates are decreasing (among the markings with
µ(O) = i). Do this for each i with mi > 0. Denote the resulting grid diagram, which is
ˆ
contained in M (I), by D(I).
ˆ
The process above results in a grid diagram D(I) with an order on the O markings
ˆ ˆ
ˆ
O1 , O2 , . . . , On

ˆ
of D(I) so that:
41

ˆ
(a) θ1 (Oi ) = 2i−1 ;
2n
ˆ
ˆ
(b) if µ(Oi )q (mod p) < µ(Oj )q (mod p) then i < j;
1
ˆ
ˆ
ˆ
ˆ
(c) if µ(Oi ) = µ(Oi+1 ) then θ2 (Oi ) − n = θ2 (Oi+1 ).
ˆ
Note that by condition (a), θ1 (Oi ) = θ1 (Oi ) for each i. This, in addition to condition
ˆ
(b), implies that for every 1 ≤ i ≤ n we have µ(Oi ) = µ(Oi ).
Now, suppose the p-tuple I is zero in every coordinate except one. Then (c) makes
ˆ
D(I) = D(I), so D(I) is in M (I) and we are finished. Suppose instead there is more
than one homology class represented among the components. Property (c) implies that if
ˆ
ˆ
ˆ
ˆ
θ2 (Oi ) − θ2 (Oi+1 ) = θ2 (Oi ) − θ2 (Oi+1 ) then µ(Oi ) = µ(Oi+1 ). Thus, since i + 1 is not
less than i, it must be that

ˆ
ˆ
µ(Oi )q (mod p) < µ(Oi+1 )q (mod p).

(3.3.3)

ˆ
ˆ
Let k = µ(Oi+1 ) and for any 1 ≤ j ≤ n, let rj be the row of D(I) containing Oj (and
ˆ
thus also Xj ). Property (a) and inequality (3.3.3) then imply that ri and ri+j are noninterleaving rows for any 1 ≤ j ≤ mk .
So, for each 0 ≤ k ≤ p − 1, we may perform row commutations on the mk rows of
ˆ
D(I) that have components in homology class k, one row at a time, until the corresponding
O markings are on the diagonal (have coordinates

2i−1 , 1 − 2i−1 ). Each of these row
2n
2n

commutations is non-interleaving, using (a), (b) and the fact that the two rows involved
in each commutation correspond to different homology classes. These row commutations
ˆ
describe an isotopy starting at the link associated to D(I) and ending at the link associated
ˆ
to D(I). Since D(I) is in M (I), this finishes the proof.
42

3.3.2

Every link is homotopic to a trivial link

In this section we show that every link in L(p, q) is homotopic to a unique trivial link. More
precisely, we show that this homotopy can be realized by a sequence of particular grid moves
(see Proposition 3.3.6).
Note that a trivial link is determined by an index set I. Any two different trivial links
cannot be in the same homotopy class, since their index sets will differ and so at least one of
their components differs in free homotopy class. Thus if a link is homotopic to some trivial
link, it is homotopic to a unique trivial link. We will show that there is a particular type of
homotopy from any link to a trivial link.
Lemma 3.3.5. Let K be a Legendrian link in L(p, q) associated to a grid diagram DK .
Suppose DK has a component with grid number more than 1. Then there exists a sequence
of commutations (both interleaving and non-interleaving) followed by a destabilization giving
a new grid diagram D such that GN (D ) < GN (DK ).
Note that in Lemma 3.3.5 we arrive at the grid diagram D without using the forbidden
column exchange in Figure 2.8. We avoid the use of such moves for contact geometric
considerations. Before proving Lemma 3.3.5 we use it to find a homotopy from any link to
a trivial link through commutations and destabilizations.
Proposition 3.3.6. Let DK be a grid diagram for the link K ⊂ L(p, q). There is a sequence
of commutations and destabilizations taking DK to a trivial link diagram. This provides a
homotopy from K to a trivial link.
Proof. We prove Lemma 3.3.5 below. By that lemma there is a grid diagram D
correK
sponding to a link K such that the grid number of each component of D
equals one, and
K
43

a sequence of commutations and destabilizations takes DK to D . Since D
∈ D(I) for
K
K
the index set I of K, we may successively interchange adjacent columns of D
until we
K
ˆ
arrive at a diagram D with the θ1 -coordinates of its O markings lined up with D(I), as in
ˆ
the proof of Lemma 3.3.4. By that proof, the link associated to D is a trivial link. This
process is a sequence of column commutations, either interleaving or non-interleaving. Since
non-interleaving commutations and destabilizations correspond to an isotopy, Lemma 3.2.1
ˆ
implies that the resulting sequence of diagrams, from DK to D, corresponds to a homotopy
from K to a trivial link.

Proof of Lemma 3.3.5. We introduce the length of arcs in a grid projection. Every grid
projection P is a union of GN (P ) horizontal arcs and GN (P ) vertical arcs. Define the
length of an arc by the number of fundamental parallelograms it traverses. For example, the
vertical arc in Figure 3.7 has length 3 and the horizontal arc has length 1.
Note that if two markings Xi and Oj share a row (resp. a column), there are two possible
arcs that could join them in a grid projection. Define len(Xi Oj ) to be the minimum of the
lengths of these two arcs. Note that if len(Xi Oj ) = 1 then Xi and Oj are in adjacent
columns (resp. rows). The converse, however, is not true since one column of the grid
diagram (a component of T − β) has p fundamental parallelograms in the same row.
For clarity, in the following figures we depict our argument by schematics rather than
depicting the link projections on a grid diagram. The schematics show only the markings
and horizontal and vertical arcs of the projection, but we remind the reader that – just as in
a grid projection – a vertical arc in the schematic could be intersecting α0 multiple times.
The arrows in the schematics indicate how commutations will affect the columns, and a
circled triple of markings indicates a destabilization.
44

ˆ
ˆ
Continuing with the proof, let D be a component of DK with GN (D) > 1. Then there
ˆ
are three markings of D, say X1 , O1 , X2 , where X1 , X2 are distinct, X1 , O1 are in the same
column, and X2 , O1 are in the same row. Furthermore, among all such triples of markings
of DK , choose a triple X1 , O1 , X2 such that len(X1 O1 ) is minimal.
Denote the column containing X1 and O1 by c1 and the row containing O1 and X2 by
r1 . Define O2 to be the O marking in the column of X2 and call this column c2 . Let X3 be
the X marking in the same row as O2 , which row we call r2 (see Figure 3.9).

Grid Projection
O1

Schematic
O1

X2
X1

X2

X1
X3

α0

O2

Figure 3.9: Decreasing Grid Number

Since X1 and X2 are distinct, we know the columns containing the pairs (X1 , O1 ) and
(X2 , O2 ) are distinct. Thus there is a sequence of commutations of columns of DK after
which we have one of two cases. In case 1, len(O1 X2 ) = 1. In case 2, which is depicted
in Figures 3.10 and 3.11, len(O2 X3 ) = 1. Each commutation (either interleaving or noninterleaving) exchanges column c2 with an adjacent column, the result being that len(O1 X2 )
decreases. As previously shown, a commutation corresponds to either a skein crossing change
or a Legendrian isotopy.
We now consider the two cases.
Case 1: By the minimality of our choice of len(X1 O1 ) we can commute row r1 with an
adjacent row, decreasing len(X1 O1 ) until len(X1 O1 ) = 1. Since len(O1 X2 ) = 1, c1 and
c2 are adjacent columns, so each of the row commutations is non-interleaving. Performing a
45

destabilization on (X1 , O1 , X2 ), the grid number drops by one.

O1

O3
X2

X1

O3
X2
X3 2
O

O1
X1

O3
O1

X

X1

X3 2
O
Figure 3.10: Decreasing Grid Number: Case 2a

Case 2: We are in one of the two situations of Figures 3.10 and 3.11. Let O3 be the
marking that shares a column with X3 .
In case 2a, len(X3 O3 ) ≥ len(X2 O2 ), so we can commute row r2 until len(X2 O2 ) = 1,
with all commutations being non-interleaving (since len(X3 O2 ) = 1). This is possible since
ˆ
X3 is not the same marking as X2 , for this would force GN (D) = 1. Note that X3 may
not be distinct from X1 . Performing a destabilization on the triple (X2 , O2 , X3 ), the grid
number decreases by one.
Otherwise, we are in case 2b where len(X3 O3 ) < len(X2 O2 ). Commute row r2 , using
only non-interleaving commutations, until len(X3 O3 ) = 1, which again is possible since
ˆ
GN (D) = 1 and len(X3 O2 ) = 1. Perform a destabilization on the triple (O2 , X3 , O3 ).
Again the destabilization drops the grid number of the diagram by one.
O1

X2
O3

X1

O1

X2
O3
X3 2
O

X1

O1
X1

X3 2
O
Figure 3.11: Decreasing Grid Number: Case 2b

46

X2
O

Remark 3.3.7. Examining the proof of Lemma 3.3.5, we see that no row commutation
performed in the reduction is interleaving.

3.3.3

The polynomial invariant

As a consequence of Proposition 3.3.6, there is a unique trivial link in each homotopy class
of links. In this section we define a complexity function on grid diagrams so that trivial
link diagrams minimize the complexity function. In classical knot theory one often considers
the unlinking number of a link diagram as part of a complexity function. We consider an
analogous number for grid diagrams.

Definition 3.3.8. Let D be a grid diagram for K. Define u(D) to be the minimum number
of skein crossing changes in a sequence of commutations and destabilizations that take D to
a trivial link diagram D(I), where the only interleaving commutations are column commutations. Proposition 3.3.6 , Remark 3.3.7, and the proof of Lemma 3.3.4 imply that there
is such a number and that if u(D) = 0 then the link associated to D is isotopic to a trivial
link.

Theorem 1.1.1. Let L be the set of isotopy classes of links in L(p, q) and let T L ⊂ L
denote the set of isotopy classes of trivial links. Define T L ∗ ⊂ T L to be those trivial
links with no nullhomotopic components. Let U be the isotopy class of the standard unknot,
a local knot in L(p, q) that bounds an embedded disk. Suppose we are given a value Jp,q (T ) ∈
Z[a±1 , z ±1 ] for every T ∈ T L ∗ . Then there is a unique map Jp,q : L → Z[a±1 , z ±1 ]
such that
47

(i) Jp,q satisfies the skein relation

a−p Jp,q (L+ ) − ap Jp,q (L− ) = zJp,q (L0 ).

(ii) Jp,q (U ) = a−p+1 .

(iii) Jp,q (U

−p −ap
L) = a z
Jp,q (L).

Remark 3.3.9. The map Jp,q is defined to be the map JM , with M = L(p, q), given by
Theorem 2.3.2 except that we define a such that ap = v. While not essential to the proof
of the theorem, the change of variable is useful for Corollary 1.1.4. To prove the theorem at
hand, by Theorem 2.3.2 we only need show that Jp,q is Laurent-polynomial valued. Note
that we have not specified what the fixed normalization on the trivial links is. Indeed, such
a choice is not canonical.

Proof. Define a function ψ which assigns to every grid diagram D in L(p, q) the triple of
numbers
ψ(D) = (GN (D), scr(D), u(D)).
Ordering these triples lexicographically, we will use ψ as a complexity function.
ˆ
ˆ
There is a grid diagram D (the diagram D(I) of Lemma 3.3.4) associated to any trivial
link K(I) such that the following holds. If DK is any grid diagram with associated link
ˆ
K, and K is in the same homotopy class as K(I), then ψ(D) ≤ ψ(DK ), equality holding
ˆ
ˆ
only if K(I) is isotopic to K. For in such a case, GN (D) ≤ GN (DK ) since D ∈ D(I).
ˆ
ˆ
ˆ
If GN (D) = GN (DK ) then DK ∈ D(I), but by definition D ∈ M , implying scr(D) ≤
48

ˆ
scr(DK ). As u(D) = 0 ≤ u(DK ) by definition, with equality only if K is isotopic to K(I),
ˆ
we see that ψ(D) < ψ(DK ) if K is not a trivial link.
Our argument inducts on the complexity function ψ. Let K be a non-trivial link in L(p, q)
and D a grid diagram associated to K. There is a sequence of u(D) skein crossing changes
that, with non-interleaving commutations and destabilizations, takes D to the trivial link
diagram in its homotopy class. Suppose the first skein crossing in this sequence is positive
and let D = D+ , D− , and D0 be diagrams as described in Section 3.2 (differing only at
the two columns of the skein crossing) with associated links K = K+ , K− , and K0 . By the
skein relation,
a−p Jp,q (K) − ap Jp,q (K− ) = zJp,q (K0 ).

(3.3.4)

But GN (D− ) = GN (D), and scr(D− ) = scr(D) since the skein crossing change is a
commutation of columns. As our choice of crossing change makes u(D− ) < u(D), we have
that ψ(D− ) < ψ(D).
Recall the defintion of the grid diagram D0 in Definition 3.2.5. D0 has the same grid
number as D. We prove below that

Lemma 3.3.10. scr(D0 ) < scr(D).

Provided Lemma 3.3.10 is true, D− and D0 have strictly lower complexity than D, so
we may assume that Jp,q (K− ) and Jp,q (K0 ) are Laurent polynomials. The skein relation
(3.3.4) then implies Jp,q (K) ∈ Z[a±1 , z ±1 ] as well. The argument is similar if the first skein
crossing change in the sequence is negative, only we use

a−p Jp,q (K+ ) − ap Jp,q (K) = zJp,q (K0 ).
49

Proof of Lemma 3.3.10. The proof will proceed as follows. Recall our definition of a map H
from the columns of a grid diagram in S 3 to a subset of the coordinate plane (see the end
of Section 3.2). Also recall that scr(D) is defined to be the number of pairs of interleaving
columns of D, the lift of D to S 3 , and these pairs need not be adjacent.
Given a skein crossing of D which is resolved to give D0 , the lifts D and D0 differ in
exactly 2p columns, each of these being a lift of a column in the given skein crossing of D.
Both D and D0 have grid number p·GN (D). Let c be the column of D between the β curves
βi , βi+1 . Then define c to be the column in D0 between βi and βi+1 . If c is not one of the
2p columns covering the skein crossing, we call c unaltered and otherwise c is called altered.
Note that a column c in D is unaltered exactly when H(c) = H(c ). So a pair of columns
{c, d} that are both unaltered in D is interleaving if and only if

c ,d

is interleaving in

D0 .
To prove the lemma then, we show that the number of interleaving pairs of columns in
D, such that one of the columns is altered, is larger than the number of interleaving pairs
in D0 . This is shown by proving inequalities (3.3.5) and (3.3.6) below. Now to the proof.
For one of the two columns in the given skein crossing of D, choose c1 as one of the p lifts
of that column in D. Choose c2 to be the lift of the other column in the skein crossing so
that c2 is adjacent to c1 . As in the outline of the proof, let c1 and c2 be the corresponding
columns in D0 .
Since D and D0 differ only in a specified way, we can refer to the same curve α0 in both
diagrams. For H = Hα0 : C (D) → R2 , let pi = H(ci ) and for H = Hα0 : C (D0 ) → R2 ,
let pi = H(ci ). If q ∈ Im H and c is some column of D (or D0 ) then define q(c) = 1 if H(c)
50

is in an interleaving region of q (recall, the interleaving regions of a point in the image of
H are as shown in Figure 3.6). Define q(c) = 0 otherwise. We will also abuse notation and
sometimes write q(H(c)) to mean q(c).
To prove Lemma 3.3.10 we first show that for any unaltered column c,

p1 (c) + p2 (c) ≥ p1 (c ) + p2 (c ).

(3.3.5)

Second we show that if d1 , d2 is a pair of adjacent altered columns in D, then
2
(p1 (di ) + p2 (di )).
(p1 (di ) + p2 (di )) ≥
i=1
i=1
2

(3.3.6)

Since c1 and c2 interleave each other in D but c1 and c2 are non-interleaving in D0 by
design, this would show that scr(D0 ) ≤ scr(D) − p.
To show that (3.3.5) holds, let c be an unaltered column of D and let q = H(c) = H(c ).
Also let Z1 , Z2 be the markings in c1 , c2 that will swap columns to give c1 , c2 . We can
choose α0 so that θ2 (Z1 ) and θ2 (Z2 ) are the x-coordinates of H(c1 ) and H(c2 ) respectively.
Without loss of generality, we may also assume that θ2 (Z1 ) < θ2 (Z2 ) and that α0 is on the
boundary of the row containing Z1 . Hence, no point in Im H has a smaller x-coordinate
than p1 = H(c1 ). In fact, with such a choice of α0 the points p1 , p2 appear as in Figure
3.12, and we get p1 , p2 by interchanging the x-coordinates of p1 , p2 .
If q is in any region of the plane other than where it appears in Figure 3.12 then it is
evident from the figure that p1 (c) + p2 (c) = p1 (c ) + p2 (c ). On the other hand, if q is as in
Figure 3.12 then p1 (c) + p2 (c) = 2 and p1 (c ) + p2 (c ) = 0. This shows that (3.3.5) holds.
We now show that (3.3.6) holds. As observed in Remark 3.2.7 our choice of α0 does not
51

p2

p2
q

q

p1

p1
H(C (D0 ))

H(C (D))

Figure 3.12: The interleaving of unaltered columns in D

change the count in (3.3.6). Given a pair d1 , d2 of interleaving columns in D, we say that
α0 is good for the pair if the x-coordinates of Hα0 (d1 ) and Hα0 (d2 ) either both come from
heights of O markings or both come from heights of X markings. Every pair of interleaving
columns has at least 2 good α-curves.

Suppose d1 and d2 are a pair of ajacent, altered columns, d1 and d2 are the corresponding
columns in D0 , qi = H(di ) and qi = H(di ). If α0 is good for d1 , d2 then the points
q1 , q1 , q2 , q2 are the corners of a rectangle in R2 , with sides parallel to the axes and with q1
(resp. q2 ) being the lower left (resp. upper right) corner. In our proof of (3.3.5), the choice
of α0 was good for c1 , c2 .
Now suppose there is a choice of α0 that is good for both pairs c1 , c2 and d1 , d2 simultaneously. With this choice of α0 the points pi , pi for i = 1, 2 appear as in Figure 3.13 with
corresponding interleaving regions. Note that for an arbitrary point a in the second octant,
we have that p1 (a) + p2 (a) ≥ p1 (a) + p2 (a). Let R be a rectangle in this octant (the sides
of R parallel to the axes) with corner points in the image of H, labeled counter-clockwise by
52

p2

p2

p1

p1

H(C (D0 ))

H(C (D))

Figure 3.13: The interleaving of altered columns in D with a common good choice of α0

r1 , s1 , r2 , s2 with r1 in the lower left corner. If
2

2
p1 (ri ) + p2 (ri ) ≥

i=1

p1 (si ) + p2 (si )

(3.3.7)

i=1

then since p1 (sj )+p2 (sj ) ≥ p1 (sj )+p2 (sj ) we see that (3.3.6) is satisfied. One easily checks
that (3.3.7) is satisfied for any rectangle in the second octant. For the example rectangle
shown on the left of Figure 3.13 we have

p1 (ri ) + p2 (ri ) = 2 and

p1 (si ) + p2 (si ) = 2.

We are left only to check the case when there is no choice of α0 that is good for both
c1 , c2 and d1 , d2 . This can only occur when the set of good α-curves for d1 , d2 is contained
in the set of α-curves that are not good for c1 , c2 . In such a situation, we have two cases.
Either di is interleaving with both c1 and c2 for i = 1 and i = 2, or di is non-interleaving
with both c1 and c2 , for i = 1 and i = 2. In the first case, since the right hand side of (3.3.6)
can be at most 4, and this equals the left hand side, (3.3.6) is satisfied.
Consider the case when di is non-interleaving with both c1 and c2 , for i = 1, 2. Then,
since d1 is interleaving with d2 , the heights of all the markings in d1 , d2 are either between
53

q2
p2

p2
q1

p1

q2+i

q1+i

p1
q2
q1

q2+i

q1+i

Figure 3.14: The interleaving of altered columns in D without a common good choice of α0
the O markings of c1 , c2 or between the X markings of c1 , c2 . Choose α0 so that the xcoordinate of p1 is minimal (as we did in Figure 3.12). With this choice we have the following:
for i ∈ {1, 2}, both x- and y-coordinates of qi = H(di ) are either numbers between the ycoordinates of p1 and p2 or are between the x-coordinates of p1 and p2 . The first case
is depicted on the left of Figure 3.14 and the second case on the right, and in the figure
i ∈ {1, 2}, and indices are taken mod 2. In both cases, the left-hand and right-hand sides of
(3.3.6) are both zero.

54

Chapter 4
Classical invariants of
Q-nullhomologous Legendrian and
transverse links

4.1

Definitions

We review the definitions given in [7] (cf. [4]) of the Thurston-Bennequin number and rotation
number, and for transverse links the self-linking number. Throughout the section all knots
and links are assumed to be Q-nullhomologous.
To begin requires the notion of a rational Seifert surface. Let K be an oriented knot in a
manifold M and let r be the order of the homology class of K in H1 (M, Z). Then, writing
ν(K) for a normal neighborhood of K, for some s ∈ Z there is a curve γ of slope (r, s) on
∂ν(K) that bounds an oriented surface Σ0 ⊂ M \ ν(K). Let Σ be the union of Σ0 and the
cone in ν(K) of γ to K. Figure 4.1 depicts a meridional cross-section of ν(K) and the cone
55

of γ to K.

cone of γ to K

K
K
γ

Figure 4.1: Constructing a rational Seifert surface

Note that the interior of Σ is an embedded surface in M and ∂Σ is an r-fold cover of K.

Definition 4.1.1. Let K be an oriented knot in M with order r in H1 (M, Z). A rational
Seifert surface for K is an oriented surface Σ with a map j : Σ → M such that j is an
embedding on the interior of Σ, j(∂Σ) = K, and j|∂Σ is an r-fold cover of K.

The previous discussion shows that every oriented (Q-nullhomologous) knot has a rational
Seifert surface. If we have a link, then we consider what Baker and Etnyre call a uniform
rational Seifert surface of the link. Such a surface meets each boundary component of the
link complement in an (r, si ) sloped curve, where r is the order of the link, rather than
in a curve with slope coming from the order of each link component. Often we will abuse
terminology and call a rational Seifert surface simply a Seifert surface. We can now define
the classical invariants for Legendrian and transverse knots.

Definition 4.1.2. Let K be an oriented (Q-nullhomologous) link with order r as above and
let j : Σ → M be a Seifert surface for K.
56

1. Given another oriented knot K , define

1
lkQ (K, K ) = Σ · K .
r

2. If in addition K is a Legendrian knot in (M, ξ) and K is a pushoff of K in the direction
of the contact framing given by ξ|K ∩ ν(K), then define the (rational) ThurstonBennequin number of K by

tbQ (K) = lkQ (K, K ).

3. For x ∈ K let dKx denote the tangent vector to K at x. For K Legendrian as above,
dK is a section of the bundle ξK . Take a trivialization j ∗ ξΣ ∼ R2 × Σ. Define the
=
(rational) rotation number of K to be the winding number of j ∗ dK in R2 under this
trivialization, divided by r:

1
rotQ (K) = winding(j ∗ dK, R2 ).
r

4. If K is a transverse knot in (M, ξ), let v be a non-zero section of j ∗ ξ. Normalize v so
that v|∂Σ defines a curve K in ∂ν(K). Define the (rational) self-linking of K to be

slQ (K) = lkQ (K, K ).

Remark 4.1.3. In general, the rotation number of a Legendrian knot and the self-linking
number of a transverse knot depend on the relative homology class of Σ. Yet once this class
57

is fixed, they do not depend on other choices made – the trivialization of j ∗ ξΣ in the case
of rotQ and the section v in the case of slQ (see [7]). The dependence on the homology of
Σ does not apply to lens spaces, however, since H2 (L(p, q)) = 0.
In the case of null-homologous links (e.g. Legendrian links in (S 3 , ξst )) these numbers
are known as the “classical invariants” of Legendrian and transverse links. In this case the
classical invariants are always integers. However, in the case of rationally null-homologous
links, these numbers are generally rational.
We recall that for any Legendrian knot K in a contact manifold (M, ξ), there is a related transverse knot T+ (K) called the positive transverse push-off of K (see [16]). Using
this construction on each component of a link, we can get the positive transverse push-off
of a Legendrian link. To construct T+ (K), find a tubular neighborhood of K that is contactomorphic, for sufficiently small ε, to Cε =

[(x, y, z)] | y 2 + z 2 < ε2 , x = x + 1 : the

quotient of an ε-neighborhood of the x-axis in (R3 , ker(dz − ydx)) by the action x → x + 1.
Here the orientation of the image of K under the contactomorphism is in the direction of
increasing x-values. The positive transverse push-off T+ (K) is defined to be the image of
{(x, ε/2, 0)} in the neighborhood of K. We can also define the negative transverse push-off
T− (K) to be the image of {(x, −ε/2, 0)}. Baker and Etnyre show the following [7]:
Lemma 4.1.4. Given a Legendrian link K, let T± (K) be defined as above. Then

slQ (T± (K)) = tbQ (K)

rotQ (K).

Remark 4.1.5. We will define T (K) to be the transverse push-off of K that satisfies
58

slQ (T (K)) = tbQ (K) + rotQ (K) . If rotQ (K) = 0 then T (K) is taken to be the positive transverse push-off.

4.2

Formulae for tbQ(K), rotQ(K), and slQ(K) from a
grid projection in L(p, q)

A method for computing the (rational) Thurston-Bennequin number of a Legendrian link in
L(p, q) via the Maslov index of a corresponding grid diagram was given in [4]. The complexity
of such computations increases quickly as the grid number of the diagram increases.
We recall that in (S 3 , ξst ) there are formulas for the classical invariants that can be
computed from a front projection (see [16]). The formulas we give here for computing the
(rational) Thurston-Bennequin, rotation, and self-linking numbers in L(p, q) are in the same
spirit. In fact, they are derived from the former.
Definition 4.2.1. Given a grid projection P for an oriented link in L(p, q) denote the writhe
of the projection by w(P ) and the number of cusps of the projection by c(P ). Also, let µ(P )
denote the algebraic intersection number of α0 with P and λ(P ) the algebraic intersection
number of P with β0 .
Let K be the link associated to a given grid projection P on a grid diagram. We recall
that for a given row (resp. column) in that diagram, P contains one of the two choices
of horizontal (resp. vertical) arcs. A projection P which is identical to P except that it
contains the other arc in this row (resp. column) corresponds to a link K that differs from
K by a Legendrian isotopy across a meridian of V α (resp. V β ). We call this isotopy a disk
slide.
59

Recall that in (S 3 , ξst ) if P is the front projection of a Legendrian link K then tb(K) =
1
w(P ) − 2 c(P ). Moreover, we note that (S 3 , ξst ) = (L(1, 0), ξU T ), and if a grid projection P
in L(1, 0) is contained in a planar subset of T then there is a slight perturbation of P giving
a front projection (see [4]).
Proposition 4.2.2. Let K be an oriented Legendrian link in (S 3 , ξst ) = (L(1, 0), ξU T ) and
let P be a grid projection of K. Let l = λ(P ) and m = µ(P ). Then

1
tb(K) = w(P ) − c(P ) − ml.
2

Proof. If P is a projection on a planar subset of Σ then we can consider it as a front
projection. In this case l = m = 0 and the proposition follows immediately.
Suppose P is any grid projection of K for which the proposition holds. We prove that
if P differs from P by a disk slide (either a disk with boundary parallel to α0 or along
one with boundary parallel to β0 ) then the proposition holds for P as well. Since every
grid projection of K is related to a planar grid projection by a sequence of disk slides, this
will prove the proposition. Inherent in this proof is the fact that disk slides correspond to
Legendrian isotopy [4].
We denote the algebraic intersection of two oriented curves γ, δ on Σ that meet transversely by γ, δ . Note that for any circle c on Σ, parallel to α0 and transverse to P , we
have m := α0 , P

= c, P

, since c separates Σ \ α0 .

Suppose P differs from P by a disk slide. Suppose further that the disk slide is along
a disk ∆ with boundary parallel to α0 . There is some orientation of ∆ such that ∂∆ =
60

Case 1

c
a

a

a

cusp

cusp

Case 2

c
a

a

a

Case 3

c
a

a

a

cusp
Figure 4.2: How tb changes with a disk slide

(−a ) ∪ a, where a is an arc of P and a is the arc that replaces it in the projection P
(see Figure 4.2). Let c be a circle that is parallel and coherently oriented to α0 and a small
distance away from (−a ) ∪ a. Then c intersects P and P tranversely. The writhe of P
differs from that of P only by the difference along double points of a and double points of
a. Thus if a is an arc with two cusps as in case 1 of Figure 4.2, then

±(w(P ) − w(P ) + 1) = c, P

=m,

where the sign on the left depends on the orientation of a with respect to c. If a has no
cusps as in case 2 of Figure 4.2 then

±(w(P ) − w(P ) − 1) = c, P
61

=m.

Finally, if a is an arc with one cusp as in case 3 then

±(w(P ) − w(P )) = c, P

=m,

with the sign depending again on the orientation of a. Also, if a is as in case 1 then
c(P ) = c(P ) − 2, if we are in case 2 then c(P ) = c(P ) + 2, and if we are in case 3, then
c(P ) = c(P ).
Since l = l ± 1 (where l = λ(P )) and m = m we can now check that

1
1
w(P ) − c(P ) − ml = w(P ) − c(P ) − m l = tb(K)
2
2

for cases 1–3.
The argument is analogous if the disk slide is along a meridional disk.
Corollary 4.2.3. Let K ⊂ (L(p, q), ξU T ) be an oriented Legendrian link with P a grid
projection for K. Let λ = λ(P ) and µ = µ(P ). Then

µλ
1
.
tbQ (K) = w(P ) − c(P ) −
2
p

Proof. Lift the projection P to a grid projection P in S 3 by lifting the Heegaard torus of
L(p, q) to Σ. Then clearly pw(P ) = w(P ) and pc(P ) = c(P ). Moreover, µ(P ) = µ(P ) and
λ(P ) = λ(P ) by definition.
It was shown in [4] that tbQ (K) =

tb(K)
p , and this completes the proof.

Corollary 4.2.4. Let K ⊂ (L(p, q), ξU T ) be a Legendrian knot with its contact framing. If
62

there is an integral surgery on K that gives a homology sphere S then µ(K)2 ≡ ±q

mod p,

where qq ≡ 1 mod p. In particular, if there is a knot in L(p, q) on which integer surgery
yields S 3 then ±q is a quadratic residue mod p.

Remark 4.2.5. We note that the last statement of the corollary is a well-known result of
Fintushel and Stern [18].

Proof. In [4] it is shown that p · tbQ (K) ≡ ±1 mod p. With Corollary 4.2.3 this implies
that µλ ≡ ±1 mod p. Since the projection of K is a closed curve, λ ≡ µq mod p, so that
µ2 q ≡ ±1 mod p.

The following proposition shows how to compute rotQ (K) given a projection P corresponding to a grid diagram for an oriented Legendrian link K in (L(p, q), ξU T ). Its proof is
similar to the proof of Proposition 4.2.2 and Corollary 4.2.3.

Proposition 4.2.6. Let µ and λ be as in Corollary 4.2.3. Note that each cusp in a grid
projection is comprised of a horizontal and a vertical arc. Define cu (P ) to be the number of
cusps whose horizontal arc is oriented against α0 and cd (P ) to be the number of cusps with
horizontal arc oriented in the direction of α0 . Then
1
(λ − µ)
rotQ (K) =
cd (P ) − cu (P ) −
.
2
p

Corollary 4.2.7. Let T+ (K) (resp. T− (K)) be the positive (resp. negative) transverse
63

pushoff of the Legendrian link K in (L(p, q), ξU T ). Then
µλ + (µ − λ)
and
p
µλ + (λ − µ)
slQ (T− (K)) = w(P ) − cu (P ) −
,
p
slQ (T+ (K)) = w(P ) − cd (P ) −

where P is the projection of K as above.

Proof. This results from Corollary 4.2.3, Proposition 4.2.6, and Lemma 4.1.4.

4.3

Bennequin-type bounds in L(p, q)

In this section we prove the lens space analogue of the version of the FWM inequality observed
by Fuchs and Tabachnikov [21]. To achieve this, we prove the analogue of a theorem of Lenny
Ng (Theorem 1 in [41]) for oriented links in the contact lens space (L(p, q), ξU T ). While the
proof follows the ideas of Ng in [41], we note that it is independent of the work of Rutherford
[54] that plays a key role in the proof given in [41]. It is an example of the power of the
point of view of grid diagrams in understanding Legendrian links.
Theorem 1.1.3. Let i be a Q-valued invariant of oriented links in L(p, q) such that

i(L+ ) + 1 ≤ max(i(L− ) − 1, i(L0 ))
and
i(L− ) − 1 ≤ max(i(L+ ) + 1, i(L0 ))
64

where L+ , L− , and L0 are oriented links that differ as in the skein relation. If slQ (T (τ )) ≤
−i(τ ) for every trivial link τ in L(p, q), then

slQ (L) ≤ −i(L)

for every link L in L(p, q). Here slQ is the maximum self-linking number among transverse
links in (L(p, q), ξU T ) that are isotopic to L.

Proof. In the proof we will abuse notation, often using P to refer to both a grid projection
and the Legendrian link it specifies. Our strategy of proof is as follows. Assume our link is a
positive transverse push-off T+ (P ). Define i(P ) = i(P ) + w(P ), where w(P ) is the writhe of
P . Then by the formula of Corollary 4.2.7, the inequality slQ (T+ (P )) ≤ −i(P ) is equivalent
to
−slQ (T+ (P )) − i(P ) = cd (P ) +

µλ + (µ − λ)
− i(P ) ≥ 0.
p

(4.3.1)

By Proposition 3.3.6, there is a minimal length sequence

α1
α2
αn
P −→ P1 −→, . . . , −→ Pn = Pτ

taking P to a grid projection Pτ associated to a trivial link diagram, where each αi is either
a skein crossing change, Legendrian isotopy, or destabilization. We will show that if some
αi increases the left side of inequality (4.3.1) then it is a skein crossing change, and that
in this case the resolution P

P0 does not increase the left side of (4.3.1). If P, P0 have

underlying diagrams D, D0 , then ψ(D) > ψ(D0 ) (recall the complexity ψ from the proof of
Theorem 1.1.1). As ψ is minimized on diagrams for trivial links, we may assume inductively
65

that inequality (4.3.1) holds for P0 , since slQ (T+ (τ )) ≤ slQ (T (τ )) ≤ −i(τ ) for any trivial
link τ .
If instead our transverse link is T− (P ) for some P then the same argument can be carried
through, replacing inequality (4.3.1) with the corresponding formula. Since every transverse
link is some transverse push-off T± (P ) of a Legendrian link, this argument is sufficient to
prove the theorem.
The expression on the left side of (4.3.1) is a Legendrian invariant, so if αi is a Legendrian
isotopy the expression is unchanged. In the case of a destabilization we make the following
claim.
Claim 1: If αi is a destabilization, then −slQ (T+ (Pi−1 )) − i(Pi−1 ) ≥ −slQ (T+ (Pi )) −
i(Pi ), so αi does not increase the left side of (4.3.1).
Proof of Claim 1. First, µ and λ are each an algebraic intersection of P with some α-curve
or β-curve respectively. Therefore they are unchanged by a destabilization.
negative crossing
x
o

x

x

upward cusp

downward cusp

Figure 4.3: On an X : N E destabilization cd (P ) increases.
It is possible that for some destabilization that is not Legendrian isotopy, cd (P ) changes.
If cd (P ) decreases, then −slQ (T+ (P )) decreases. The destabilization was an isotopy, so i(P )
remains unchanged and we see that the left side of (4.3.1) does not increase. If cd (P ) increases
the destabilization is of type X:NE or O:SW. Figure 4.3 depicts when the destabilization is
type X:NE. In such a case, cd (P ) increases by 1, but w(P ) does also. Thus slQ (T+ (P )) does
66

not change, and so −slQ (T+ (P )) − i(P ) also does not change.
We remark that Claim 1 also holds for the case of a negative push-off T− (P ). In the
proof one only need consider cu (P ) instead of cd (P ).
Thus we only need concern ourselves with skein crossing changes. Suppose α1 is a skein
crossing change. If the pair of columns involved in α1 are a positive (resp. negative) skein
crossing in P , then use disk slides if necessary so that, at the skein crossing, P appears as
in Figure 4.4 (resp. as in Figure 4.5). Since disk slides are Legendrian isotopy, this does not
alter the left side of inequality (4.3.1).
x
x
o
x
x

o
P

o
o
P = P−

x
x
o
o
P0

Figure 4.4: A skein crossing change and resolution at a positive skein crossing.
If α1 does not increase the left hand side of (4.3.1), then we are finished by induction.
Otherwise the left side of (4.3.1) does increase in passing from the projection P to P− (resp.
P+ ). Lemma 4.3.1, which we prove below, shows that in this case −slQ (T+ (P )) − i(P ) ≥
−slQ (T+ (P0 )) − i(P0 ). As we remarked before, there is a complexity ψ which is minimized
by diagrams for trivial links, such that if D0 is the grid diagram for P0 and D the diagram
for P , then ψ(D0 ) < ψ(D). Thus we may assume that P0 satisfies (4.3.1), so by Lemma
4.3.1, P does also.
Lemma 4.3.1. Let P be a grid projection containing a skein crossing as in Figure 4.4
or as in Figure 4.5, and let P be the grid projection obtained by a skein crossing change
67

at that crossing. Further, suppose −slQ (T+ (P )) − i(P ) < −slQ (T+ (P )) − i(P ). Then
−slQ (T+ (P )) − i(P ) ≥ −slQ (T+ (P0 )) − i(P0 ).

x
x
o
x

o

x

P
o

o
P = P+

x
x
o
o
P0

Figure 4.5: A skein crossing change and resolution at a negative skein crossing

Proof of Lemma 4.3.1. Firstly, λ is not changed in passing from P to either P or P0 . The
columns depicted in Figure 4.4 (resp. Figure 4.5) are adjacent, so there is a β-curve, say βi ,
between the two columns. The intersection of the projection with some β-curve determines
λ, but the value of λ is independent of the choice of β-curve for this intersection. Choose any
β-curve other than βi (at least one exists, because the existence of a skein crossing implies
at least two columns in the grid diagram). Since the horizontal arcs of the projections P ,
P , and P0 are identical outside of the two columns shown, λ(P ) = λ(P ) = λ(P0 ).
Also, consider the pair of vertical arcs depicted in each projection P , P , and P0 . The
sum of the lengths of these two arcs is the same in all three projections. As P , P and P0
are identical elsewhere, µ(P ) = µ(P ) = µ(P0 ).
Note that the number and nature of cusps in P are the same as those in P and P0 ,
so cd (P ) = cd (P ) = cd (P0 ). Therefore, the equality in (4.3.1) and the supposition that
−slQ (T+ (P )) − i(P ) < −slQ (T+ (P )) − i(P ) imply that i(P ) > i(P ).
68

However, by our assumption on the invariant i, we see that

i(L+ ) ≤ max i(L− ), i(L0 )
and
i(L− ) ≤ max i(L+ ), i(L0 ) .

Since P = P− or P = P+ (depending on the sign of the skein crossing in question),
i(P ) > i(P ) implies that i(P ) ≤ i(P0 ). But that implies that −slQ (T+ (P )) − i(P ) ≥
−slQ (T+ (P0 )) − i(P0 ), finishing the proof.
Theorem 1.1.3 has the following applications.
Corollary 1.1.4. Let Jp,q denote the HOMFLY polynomial invariant in L(p, q), normalized so that if τ is a trivial link with no nullhomotopic components, or is the unknot, then
Jp,q (τ ) = a

p·slQ (T (τ ))+1

. Given an oriented link L in L(p, q), set e(L) to be the lowest

degree in a of Jp,q (L). If Lt is a transverse representative of L in (L(p, q), ξU T ), then

slQ (Lt ) ≤

e(L) − 1
.
p

Proof. The defining skein relation of Jp,q says that

e(L+ ) ≥ min e(L− ) + 2p, e(L0 ) + p ,

implying
−e(L+ ) + 1 ≤ max −e(L− ) + 1 − 2p, −e(L0 ) + 1 − p .
69

Dividing by p and then adding 1 we have

−e(L+ ) + 1
+ 1 ≤ max
p

−e(L− ) + 1
−e(L0 ) + 1
− 1,
p
p

.

−e(L+ ) + 1
−e(L0 ) + 1
+ 1,
p
p

.

A similar computation shows

−e(L− ) + 1
− 1 ≤ max
p

Letting i(L) =

−e(L)+1
, we see that i satisfies the first hypothesis of Theorem 1.1.3.
p

Moreover, by our choice of normalization e(τ ) = p · slQ (T (τ )) + 1 for any trivial link τ and
so −i(τ ) = slQ (T (τ )). Since all conditions of Theorem 1.1.3 are met, we are done.

Corollary 1.1.5. If τ is a trivial link in L(p, q), then T (τ ) has maximal self-linking
number among all transverse representatives of τ . If τ is a trivial knot, then the Legendrian
knot associated to its grid number one diagram has maximal Thurston-Bennequin number.

Proof. Let τt be some transverse representative of τ . Corollary 1.1.4 says that slQ (τt ) ≤
e(τ )−1
= slQ (T (τ )).
p
Now consider a Legendrian knot K that is associated to a grid number one diagram and
let K be another Legendrian knot with the same knot type as K. Suppose that tbQ (K) <
tbQ (K ). In [7] it is shown that K and K are Legendrian isotopic after each has been
positively and negatively stabilized some number of times, and therefore tbQ (K) − tbQ (K )
is an integer. So we have tbQ (K) + 1 ≤ tbQ (K ).
70

By Corollary 4.2.6, if P is a grid projection for a grid diagram of K, then

1
λ−µ
rotQ (K) = (cd (P ) − cu (P )) −
.
2
p

Since we are in the grid number one case, P has one vertical and one horizontal arc. Also
we can choose P so that 0 < µ(P ) < p and 0 < λ(P ) < p. This choice of P has no cusps at
all, implying that rotQ (K) < 1.
By Corollary 4.1.4 the transverse pushoff T (K) of K has self-linking slQ (T (K)) =
tbQ (K) + rotQ (K) . So since rotQ (K) < 1, it must be that slQ (T (K)) < tbQ (K ).
This contradicts the fact that slQ (T (K)) is maximal, since one of the positive or negative
transverse pushoffs of K has self-linking number at least as large as tbQ (K ).

4.4

Examples and Computations

In this section we compute the polynomial Jp,q (L) for two examples in L(5, 1). We normalize
on the trivial links as in Corollary 1.1.4. The first example we compute is a nullhomologous
knot B which is depicted via a grid diagram at the top of Figure 4.6. The second example
L, also in L(5, 1) and shown in Figure 4.7, is homotopically nontrivial, having µ(L) = 2
mod 5. The second example gives rise to a family of Legendrian links {Ln }∞ . For every
n=1
n, T (Ln ) = T+ (Ln ), and we show that the family T (Ln ) has arbitrarily negative self-linking
numbers that are maximal in their link type.
Example 1. In Figure 4.6 we show the skein tree of B. Note that the relevant skein
crossing in B is positive. The skein crossing change (the left branch in the tree) takes B to
the unknot K0 . Resolution of the skein crossing (the right branch) gives a link B0 .
71

To see that B0 is isotopic to a trivial link, note that its index set is (0, 1, 0, 0, 1) and that
if we label the O markings O1 , O2 so that θ1 (O1 ) < θ1 (O2 ), then µ(O1 ) = 1 and µ(O2 ) = 4.
Since 1 · 1 < 4 · 1( mod 5), we see that the grid diagram shown for B0 is the exactly the
trivial link diagram D(0, 1, 0, 0, 1).
B
4

0

1

2

3

o

x

o
0
B− = K0
0
1

4

x
1

0

2

3

4

2

3

4

4

x

0

3

4

1

2

3

o
0

0

4
x

o
2

0
B0

o
o x
1

4

x
1

2

3

4

0

Figure 4.6: Example: A link B in L(5, 1) and its skein tree.

Recall the formulas for the classical invariants, in particular the result of Proposition
4.2.7. Letting B0 also denote the corresponding Legendrian link in (L(5, 1), ξU T ), note that
4
tbQ (B0 ) = − 4 and rotQ (B0 ) = 0. So T (B0 ) = T+ (B0 ) and slQ (T (B0 )) = − 5 . Since B0
5
is a trivial link, we have that J5,1 (B0 ) = a−4+1 = a−3 . And thus

J5,1 (B) = a10 J5,1 (K0 ) + a5 zJ5,1 (B0 )
= a10 a−4 + a5 z a−3
= a6 + a2 z.

Example 2. In the second example we have a negative skein crossing associated to
the first and second columns of the grid diagram for L. Via grid moves that correspond
72

to isotopy (a non-interleaving row commutation, destabilization, a non-interleaving column
commutation, then another destabilization) the diagram shown for L+ in Figure 4.7 is taken
to a grid number one diagram of the trivial knot K2 . So L+ is isotopic to the trivial knot
K2 .
L
4

0

1

2

3
x

o
o
0

4
x

1

2

o
4

3

x
0

L+
4
o

0

1

2

4

4
o

x
o

0

3
x

1

2

3

0

1

L0
2

o

o x
4
0

0

1

2

3

3
x

4

x
o x
4
0

Figure 4.7: Example: A link L in L(5, 1) and its skein tree.

The diagram shown in the figure of L0 (after a destabilization) has two grid number one
components: both of type DK . The O markings are adjacent and on a negative slope, so
1
we have the diagram D(0, 2, 0, 0, 0) and L0 is trivial.
Since slQ (T (K2 )) = 1 and slQ (T (L0 )) = − 4 , we compute
5
5
J5,1 (L) = a−10 J5,1 (K2 ) − a−5 zJ5,1 (L0 )
= a−10 (a2 ) − a−5 z(a−3 )
= a−8 (1 − z).

Consider the family of links {Ln }n≥0 where Ln is the link associated to the grid diagram
73

in Figure 4.8 with grid number n + 2. Then L1 is the knot L in Example 2 above and L0 is
isotopic to the trivial 2-component link in the skein tree of L. The link Ln is a knot if n is
odd and a 2-component link if n is even.
The first two columns of this grid diagram make a negative skein crossing. It is not
difficult to see that for n ≥ 2, commutation of these columns of Ln gives a link isotopic to
Ln−2 and the resolution of the same columns gives a link isotopic to Ln−1 . Therefore

J5,1 (Ln ) = a−10 J5,1 (Ln−2 ) − a−5 zJ5,1 (Ln−1 ).

We know that J5,1 (L0 ) = a−3 and J5,1 (L1 ) = a−8 (1 − z) by our computations above.
Define fn recursively: let

f1 = 1 − z,

f0 = 1,

and define fn = fn−2 − zfn−1 for n ≥ 2. Then the skein relation implies that J5,1 (Ln ) =
a−5n−3 fn . Note that the recursive definition of fn implies that it is not zero for any n.
Using Corollary 1.1.4 we show that T (Ln ) has maximal self-linking number for each n. We
will see below that rotQ (Ln ) = −n, and so T (Ln ) = T+ (Ln ) for all n.
4

0
o

1

2

o

n columns

3
x
o

x
o

..

.

x
o

0

1

2

3

4

Figure 4.8: Grid diagram associated to the link Ln .
74

4

x

x
0

Let T (Ln ) = T+ (Ln ) denote positive transverse pushoff of Ln . By Corollary 1.1.4

slQ (T (Ln )) ≤

e(Ln ) − 1
4
= −n − .
5
5

If we choose Pn to be a grid projection of Ln , then

slQ (T (Ln )) = w(Pn ) − cd (Pn ) −

µλ + (µ − λ)
.
5

Let Pn be the grid projection depicted in Figure 4.8. Then w(Pn ) = −n − 2, cd (Pn ) = 0,
and cu (Pn ) = 2(n + 2). Moreover, µ(Pn ) = 2 and λ(Pn ) = −8. Therefore, tbQ (Ln ) =
4
−n − 2 − (n + 2) + 16 = −2n − 5 and rotQ (Ln ) = −n − 2 + 10 = −n. As a consequence,
5
5
4
slQ (T (Ln )) = −n − ,
5

showing that each T (Ln ) maximizes its self-linking number.

75

Chapter 5

Fibered knots and the universally
tight contact structure

In this chapter we recall briefly an association between fibered links in a 3-manifold and
contact structures on that manifold. We will review a bit of the history that explores which
fibered links are associated to certain contact structures. Hedden and Plamenevskaya have
a recent result [27], showing that, given a fibered knot associated to a contact structure with
non-trivial Ozsv´th-Szab´ contact invariant, the dual knots to “sufficiently large” surgeries
a
o
on that knot are also associated to a contact structure with non-zero contact invariant. As a
consequence, the dual to any framed knot arising from Berge’s double primitive construction
is associated to a tight contact structure on the lens space obtained by the surgery. We
consider certain families of these dual knots in L(p, q), and we use results from Section 4.2
to determine whether these links are associated with the universally tight contact structure
ξU T on L(p, q).
76

5.1

Background

Given a rationally null-homologous link B in a closed, oriented 3-manifold Y , a rational open
book decomposition (B, π, F ) of Y is a fibration π : Y \ν(B) → S 1 with compact fiber F , and
ν(B) is a small tubular neighborhood of B in Y . Note that we do not require that F meet
∂ν(B) in a longitude of B, i.e. F is a rational Seifert surface for B. If B is not connected,
we will only consider the case that F is a uniform Seifert surface. In the case that F is an
honest Seifert surface, (B, π, F ) is called an integral open book decomposition.
In what follows, we will often suppress the map π from the notation, denoting (B, π, F )
as (B, F ). Alexander showed [1] that every closed oriented Y 3 has an integral open book
decomposition.
From any integral open book decomposition (B, F ) of Y , with B oriented as the boundary
of F , a construction of Thurston and Winkelnkemper [56] gives a contact 1-form α on Y
with the properties:
(a) α(v) > 0 for any v tangent to B (and coherently oriented with B);
(b) dα is a volume form on the interior of F .
We say that if ξ = ker α for some contact 1-form α satisfying (a) and (b), then (B, F )
supports ξ. By [56], every fibered knot (B, F ) supports a contact structure, in fact it is
unique. In [3], this was extended to rational open books, for if (B, F ) is a rational open
book decomposition, then we can still require that for a contact structure ξ = ker α to be
supported, the contact form α satisfy (a) and (b). It is shown in [3] that a rational open
book (B, F ) supports a unique contact structure, by determining an integral open book from
(B, F ) (called its integral resolution) and showing that (B, F ) supports ξ if and only if its
77

integral resolution does.
The exact correspondence between open book decompositions of Y and contact structures
on Y was elucidated by Giroux: open book decompositions and contact structures are in
bijective correspondence, if one considers open book decompositions of Y up to positive Hopf
stabilization, and contact structures up to isotopy [23].
Given a fibered knot K in Y , one could ask which contact structure on Y is supported
by K. When Y = S 3 , an interesting result of Hedden (in [26]) shows that the answer
is connected to strongly quasipositive knots and the Ozsv´th-Szab´ concordance invariant
a
o
arising in knot Floer homology. We quickly recall the definitions of a strongly quasipositive
knot and of τ in order to state Hedden’s theorem.
Let σ1 , σ2 , . . . , σn−1 denote the standard generators of the braid group Bn on n strands.
For any i < j ≤ n, let σi,j define the braid σi,j = (σi . . . σj−2 )σj−1 (σi . . . σj−2 )−1 in Bn .
ˆ
A knot K is called strongly quasipositive if K may be realized as the closure β of the braid
m
β=
k=1

σi ,j .
k k

We note that the maximum Euler characteristic of a strongly quasipositive knot is known.
Suppose K is such a knot, realized as the closure of some β ∈ Bn as above, where β is the
product of m elements σi,j . Then K has a Seifert surface Σ constructed from n parallel
disks with m half-twisted bands attached (the one corresponding to σi,j being attached to
the ith and j th parallel disk). Now, from the “Legendrianization” of the closure of β (see
[26]) which we call L, one sees that

tb(L) + |rot(L)| = −n + m.
78

So the Bennequin inequality is sharp on K and the Seifert surface Σ has maximum Euler
characteristic.
Let F(Y, K, i) denote the filtration that a null-homologous knot K ⊂ Y induces on the
Ozsv´th-Szab´ chain complex CF (Y, s) associated to Y and s, a spinc -structure on Y . In
a
o
the setting where Y = S 3 , the concordance invariant is defined by

τ (K) = min j ∈ Z | i∗ : H∗ (F(Y, K, j)) → HF (S 3 ) ∼ Z is non-trivial.
=

In [26] Hedden proves the following. Let ξ(K, F ) denote the contact structure corresponding to the fibered knot (K, F ).
Theorem 5.1.1. Let K ⊂ S 3 be a fibered knot with fiber F . Then

(K, F ) is strongly quasipositive ⇐⇒ ξ(K, F ) = ξst ⇐⇒ τ (K) = g(K),

where ξst is the standard tight contact structure on S 3 and g(K) is the Seifert genus of K.
Remark 5.1.2. The method of Hedden’s proof uses that ξ(K, F ) = ξst ⇐⇒ c(ξ(K, F )) =
0, where c(ξ) is the Ozsv´th-Szab´ contact invariant of ξ, which equivalence uses the uniquea
o
ness of the tight contact structure on S 3 .
In [46], Ozsv´th and Szab´ show that any knot K for which some positive integral surgery
a
o
yields a lens space has the property that τ (K) = g(K). Thus, since the work of Ni shows
that all such knots are also fibered [43], Theorem 5.1.1 above tells us that every knot with
a positive integral lens space surgery supports the tight contact structure on S 3 .
The discussion makes clear that if a fibered knot in S 3 supports the tight contact structure
ξst then the Bennequin inequality is sharp for that knot. In fact, more generally one can
79

see from the construction of Thurston and Winkelnkemper that if (K, F ) is an integral open
book decomposition of Y , then K is transverse to ξ(K, F ) and slξ(K,F ) (K) = −χ(F ). Thus
if ξ(K, F ) is tight then the Eliashberg-Bennequin inequality implies that F has maximum
Euler characteristic among Seifert surfaces of K.
In [17], Etnyre and Van Horn-Morris considered whether, given a contact structure ξ on
Y , and an integral open book (K, F ) of Y , the equality slξ (K) = −χ(F ) is sufficient to say
that K supports ξ. The question was generalized in [7] to the setting of a rational open
book decompostion, with K being Q-nullhomologous. The answer, stated below, gives a
nice geometric meaning to the sharpness of the Eliashberg-Bennequin inequality in (Y, ξ).

Theorem 5.1.3. Let (K, F ) be a fibered, transverse rationally null-homologous link in a
contact 3-manifold (Y, ξ) such that ξ restricted to the exterior of K is tight. Suppose that
[K] has order r in H1 (Y ). Then r · slξ (K) = −χ(F ) if and only if either ξ = ξ(K, F ) or is
obtained from ξ(K, F ) by adding Giroux torsion along incompressible tori in the complement
of K.

5.2

Surgeries on fibered knots, contact structures and
Berge knots

One of the fundamental constructions in low-dimensional topology is that of Dehn surgery.
If a knot K ⊂ M has Dehn surgery that yields some 3-manifold N , then the core of the
filling is a knot in N , which we call the dual knot to this surgery. Note that this dual knot
has a surgery that yields M .
If the surgery coefficient on K was an integer p, then [K ] has order p in H1 (N ). Fur80

thermore, the complement of a tubular neighborhood of K in M and of K in N are homeomorphic. Thus, if K was fibered in M , then K is fibered in N , with the boundary of a
fiber being some (p, s)-sloped curve in the boundary of a neighborhood of K . Thus K supports a rational open book decomposition of N , and we may consider the associated contact
structure.
Recently, Hedden and Plamenevskaya [27] considered this situation when K is fibered,
with fiber surface F , and supports a contact structure ξ on M with c(ξ) = 0. They were
able to prove the following.
Theorem 5.2.1. Suppose that K ⊂ M is fibered, with fiber surface F , let g be the genus of
F , and suppose that K supports a contact structure ξ on M such that c(ξ) = 0. If K ⊂ N
is the dual to p-surgery on K, and p ≥ 2g, then K supports a contact structure ξ on N
with the property c(ξ ) = 0.
Theorem 5.2.1 has an application to a class of knots arising in the Berge Conjecture,
which attempts to characterize all knots in S 3 which admit a surgery resulting in a lens
space. This problem was first addressed by Moser [38], who showed that all torus knots
admit such a surgery. By the cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen
[14], the remaining knots would need to admit integer surgeries giving a lens space. A
conjectured solution was provided through the work of John Berge [11]. He describes knots
that are doubly primitive with respect to a standard genus 2 Heegaard splitting of S 3 and
shows that all such knots have a surgery that yields a lens space. It is conjectured that this
completes the list.
Recent work on the Berge conjecture has indicated that it may be more tractable to
consider the “inverse” problem: what knots in some given lens space L(p, q) admit an integer
81

surgery that yields S 3 ? When K ⊂ S 3 is a Berge knot, we understand K to carry information
about its framing on the genus 2 surface. When we mention the dual to some Berge knot, this
framing is part of the data. The surgery so described by this framing produces a dual to the
Berge knot that is 1-bridge in a genus 1 Heegaard splitting of the lens space. Furthermore,
this dual admits a grid number one diagram (see Lemma 1 and Theorem 2 in [11]).
As mentioned above, Berge knots are fibered. Furthermore, they each are the binding
of an open book decomposition of S 3 that supports the standard (tight) contact structure,
by Theorem 5.1.1. Using monopole Floer homology, it was shown by Kronheimer-MrowkaOzsv´th-Szab´ [35] that if p-surgery on a knot in S 3 yields a lens space, then 2g − 1 ≤ p,
a
o
where g is the Seifert genus of the knot. Combining this with work of Greene that solves
the lens space realization problem [24], we see that the surgery coefficient on a Berge knot
must be at least twice the Seifert genus of the knot. Thus, we may apply Theorem 5.2.1, and
conclude that the dual knot K in L(p, q) supports a contact structure ξ that has c(ξ ) = 0,
and so ξ is tight.
Below we use Theorem 5.1.3 and the formulae developed in Section 4.2 to see which duals
of Berge knots support the universally tight contact structure on that lens space. The moral
of our results is that one should not expect a Berge dual to support the universally tight
contact structure. Berge’s list separated double primitive knots into 3 types: knots in a solid
torus, knots on a genus 1 fiber surface, and 4 “sporadic families” of knots. Our computations
will deal only with the knots on a genus 1 fiber surface.

82

5.3

Computations

The only genus 1 fibered knots in S 3 are the right- and left-handed trefoils and the figure
eight knot. We remark that every embedded closed curve on the Seifert surface of such knots
may be represented either as the closure of a positive braid or a negative braid. This was
shown to be the case in Ken Baker’s thesis [6], and we illustrate the case of the right-handed
trefoil in Figures 5.3 and 5.4. The maximum Euler characteristic of such closures, as they
are strongly quasipositive knots (or mirrors thereof), can be found as the Euler characteristic
of the quasipositive Seifert surface constructed above.

g1
g2

Σ

Figure 5.1: Berge knots on a Seifert surface of the trefoil
Consider a knot embedded on the Seifert surface of a right-handed trefoil, with fiber
surface Σ shown in Figures 5.3 and 5.4. To see how Σ (and a Berge knot) sits on a genus
2 Heegaard surface, refer to Figure 5.1. Every Berge knot in this family is homologous to
a[g1 ] + b[g2 ], where g1 , and g2 are the cores of the handles of Σ as shown in Figure 5.1, and
a and b are relatively prime. If K is such a knot, with [K] = a[g1 ] + b[g2 ] on Σ, then K is
given a framing by Σ, and this is the framing for the surgery. It is shown in [11] that the
prescribed surgery on K yields L(p, q), where p = a2 + ab + b2 and q ≡ a2 b−2 (mod p).
83

Proposition 5.3.1. Let K ⊂ S 3 be a double primitive knot on a Seifert surface Σ of the
right-handed trefoil as described above with [K] = a[g1 ] + b[g2 ]. Also let p = a2 + ab + b2 and
q ≡ a2 b−2 (mod p). If K ⊂ L(p, q) is the dual knot to K, then K supports the universally
tight contact structure on L(p, q) if and only if a = 1 or b = 1.
Remark 5.3.2. Note that if a = 1 or b = 1 then K is a torus knot.
Proof. The maximum Euler characteristic of a Seifert surface for K is χ(K) = 2a + 2b −
a2 − ab − b2 , as Figures 5.3 and 5.4 indicate that K is the closure of a negative braid with
index a + b and ab + a2 − a + b2 − b crossings.
We will show by computation that the maximum self-linking number of K cannot make
the Bennequin bound sharp for K . For our computations, we use the following lemma.
Lemma 5.3.3. Suppose r, s are integers with 0 < r < s. Then r2 + r + 1 ≡ 0 (mod s), if
and only if there is a pair of integers a, b with (a, b) = 1 such that s = a2 + ab + b2 and
r≡

a2
b2

(mod s).

Proof. First suppose that s = a2 + ab + b2 and r ≡

a2
b2

(mod s) for some relatively prime

pair (a, b). Since r3 − 1 ≡ b−6 (a6 − b6 ) = b−6 (a − b)(a3 + b3 )(a2 + ab + b2 ), we see that
r3 ≡ 1 (mod s). A similar calculation shows that

a 3 ≡ 1, so r2 ≡ a (mod s). As a
b
b

result,

r + r2 + 1 ≡
≡

a2
b2

+

a
+1
b

1
a2 + ab + b2 ,
2
b

which is a multiple of s.
84

Now, for any 0 < r < s, suppose that r2 +r +1 = ms, for some multiple m. Then ms is a
number properly represented by the positive definite quadratic form f (x, y) = x2 + xy + y 2 .
A number is so represented if and only if its prime decomposition is of the form

ms = 3ε

pk ,
i

where ε is 0 or 1 and each pi is a prime congruent to 1 (mod 3) (see, for example, [44,
Chapter 3 (p176)]). Since s must therefore have a prime decomposition of the same form, s
is also properly represented by the quadratic form f .

Let µ ≡ µ(K ). We know from Corollary 4.2.4 and the proof of Lemma 5.3.3 that

µ2 ≡ ±q −1 ≡ ±q 2 (mod p).

In fact, an examination of Berge’s proof that L(p, q) is the lens space obtained from this
surgery on K reveals that µ ≡ ±q(mod p) (see the last two paragraphs of the proof of
Lemma 3 [11]). Let Kl be the Legendrian representative of K in the universally tight
contact structure corresponding to a grid number one diagram. We get that

p · slQ (T+ (Kl )) ≡ −µλ + λ − µ ≡ −µ2 q + µq − µ
p · slQ (T− (Kl )) ≡ −µλ − λ + µ ≡ −µ2 q − µq + µ

by Corollary 4.2.7. As a consequence, we have that

p · slQ (T+ (Kl )) ≡ −1 ± q 2
85

q,

and,

which is congruent to either 2q 2 or 2q since, due to Lemma 5.3.3, q 2 + q + 1 ≡ 0(mod p).
Similarly, we see that p · slQ (T− (Kl )) is congruent to either 2q or 2q 2 .

Suppose that K ⊂ L(p, q) is the binding of an open book decomposition that supports
ξU T . Then by Theorem 5.1.3,

p · slQ (K ) = −χ(K ) = −χ(K) = a2 + ab + b2 − 2a − 2b.

The maximum self-linking number of K is realized by either the positive or negative transverse push-off of a Legendrian realization of K , and so either 2q 2 ≡ −2(a + b)(mod p) or
2q ≡ −2(a + b)(mod p). Recall that q ≡ a2 b−2 and q 2 ≡ ab−1 . Since p cannot be even, 2
is invertible. Moreover, q 3 ≡ 1(mod p), and so either of these cases implies

1 ≡ −(a + b)3 = −(a3 + 3a2 b + 3ab2 + b3 ) ≡ −(a2 b + ab2 )(mod p).

So 0 ≡ 1 + a2 b + ab2 ≡ 1 − b3 (mod p), implying that either b = 1 or b2 + b + 1 ≡ 0(mod p),
the latter only being possible if a = 1.

To see that we have the equality, p · slQ (T (Kl )) = −χ(K ), when b = 1, first recall
that a grid number 1 diagram admits a projection with no cusps, and with 0 ≤ µ < p and
0 ≤ λ < p. Let w be the writhe of this grid projection for K . Now since b = 1, we have
p = a2 + a + 1, µ = q = a2 , and λ ≡ µq = a4 . But 0 ≤ λ < p, so λ = a. We now use
Corollary 4.2.7 again and get that the equality holds if

(a2 + a + 1)w − a3 − a + a2 = a2 − a − 1,
86

(5.3.1)

since b = 1 implies χ(K ) = −a2 + a + 1. Equation 5.3.1 holds if w = a − 1. To see that
the writhe of our chosen projection is a − 1, we use that q = a2 , which implies that a grid
number one diagram with µ = ja will have its X marking in the (p − j)th fundamental
parallelogram (this follows from −ka2 ≡ ka + k(mod p) for k = ja). Thus, when µ = a2 ,
and since we have chosen the projection with λ > 0, the horizontal arc intersects the vertical
arc in every parallelogram it crosses, of which there are a − 1.
Finally, we examine Berge knots that arise as simple closed curves on the Seifert surface
Σ of a figure 8 knot. Similar to the previous setting, all such knots are homologous on Σ to
a[g1 ] + b[g2 ], where (a, b) = 1, and g1 and g2 are the cores of the handles of Σ depicted in
Figure 5.2. If K is such a knot, the lens space obtained as Σ-framed surgery on K is L(p, q),
where p = b2 − ab − a2 and q = a2 b−2 . We obtain the following result.

g1
g2

Σ

Figure 5.2: Berge knots on a Seifert surface of the figure 8 knot
Proposition 5.3.4. Let K ⊂ S 3 be a double primitive knot on a Seifert surface of the figure
8 knot, as described above, with [K] = a[g1 ] + b[g2 ]. Also, suppose that b ≥ 13 and a > b2 ,
and let p = b2 − ab − a2 and q ≡ a2 b−2 (mod p). If K ⊂ L(p, q) is the dual knot to K,
then K does not support the universally tight contact structure on L(p, q).
87

Remark 5.3.5. For lower values of a, b, it is, at best, quite rare that K supports the
universally tight contact structure. For example, among relatively prime pairs (a, b) with
a ≤ 75 and b ≤ 40, there are at most 9 such pairs.

Proof. To calculate the maximum Euler characteristic of a surface (rationally) bounded by
K , we appeal to [6, Appendix B], where we see that K is the closure of a positive braid
with index a + b and ab + (a + b)(a + b − 1) twists. Thus, the maximal Euler characteristic
of a Seifert surface for K is

χ(K) = χ(K ) = 2(a + b) − a2 − 3ab − b2 ≡ 2(a + b) − 2ab − 2b2 (mod p).

Lemma 5.3.6. Given relatively prime a, b, let p = b2 − ab − a2 , q ≡ a2 b−2 , and r ≡
ab−1 . Then q 2 − 3q + 1 ≡ 0(mod p) and r2 + r − 1 ≡ 0(mod p).

Proof. The proof is straightforward, noting that b is invertible mod p, since (a, b) = 1.

As a consequence of Lemma 5.3.6, we discover that r−1 ≡ r + 1, q −1 ≡ r + 2, and
q −1 ≡ 3 − q. Some calculations then show that −q ≡ r − 1, and so r3 − 1 ≡ −2q and
r3 + 1 ≡ 2r. As in the proof of the previous proposition, Corollary 4.2.7 implies that if Kl
is the Legendrian realization of K in (L(p, q), ξU T ) corresponding to the grid number one
diagram, then
p · slQ (T (Kl )) ≡ −µ2 q ± µq

µ(mod p),

depending on whether T (Kl ) = T± (Kl ). Berge’s work again implies that µ ≡ ±q, as before.
88

Combining this with the consequences of Lemma 5.3.6, we get

p · slQ (T (Kl )) ≡





−2q 2 , if T (K ) = T+ (K )
l
l


−2q 2 r, otherwise.


Suppose that K supports the contact structure ξU T . Then p·slQ (T (Kl )) = −χ(K ). Thus,
since p is odd, and so 2 is invertible mod p, we have

ab + b2 − (a + b) ≡



 2
−q , if T (K ) = T (K )

+ l
l

 2
−q r, otherwise.


But a + b ≡ a(1 + r−1 ) ≡ aq −1 . So if T (Kl ) = T+ (Kl ), we get that −q 3 ≡ a(b − 1). Using
that q 2 − 3q + 1 ≡ 0 one then arrives at 8a − 5b ≡ a(b2 − b)(mod p). However, since a > b2
and b ≥ 13, we get that p = a2 + ab − b2 and

0 < 8a − 5b < 8a < (b2 − b)a < ab2 < a2 < p,

a contradiction.
On the other hand, if T (Kl ) = T− (Kl ), then −q 3 r ≡ a(b − 1). This would require that
−13a + 8b ≡ a(b2 − b)(mod p). But now we have a contradiction, since

p > p − 13a + 8b > p − 13a > a(b2 − b) > 0,

the second to last inequality resulting from the fact that a > b2 , and b > 13.

89

a
b

a

b
a

b
b a

(A)

(B)
a
{

b
a
{

a
b

b

Figure 5.3: In surface (A) the embedded curve obtained from the weighted edges as indicated
is already a negative braid. In surface (B), this is not the case. By isotopy of the surface,
we get the top-left surface in Figure 5.4, where it shows how to get a negative braid closure.

90

b
a

a
b

Figure 5.4: The isotopy of Σ above starts at the top-left figure and ends at the bottom-right
figure. The isotopy from the top-right figure to the bottom-left figure is a “Reidemeister
I move” of one of the bands in the closure, which move cancels the twisting. From the
bottom-left figure to the bottom-middle figure, one performs a finger move through the full
twist.

91

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