FAMILIES OF KNOT FLOER HOMOLOGY THEORIES AND DEEPLY SLICE KNOTS

By

Tristan Wells

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2024

ABSTRACT

In this dissertation, we present the culmination of two projects, after an overview of the

primary tool involved in the research, Heegaard Floer theory. In this overview, we discuss

the origins of Heegaard Floer homology, an invariant associated to a Spinc 3-manifold, as

well as its flavors. We then present multiple flavors of knot Floer homology, a refinement of

that theory.

The first project is a structural theorem for a family of knot invariants due to Dowlin.

L-space knots are knots which admit surgeries that have simple Heegaard Floer homology

and thin knots are ones whose knot Floer homology is concentrated in a single δ-grading.

Each class of knots has well known knot Floer complexes. As such, we show that for L-space

knots and thin knots, the theories that Dowlin constructed are a change of coefficients from

an older theory, the minus flavor of knot Floer homology. Many supporting examples are

shown in its final section. The proof uses a popular cancellation lemma for chain complexes

with the special shapes involved.

The second project is a collaboration with McConkey, St. Clair, and Zhang.

In this

dissertation, we show that the Whitehead double of the dual knot to 1/n surgery on the

knot 61 in the 3-sphere is deeply slice in a contractible 4-manifold. That is, it bounds a

smoothly embedded disc in the manifold, but not in a collar neighborhood of its boundary,

the surgered manifold. This is partial progress in answering one of the Kirby questions

regarding nullhomotopic deeply slice knots, as referenced in earlier work of Klug and Ruppik.

To prove our theorem, we make use of the immersed curves perspective of bordered Floer

homology and knot Floer homology, which we introduce in previous sections.

Copyright by
TRISTAN WELLS
2024

ACKNOWLEDGEMENTS

I always wonder how it is that the preface or acknowledgements section of a work I read

comes to such great length. How can it be that there are that many on a writer’s mind when

assuming a reflective stance on a job “well-done”? Yet, as I ponder this myself, a slew of

names clamber for attention in my thoughts.

First and foremost, I am deeply indebted to both my mother and my father for their

unending emotional, mental, and financial support, deliberate or subconscious.

I cannot

imagine a life without such a formidable structure to lean on. The remainder of my family

comes next to mind, where perhaps not a single mote of discouragement can be found. My

nearby grandparents in Grand Rapids provided so much that I hesitate to move away.

Next are those that I became close to along the way. One of my closest friends, Dan, has

been solidly supportive in every way, and it has been an immense pleasure to spend these

six years together as friends, roommates, travel partners, confidants, colleagues, and almost

brothers. Something similar goes for my other roommates over the years: Bridget, Brita,

Sydney, David, Jacob, Joe, and Steven. Then there are so many affiliated with the them,

the university, or otherwise that I shant attempt to recall them here, save for my academic

brethren: Chen and Chris S., who have been particularly integrated into both my work and

home lives. Other colleagues who struggled with me include Danika, Chris P., Dean, Ivan,

Rithwik, and Rob.

Third is the entirety of the support from faculty and staff at Michigan State University,

from the janitorial staff to the administrative staff, and from Andy and Tsveta for all things

teaching to, of course, my advisory committee: Effie, Teena, Matt S., and my advisor Matt

H. Without Matt H., to guide my mathematical upbringing and all of their support in

navigating this strange sort of professional world, I would have been lost long ago.

Finally come my personal friends. As one who frequents the world of online gaming, I

have established a network of lifelong friends from both before my time at MSU and onward.

These are not limited to but must include: Ryan “Mirtsauce”, Austin “Spandex”, “Megavin”,

iv

Caleb “Inky”, JP “Hobo”, Nick “Bawitdaba”, Mal, Irvine, Dallas, Taylor, Tobias, and my

wonderful girlfriend, my dearest Marissa “PiercingPath”, who checked in with me every step

of the way.

It was my hope to incorporate my voice as much as possible in this dissertation, but

technical writing asserts its writing patterns. With luck, these meager paragraphs will lend

a hand to what I hope is a gentle introduction for any reader before the onslaught of math-

ematical rigor and depth obscures my voice entirely. For this opportunity to write my own

thanks, I am very grateful. I guess it is not so surprising authors spend time on the acknowl-

edgements.

v

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2

HEEGAARD FLOER THEORY . . . . . . . . . . . . . . . . . . .
2.1 Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Knot Floer Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 3

A STRUCTURE THEOREM FOR A FAMILY OF KNOT FLOER
HOMOLOGY THEORIES . . . . . . . . . . . . . . . . . . . . . .
3.1 Surgery and L-Space Knots
. . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Thin Knots and the δ-Grading . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Family HF Kn(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Structure Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Examples

CHAPTER 4

DEEPLY SLICE KNOTS . . . . . . . . . . . . . . . . . . . . . . .
4.1 Topological Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Bordered Floer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Invariants
4.4 Examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

5
5
13

19
20
22
23
27
31

34
34
41
51
53
57

62

vi

CHAPTER 1

INTRODUCTION

Beginning in as simple terms as possible, the content of this dissertation is low dimen-

sional topology, which is the study of manifolds of dimension four or less. For the uniniti-

ated, an n-manifold can be thought of simply as a space which appears like “ordinary” space

nearby. The easiest example is the Earth; a human walking on the Earth appears to have

two primary directions to move in: East/West or North/South, like the ant in Figure 1.1.

Consequently, we say that the 2-dimensional sphere is a 2-manifold. One of the main goals

of low dimensional topology is to classify manifolds as best we can. To do this, we construct

invariants, which are labels that do not change when looking at an equivalent manifolds,

much like labeling objects by color. The better the invariant, the finer the classification of

the objects.

There is a similar process for other topological objects, like knots. Roughly speaking,

a knot is a twisted up piece of string tied end to end. One can easily imagine a compli-

cated such object, like the first knot with six crossings displayed in Figure 1.1. Since the

1920s, topologists have been interested in knots, whether it is to tabulate them or discover

implications for other manifolds. A formal definition appears in Chapter 2. The purpose of

this dissertation is to study a particular family of knot invariants in Chapter 3, and apply

another kind of knot invariant in Chapter 4.

More formally, in 2001, Ozsváth and Szabó introduced topological invariants of a 3-

manifold paired with extra data, a Spinc structure. To a pair, (Y, s), they associate the

Heegaard Floer homology groups (cid:100)HF (Y, s), HF −(Y, s), HF +(Y, s), and HF ∞(Y, s) [OS04d].
Henceforth, HF ◦(Y, s) will refer to the groups agnostic of flavor. In a follow-up publication,

Ozsváth and Szabó explore further the properties of these invariants [OS04c]. The Heegaard

Floer homology groups enjoy many properties. First, the construction of the chain groups is

combinatorial, coming from a Riemannian surface together with a set of curves and marked

points, called a Heegaard diagram. As a result, computing the groups is reasonable. Second,

1

(a)

(b)

Figure 1.1 (a) A picture of the ant choosing between two directions on Earth’s surface (a
2-sphere), and (b) a diagram depicting a possibly complicated twisted up piece of string,
tied end to end, called a knot.

there are a number of exact triangles for the Heegaard Floer homology groups of 3-manifolds

that differ by surgery on knots, yielding additional powerful computational tools [OS04c].

Finally, and most importantly for this dissertation, Ozsváth and Szabó in [OS04b], and

independently Rasmussen in [Ras03], observed that a filtration of the chain complex for

Heegaard Floer homology arising from the presence of a null-homologous knot in (Y, s) leads

to a knot invariant called knot Floer homology.

At its core, knot Floer homology is the filtered chain homotopy type coming from a

filtration associated to a null-homologous knot in Y . Roughly, the filtration comes from the

Spinc structures in the 3-manifold with boundary obtained by removing a neighborhood of

the knot in Y . An extremely important property of knot Floer homology is that it categorifies

a much older invariant, the Alexander polynomial. That is, the Alexander polynomial of a

knot is the graded Euler characteristic of its knot Floer homology:

∆K(t) =

(cid:88)

grU ,A

(−1)grU tArank(cid:92)HF K grU (K, A).

Since its advent, knot Floer homology has received extensive study, and is an essential com-

2

ponent to both projects presented here. Thus, the second chapter (Chapter 2) is dedicated

to an overview of this theory.

Another knot invariant of active research is Khovanov homology, constructed by Kho-

vanov in [Kho00]. Like knot Floer homology, the Khovanov homology groups can be com-

puted combinatorially, although this time from a knot diagram. Also like knot Floer homol-

ogy, Khovanov homology categorifies an older knot invariant, the Jones polynomial. That is,

the Jones polynomial of a knot is the graded Euler characteristic of its Khovanov homology:

VK(q) =

(cid:88)

(−1)iqjrankKhi,j(K).

i,j

Of particular interest are spectral sequences between invariants like Khovanov homology and

Floer-theoretic invariants. In an effort to construct a spectral sequence between these two

particular theories, Dowlin constructed a family of knot invariants which are obtained by

taking quotients in the ground ring from the minus flavor of knot Floer homology, dubbed

HF Kn(K) [Dow18]. In Chapter 3, we prove a theorem regarding the structure of this family

of invariants for two classes of knots:

Theorem 1.0.1. If K ⊂ S3 is an L-space knot or a thin knot, then

HF Kn(K) ∼= (cid:0)HF K −(K) ⊗
Q[U ]

Q[U ]
(U n)

(cid:1) ⊕ Tor

Q[U ]
1

(HF K −(K); Q[U ]).

To do this, we note that chain complexes arising from L-space knots and thin knots have

special shapes [OS05a, Pet13]. L-space knots are ones which yield an L-space after Dehn

surgery along the knot, a process discussed in Section 3.1.1. L-spaces are 3-manifolds whose

Heegaard Floer homology is “simplest,” i.e.

rank(cid:100)HF (Y ) = |H1(Y ; Z)|.

Thin knots are ones whose knot Floer homology is concentrated in a single δ-grading, where

δ = 1

2(grU − grV ). Applications of a cancellation lemma on the level of chain complexes
together with the universal coefficients theorem are the essential components of the proof.

3

Finally, Chapter 4 concerns a collaborative project with McConkey, St. Clair, and Zhang,

where we outline a method of constructing deeply slice knots. These are knots in a 3-

manifold Y which are smoothly slice in a 4-manifold with boundary Y , but not slice in a

collar neighborhood of the boundary. That is, one can find a smoothly embedded disk in

the 4-manifold that goes “deep” into the 4-manifold, but not otherwise. The main theorem

in Chapter 4 is the following.

Theorem 1.0.2 (McConkey, St. Clair, W., Zhang). For the first 6-crossing knot, K = 61,

the Whitehead double of the dual knot to 1/n surgery along K, D+(µ1/n(K)), is deeply slice

in a contractible 4-manifold with boundary S3

1/n(K).

The tools used in proving this theorem are extensive. We apply the results of Chen,

Hanselman, Rasmussen, and Watson in various papers where they develop an immersed

curves package for bordered Heegaard Floer homology [Che23, CH23, HRW23, HRW22].

Since background on this theory is involved, we present an overview in Section 4.2.

4

CHAPTER 2

HEEGAARD FLOER THEORY

The story underlying this dissertation began with the Heegaard Floer homology of

closed Spinc 3-manifolds, a theory which was refined in numerous ways for 3-manifolds with

boundary. A capstone of these efforts, central to this dissertation, is the immersed curves

perspective, which arose some two decades later.

2.1 Heegaard Floer Homology

The general machinery needed to construct the Heegaard Floer homology groups is rather

extensive. For the purpose of self-containment, they are outlined here. However, proofs will

be omitted and can be found in [OS04d] and [OS04a].

2.1.1 Heegaard Diagrams

The primary objects of study for Heegaard Floer homology are closed, oriented 3-manifolds.

An essential object in Heegaard Floer theory is the idea of a Heegaard diagram represent-

ing a Heegaard splitting. A Heegaard splitting is a description of a closed 3-manifold as

the union of two genus g handlebodies identified along their common boundary. It is well

known that every closed 3-manifold admits a Heegaard splitting. A Heegaard diagram is a

2-dimensional way of encoding the information of a Heegaard splitting, which additionally

gives a full handle decomposition of the 3-manifold.

Definition 2.1.1. A genus g Heegaard diagram for the 3-manifold Y = Hα ∪ Hβ is a

closed, genus g surface F together with a set of g embedded simple closed curves α =

{α1, ..., αg}, called the α-curves, which are linearly independent in H1(F ; Z), and another

set of g embedded simple closed curves β = {β1, ..., βg}, called the β-curves, which are

linearly independent in H1(F ; Z) and intersect α transversely.

To see the Heegaard splitting arising from the Heegaard diagram, thicken F to F × [0, 1]

and attach a 3-dimensional 2-handle along each α-curve in F ×{0}. The result is a 3-manifold

with two boundary components, one which is F and the other is a 2-sphere. Then there

5

Figure 2.1 Top left: Genus 0 Heegaard diagram for S3. Top right: Genus 1 Heegaard
diagram for S3 (stabilized from the left). Bottom left: Genus 2 Heegaard diagram for S3.
Bottom right: genus 1 Heegaard diagram for S1 × S2.

is a unique way to attach a 3-ball to the 2-sphere boundary (up to orientation-preserving

homeomorphism), yielding the handlebody Hα. Using the β-curves in F × {1} similarly

yields the handlebody Hβ. Regarding the α-curves as the belt circles of 3-dimensional 1-

handles and the 3-ball as a 3-dimensional 0-handle yields a handle decomposition of Y . Some

examples are shown in Figure 2.1.

Given a handle decomposition for Y , one can construct a Morse function f : Y → R

whose index i critical points (which are non-degenerate, isolated, and finite) correspond to

attaching i-handles, and vise versa. The handlebodies in the Heegaard splitting of Y are
Hα = f −1(cid:0)[0, 3/2](cid:1) and Hβ = f −1(cid:0)[3/2, 3](cid:1) and the Heegaard surface is F = f −1(cid:0){3/2}(cid:1).
More in depth descriptions of this process can be found in [GS99] and [Mil63], while a nice

schematic is shown in Figure 2.2.

6

Figure 2.2 A schematic for the Morse function f : S3 → R. The Heegaard surface,
Σg = f −1(3/2), is shown in Figure 2.1. The attaching circles for Hα are in red, while the
attaching circles for Hβ are in blue. A gradient flow line (from top to bottom) is shown in
green.

2.1.2 Heegaard Floer Chain Complex

Heegaard Floer homology can be interpreted as “infinite-dimensional Morse homology,”

which is nicely described in the lecture notes of Hutchings [Hut02]. Given a generic auxiliary

metric on Y and a Morse function f : Y → R, consider the gradient flow lines of the gradient

vector field −∇f with respect to that metric. Then the Morse chain complex is roughly the

graded Z-module generated by critical points with the differential that counts the number

of unparametrized flow lines between critical points that differ in index by one. Some flow

lines are shown in Figure 2.2.

Recall that the Heegaard Floer groups are invariants of a 3-manifold Y together with a

Spinc structure, s. To account for s, a modification of the Heegaard diagram is in order.

7

Definition 2.1.2 (Heegaard Diagram). A pointed genus g Heegaard diagram for a closed,

oriented 3-manifold Y is a tuple (Σg, α, β, w) where

• Σg is a closed, genus g surface,

• α is a collection of g pairwise disjoint embedded simple closed curves on Σg which are

linearly independent in H1(Σg; Z),

• β is a collection of g pairwise disjoint embedded simple closed curves on Σg which are

linearly independent in H1(Σg; Z) and transverse to α,

• w is a basepoint in Σg \ (cid:0)α ∪ β(cid:1), and

• Y can be constructed as above from the data (Σg, α, β) as in Definition 2.1.1.

Now we apply an analog of Morse homology, Lagrangian Floer homology, to the sym-

metric product of the Heegaard diagram. It is well known that Symg(Σg), the set of un-

ordered g-tuples of points on Σg, is a 2g-dimensional symplectic manifold which inherits

a complex structure from a complex structure on Σg via the holomorphic quotient map

Σ×g

g → Symg(Σg). Then Tα = α1 × α2 × ... × αg and Tβ = β1 × β2 × ... × βg are two
Lagrangian submanifolds of Symg(Σg). Since α and β intersect transversely, Tα and Tβ

intersect transversely. In the Morse theoretic picture, the intersection points of the Tα and

Tβ can be thought of as a g-tuple of gradient flow lines for f on Y , which pair up the index

2 and index 1 critical points.

Like the Morse chain complex, the Heegaard Floer chain complex is generated by these

intersection points, and the differential counts “flow lines,” in the form of pseudo-holomorphic

discs between intersection points of index differing by 1. To see if there is such a disc between

intersection point x ∈ Tα ∩ Tβ and y, one checks the homology class of the loop created by

the gradient flow lines from x to y and back again. Through the equivalence

H1(Y ; Z) ∼=

H1(Σg; Z)
span{[α], [β]}

,

8

Figure 2.3 A Whitney disc thought of as the unit disc in the complex plane.

Figure 2.4 A figure depicting a loop between x and y with ϵ(x, y) = 0, in green.

the desired loop can be seen on Σg as a collection of arcs in α ∪ β connecting the components

of x and y in a particular way. For an example, see Figure 2.4. Given x and y, let ϵ(x, y)

be the image of a loop connecting x and y in H1(Y ; Z). We say x ∼ y if ϵ(x, y) = 0. Thus,

there is a (possibly empty) equivalence class of intersection points in Tα ∩Tβ for each element

of H1(Y, Z).

The group H1(Y, Z) is in one-to-one correspondence with the Spinc structures on Y .

Through Turaev’s equivalence, on can regard the latter as homology classes of vector fields

on Y outside a Euclidean ball [Tur97]. While many details of this equivalence are omitted,

the choice of basepoint w distinguishes a flow line in Y of −∇f from the index 3 critical

point to the index 0 critical point, so that an intersection point x ∈ Tα ∩ Tβ along with w

fixes a Spinc structure on Y . Thus, we have a map sw : Tα ∩ Tβ → Spinc(Y ). There is a

9

Whitney disc between x and y when sw(x) = sw(y), or equivalently, when ϵ(x, y) = 0.

Here, by a Whitney disc between x and y, we mean a map from the unit disc in C to the

symmetric product, u : D → Symg(Σg) such that u(−i) = x, u(i) = y, and the imaginary

arc, eα, of the boundary of the disc is mapped to an arc in Tα and the real arc, eβ, is mapped

to an arc in Tβ, as in Figure 2.3. Let π2(x, y) be the set

(cid:20)

(cid:0)D, eα, eβ, −i, i(cid:1), (cid:0)Symg(Σg), Tα, Tβ, x, y(cid:1)

(cid:21)
.

This set π2(x, y) is empty if ϵ(x, y) ̸= 0. Given ϕ ∈ π2(x, y), let M(ϕ) be the moduli space

of holomorphic representatives of ϕ. Let µ(ϕ) be its expected dimension, called the Maslov

index of ϕ. M(ϕ) admits a natural R action by translation by looking at a Whitney disc as

a vertical strip. (cid:99)M(ϕ) = M(ϕ)/R is a compact manifold of dimension 0 when µ(ϕ) = 1. Let
nw(ϕ) denote the algebraic intersection between ϕ and the subvariety {w} × Symg−1(Σg).

We now have everything in place to define the Heegaard Floer complex:

Definition 2.1.3. Given a closed, smooth 3-manifold Y together with a Spinc structure s

and a Heegaard diagram (Σg, α, β, w) for Y , the Heegaard Floer chain complex of (Y, s) is

freely generated over F[U, U −1] by all intersection points x ∈ Tα ∩ Tβ such that sw(x) = s,

denoted

CF ∞(Y, s; F[U, U −1]) :=

(cid:77)

F[U, U −1],

x∈Tα∩Tβ
sw(x)=s

with boundary map defined on generators by

∂∞(x) :=

(cid:88)

(cid:88)

# (cid:99)M(ϕ)U nw(ϕ)y.

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1

Theorem 2.1.4. [OS04d] CF ∞(Y, s; F[U, U −1]) is a chain complex, i.e. (∂∞)2 = 0.

Since we will see that the homology of this complex is an invariant later, we ignore the

dependence of the above definition on the Heegaard diagram (a slight abuse of notation).

Notice that when F = Z/2Z, (cid:99)M(ϕ) is either 0 or 1. Henceforth, F will be taken to be

10

the field of two elements to avoid discussion of signs. The Heegaard Floer groups of (Y, s)

are the homology groups of various subcomplexes of CF ∞(Y, s). After a discussion about

gradings, we present the theorems (without proof) necessary to establish the homology of

CF ∞(Y, s, F[U, U −1]) as a 3-manifold invariant.

2.1.3 Gradings and Flavors

The complex in Definition 2.1.3 comes equipped with a relative integral grading, called

the Maslov grading, defined on generators with π2(x, y) ̸= 0 by:

gr(x) − gr(y) = µ(ϕ) − 2nw(ϕ).

For an arbitrary choice of grading 0, we can now define the subcomplex CF −(Y, s) ⊂

CF ∞(Y, s) generated by all x with negative Maslov grading, and its quotient complex

CF +(Y, s).

As F[U ]-modules, they each admit an action via multiplication by U which lowers the

Maslov grading by 2. The kernel of this action yields another subcomplex, (cid:100)CF (Y, s), which
can be thought of as the subquotient where the differential only counts discs which have no

algebraic intersection with the subvariety {w} × Symg−1(Σg), i.e. nw(ϕ) = 0. Each variant,

dubbed CF ◦(Y, s) when not specifying the flavor, has an induced differential, and their

associated homology groups HF ◦(Y, s) are the homology groups of the corresponding chain

complex. The following essential theorem holds:

Theorem 2.1.5 ([OS04d]). The invariants HF ◦(Y, s) thought of as modules over F[U ], are

topological invariants of Y and s in that they are independent of choice of Heegaard diagram

and choice of path of almost complex structures Symg(Σg).

The preceding theorem is proven by showing invariance under pointed Heegaard moves

consisting of isotopies of curves (maintaining some kind of admissibility conditions), handle

slides, and (de)stabilizations, all of which occur in the compliment of the basepoint w.

11

2.1.4 Examples

It is illuminating to consider a common example computation of Heegaard Floer homology

for S3, omitting only the most technical details. Using the genus 1 diagram in Figure 2.1 will

allow us to look for discs in the 1-fold symmetric product of the torus, which is still just the

torus. In general, counting discs using the data of the Heegaard diagram is quite difficult.

Other tools have been developed to make computations easier and more combinatorial, such

as the methods in [SW10], [MOT09], and [LOT14], the latter of which is closely related to

methods in Section 4.2.

Let us look more closely at a genus 1 Heegaard diagram for S3 in Figure 2.5. Since Σg is

just the torus (g = 1), Tα and Tβ are simply the curves α and β respectively. The Heegaard

Floer chain complex (cid:100)CF (S3) is generated by the intersection points a, b, and c of α and β.
We can see immediately from considering the loops on Σg that ϵ(a, c) = ϵ(b, c) = ϵ(b, c) = 0,

which aligns with the fact that S3 admits only a single Spinc structure, corresponding to 0 in

its first homology. Although discussion on how to determine the Maslov index of a disc has

been omitted, it is the case that any disc from a to c does not have Maslov index 1. However,

the discs labeled ϕ1 and ϕ2 have µ(ϕ1) = µ(ϕ2) = 1, and will be counted in differentials.

Following the formulae in Definition 2.1.3, the differentials are

(cid:98)∂(a) = 0

(cid:98)∂(b) = c

(cid:98)∂(c) = 0

∂−(a) = 0

∂−(b) = U a + c

∂−(c) = 0.

Taking homology for the hat flavor, we see that the generators b and c have an arrow

canceling them, so (cid:100)HF (S3) ∼= Z/2Z = F2, generated by a. For the minus flavor, we again
see that b cancels with the linear combination U a + c, leaving a as the sole generator of

HF −(S3) ∼= F2[U ].

12

Figure 2.5 A "standard" genus 1 pointed Heegaard diagram for S3, where the beta (blue)
curve β is perturbed to give more intersection points.

2.2 Knot Floer Homology

Having introduced the machinery of Heegaard Floer homology, most of the tools needed

to construct a Heegaard Floer-theoretic invariant for knots in S3 are in place. First, let us

establish some preliminaries. A null-homologous knot in a 3-manifold Y is an isotopy class of

embeddings K : S1 (cid:44)→ Y such that [K] = 0 ∈ H1(Y ; Z). The manifold of primary concern in

this paper is S3, in which all circle embeddings are null-homologous, henceforth, “knot” will

mean an isotopy class of embeddings of S1 in S3, and will be denoted simply by K. Often,

a knot is depicted by projecting it to the 2-sphere and recording crossing information at the

double points (for a generic projection). This is called a knot diagram, and a few examples

of knot diagrams are shown in Figure 2.6. One important knot is the so-called unknot, U,

which is the only knot which has a knot diagram with no crossings and is isotopic to the

unit circle in the equator S2 of S3. A long-standing goal of low-dimensional topologists

and knot theorists is to tabulate and classify all the knots in S3, among discerning their

other properties. One such tool, which also yields interesting results in 3- and 4-manifold

topology is knot Floer homology, a refinement of Heegaard Floer homology to an invariant

pairs (S3, K). As in Section 2.1, many proofs and details may be omitted, but found in

[OS04b] or [Ras03].

13

Figure 2.6 Left: A knot diagram of the unknot. Middle: Another knot diagram of the
unknot. Right: A knot diagram of the trefoil.

2.2.1 Doubly-pointed Heegaard Diagrams

To refine the Heegaard Floer homology groups to invariants of the pair (S3, K), we need

to add extra data to Heegaard diagrams for S3. In particular, we define the following:

Definition 2.2.1 (Doubly-pointed Heegaard diagram). A doubly-pointed Heegaard diagram

for S3 compatible with a knot K (cid:44)→ S3 is a tuple (Σg, α, β, w, z) where

• (Σg, α, β, w) is a Heegaard diagram for S3 as in Definition 2.1.2, and

• z is another basepoint in Σg \ (cid:0)α ∪ β(cid:1) such that K can be recovered from w and z.

The recovery of the knot is as follows: Using the Heegaard splitting of S3 into Hα and Hβ

as in Section 2.1.1, connect w to z by a curve on Σg \α and push the curve into Hα. Similarly,

connect z to w by a curve on Σg \ β and push it into Hβ. The result is an embedding of K

into the described 3-manifold, in this case, S3. From a Morse theoretic perspective, the knot

K can be thought of as a union of two gradient flow lines of −∇f from the index 3 critical

point to the index 0 critical point, specified by the z and w basepoints, where f : S3 → R is

the Morse function giving rise to the Heegaard diagram (Σg, α, β, w, z). It is well known that

any knot in S3 (or any 3-manifold) admits a compatible doubly-pointed Heegaard diagram.

Figure 2.7 depicts a common Heegaard diagram for the trefoil.

2.2.2 Knot Floer Complexes

The additional basepoint not only records the isotopy type of the knot, but also induces a

filtration on the complex CF ∞(S3), called the Alexander filtration: Given generators x and y

14

Figure 2.7 A doubly-pointed Heegaard diagram compatible with the trefoil in S3. The
green arcs make up the knot.

for CF ∞(S3) and a disc ϕ ∈ π2(x, y), their Alexander filtration level differs by nz(ϕ)−nw(ϕ).

Definition 2.2.2. Given a knot K in S3 and a doubly-pointed Heegaard diagram H =

(Σg, α, β, w, z) compatible with K, the full knot Floer complex of the pair (S3, K) is freely

generated over F[U, V ] by all intersection points x ∈ Tα ∩ Tβ, denoted

CF KU,V (K) :=

(cid:77)

F[U, V ],

x∈Tα∩Tβ

with boundary map defined on generators by

∂U,V (x) :=

(cid:88)

(cid:88)

# (cid:99)M(ϕ)U nw(ϕ)V nz(ϕ)y.

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1

As before, we will see that the invariants derived from this complex do not depend

on the Heegaard diagram, so we often (slightly abusively) write CF KU,V (K) in place of

CF KU,V (H). In fact,

Theorem 2.2.3 ([OS04b]). The chain homotopy type of CF KU,V (K) is a topological invari-

ant of (S3, K) in that it is independent of choice of Heegaard diagram.

As before, the proof checks invariance under Heegaard moves. Also like before, the

modules carry grading information. The obvious first choice is to allow the naive bi-grading

by U and V on the ring F[U, V ] to descend to a relative grading on the full knot Floer

15

complex. The bi-grading gr = (grU , grV ) is such that gr(U ) = (−2, 0) and gr(V ) = (0, −2).
2(grU − grV ). The inherited relative
Then the Alexander grading is defined to be A = 1

gradings on CF KU,V (K) for ϕ ∈ π2(x, y) mimic the relative grading on CF ∞(S3):

grU (x) − grU (y) = µ(ϕ) − 2nw(ϕ)

grV (x) − grV (y) = µ(ϕ) − 2nz(ϕ).

Setting V = 1 and remembering only grU

recovers the definition of CF −(S3) as in Section

2.1.4. Choosing the grading of 1 in HF −(S3) ∼= F[U ] to be 0 establishes an absolute U

grading on CF KU,V (K). This process is simply forgetting the new basepoint, z. Forgetting

w instead has a symmetrical effect, and establishes the absolute V grading on CF KU,V (K).

This symmetry will be an essential computational tool much later, in Chapter 4.

2.2.3 Flavors

As is the case in Heegaard Floer homology, the filtered complexes also enjoy many vari-

ations. Since the chain homotopy type of CF KU,V (K) is invariant, all of the following

restrictions are as well. First, setting U = V = 0 gives a chain complex over F with the

restriction in the differential that one only counts discs which do not intersect either base-

point. We denote this complex, the knot Floer homology, by (cid:92)CF K(K) = (cid:76)

F⟨x⟩ with

the induced differential

x∈Tα∩Tβ

(cid:98)∂(x) :=

(cid:88)

(cid:88)

# (cid:99)M(ϕ)y.

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1
nw(ϕ)=nz(ϕ)=0

Another choice is to only set V = 0, yielding a chain complex over F[U ] with the re-

striction in the differential being that one only counts discs which do not intersect the z

basepoint, but can intersect the w basepoint. We denote this complex, the minus flavor of
knot Floer homology, by CF K −(K) = (cid:76)

F[U ] with the induced differential

∂−(x) =

x∈Tα∩Tβ

(cid:88)

(cid:88)

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1
nz(ϕ)=0

16

# (cid:99)M(ϕ)U nw(ϕ)y.

In both cases, it is more common to prescribe the bi-grading on the modules with grU

and

A, which sacrifices no information since any two gradings among grU
the third. These complexes are Z ⊕ Z-filtered complexes by powers of U and V , and it

, and A determine

, grV

is most convenient to present a filtered complex information in the integral lattice with the

Alexander filtration on the vertical axis and the algebraic filtration coming from the U action

on the horizontal axis, as in Figure 2.8.

2.2.4 Examples

We will now flesh out an example using the Heegaard diagram for S3 in Figure 2.5.

First, we refine the Heegaard diagram with the extra data coming from the trefoil knot in

S3, which we denote T2,3 since it’s the p = 2, q = 3 torus knot. The resulting diagram and

knot are shown in Figure 2.7. Using this, we can compute the modules and differentials for

various flavors of knot Floer homology. Let us consider CF K −(T2,3) and (cid:92)CF K(T2,3). Each
are modules over their respective rings with three generators, a, b, and c. The differentials,

following the definitions above are

∂U,V (a) = 0

∂U,V (b) = U a + V c

∂U,V (c) = 0

(cid:98)∂(a) = 0

(cid:98)∂(b) = 0

(cid:98)∂(c) = 0

∂−(a) = 0

∂−(b) = U a

∂−(c) = 0.

It is convenient to look at these complexes in the plane in order to take homology and

keep track of gradings, using Figure 2.8. Now it is plain to see that (cid:92)HF K(T2,3) ∼= F(0,1) ⊕
(cid:98)gr = (grU , A). Similarly, HF K −(T2,3) ∼= F(0,1) ⊕ F[U ](−2,−1), where
F(−1,0) ⊕ F(−2,−1), where
the F summand is generated by a and the F[U ] summand is generated by c and powers of U

times c.

17

U −1a

U −1b

a

U −2b

U −2c

U −1c

b

c

U 2a

U a

U b

U c

(a) CF K∞(T2,3)

a

b

c

U a

U 2a

U 2b

U 3a

U b

U c

a

b

c

(b) gCF K −(T2,3)

(c) (cid:92)CF K(T2,3)

Figure 2.8 A bunch of versions of the knot Floer complex for the trefoil.

18

CHAPTER 3

A STRUCTURE THEOREM FOR A FAMILY OF KNOT FLOER
HOMOLOGY THEORIES

The relationship between the Floer-theoretic knot invariants introduced by Ozsváth

and Szabó [OS04b] and Rasmussen [Ras03] and the quantum knot invariants of Khovanov

[Kho00] and Khovanov and Rozansky [ML04, KR08] has been an active topic in topology

for the past twenty years. These invariants categorify many knot invariants, like the Jones

polynomial and the Alexander polynomial. That is, the graded Euler characteristic of these

theories returns the respective polynomials. The first such relationship came from a spectral

sequence from the Khovanov homology of a link to the Heegaard Floer homology of its

double branched cover, constructed by Ozsváth and Szabó in [OS05b]. Since then, a slew of

spectral sequences of that form have been discovered, yielding a variety of rank inequalities

between the associated homologies as well as indicating the possibility of more interesting

relationships.

The following was a long-standing conjecture of Rasmussen from [Ras05]:

Theorem 3.0.1 (Rasmussen’s Conjecture, [Dow24]). For a knot K ⊂ S3, rk(Kh(K)) ≥
rk((cid:92)HF K(K)).

In [Ras05], Rasmussen suggests that constructing a spectral sequence is a promising way

to address this conjecture, a strategy successfully employed by Dowlin in [Dow18, Dow24].

It is natural to want to complete the diagram in Figure 3.1 with spectral sequences from the

Khovanov-Rozansky homologies, KRn(K), to a proposed Floer-theoretic analog. In [Dow18],

Dowlin defines a candidate family of knot invariants, called HF Kn(K), and presents the

generalized conjecture.

Conjecture 3.0.2. [Dow18] For a knot K ⊂ S3, there are spectral sequences KRn(K) ⇒
HF Kn(K) and KRn(K) ⇒ (cid:92)HF Kn(K).

Motivated by Conjecture 3.0.2, we study HF Kn(K), and make the following conjecture.

19

H(K)
...

KRn(K)
...

KR3(K)

HF K(K)
...

HF Kn(K)
...

HF K3(K)

KR2(K) = Kh(K)

HF K2(K)

Figure 3.1 Dotted arrows are conjectured, while solid arrows have been proven in suitable
versions.

Conjecture 3.0.3. For any knot K (or more generally, link) in S3, HF Kn(K) can be

obtained by a change of coefficients from HF K −(K).

Although further investigation is required to prove Conjecture 3.0.3, much can be said

about two wide classes of knots for which the knot Floer homology (among other invariants)

is already well understood and computable. In particular, we focus our attention on L-space

knots and Floer homologically thin knots.

3.1 Surgery and L-Space Knots

3.1.1 Dehn Surgery

One ubiquitous way to describe a 3-manifold is via Dehn surgery on a knot or link in S3.

Since knots are smooth embeddings, any K ⊂ S3 has a tubular neighborhood, ν(K), which,

after cutting S3 open by removing ν(K), can be identified with the solid torus, a genus 1

handlebody whose boundary is the “standard” torus, denoted T 2. This can be seen in Figure

3.2. We then have two 3-manifolds with boundary: ν(K) and the knot exterior, S3 \

◦
ν(K),

sometimes denoted XK. Their boundaries are ∂(S3 \ ν(K)) = ∂ν(K) = T 2. For further

details, see [Sav11].

Now that there has been some cutting, it is only natural to do some gluing. It is im-

mediately clear that identifying the common boundary of S3 \ ν(K) and ν(K) will return

S3, but other gluings via homeomorphisms of T 2 are also possible. We say Y is obtained

20

Figure 3.2 Left: A knot in S3 with a regular tubular neighborhood. Right: the standard
solid torus, with preferred basis for H1(∂(ν(K)) = T 2 labeled as λ and µ.

from surgery along K in S3 if it was obtained by removing a neighborhood of K and gluing

the boundaries of S3 \ K and ν(K) via a homeomorphism of T 2, h : T 2 → T 2. That is,

Y = S3 \ K ∪h ν(K). This idea can be extended to links by doing knot surgery on each

component consecutively.

The resulting manifold depends entirely on h. Even further, one only needs to keep track

of the image of the meridian of K, called µ, thought of as {point} × ∂D ∈ T 2. Up to isotopy,

any simple closed curve on T 2 can be given by a pair of coprime integers (p, q), keeping

track of its homology class in H1(T 2; Z) in the basis given by a certain longitude of K in

S3 \ ν(K) and the meridian of K. This longitude, called the Seifert longitude, λ, is the

curve on ∂(S3 \ ν(K)) which is homologically trivial in S3 \ ν(K). Then Y depends only on
(cid:2)h({point} × ∂D)(cid:3) = pµ + qλ ∈ H1(∂S3 \ ν(K)), where either p or q could be 0. It is well
known that any closed orientable 3-manifold can be obtained by surgery on some link in S3,

and we often write Y = S3

p/q(K) for the reduced fraction p/q ∈ Q ∪ {∞} to indicate the

surgery.

3.1.2 L-Space Complexes

The Heegaard Floer homology of L-spaces are particularly well understood. A rational

homology 3-sphere Y is an L-space if rk(cid:100)HF (Y ) = |H1(Y )|. A knot is a positive (or negative)
L-space knot if it admits a positive (or negative, respectively) surgery (p/q > 0) yielding an

L-space. We think of L-spaces as having the “simplest” Heegaard Floer homology possible.

21

U k1

x1

x2

V k2

x3

U k3

. . .

U km−2

xm−1

V km−1

xm

x1

V k1

x2

U k2

x3
...

V km−2

Figure 3.3 Plotted roughly in the integral lattice given by grU
positive staircase complex, while the right shows a negative staircase complex.

, the left shows a

U km−1

xm

xm−1
and grV

A summand frequently appearing in the full knot Floer complex of highly studied knots is

the so-called staircase complex:

Definition 3.1.1 (Staircase). A chain complex C =

F[U, V ] freely generated by {xi}m
i=1
is a staircase complex if it is in one of the forms in Figure 3.3 for some positive integers ki

m
(cid:76)
i=1

and odd integer m ≥ 3, where each xi is the generator of F[U, V ] corresponding to the i-th

summand of C.

With the following theorem, we see that staircase complexes completely characterize

L-space knot Floer homology:

Theorem 3.1.2 ([OS05a]). If a knot K is a positive (or negative) L-space knot, then

CF K −(K) is chain homotopy equivalent to a positive (or negative, respectively) staircase

complex.

The full knot Floer complex for the trefoil, CF KU,V (T2,3) as in an example in Section

2.2.4, is a staircase complex.

3.2 Thin Knots and the δ-Grading

Recall from Section 2.2.2 that full knot Floer complexes are birgraded modules by

(grU , A). There is a combination of these gradings, called the δ-grading, given by

δ = A − grU .

22

This can be thought of as a relative δ-grading on CF KU,V (K) in the following way. Given

a doubly-pointed Heegaard diagram for K ⊂ S3 and ϕ ∈ π2(x, y), the relative δ-grading is

δ(x) − δ(y) = nz(ϕ) + nw(ϕ) − µ(ϕ).

A knot K is Floer homologically thin, or just thin, if its knot Floer homology is supported

in a single δ-grading. For example, we know from the trefoil example in Section 2.2.4 that

(cid:92)HF K(T2,3) ∼= F(0,−1) ⊕ F(1,0) ⊕ F(2,1).

The δ-gradings of 1 ∈ F in each summand are all −1, so the knot Floer homology of the

right-handed trefoil is thin.

Like L-space knots, thin knot Floer complexes have a nice form. There is another model

summand to include:

Definition 3.2.1 (Box). A chain complex C =

box complex if it is in the form

4
(cid:76)
i=1

F[U, V ] freely generated by {xi}4

i=1

is a

U k

U k

x2

V l

x4

x1

−V l

x3

for some positive integers k and l, where each xi is the generator of Q[U, V ] corresponding

to the i-th summand of C.

Theorem 3.2.2 ([Pet13]). If a knot K is Floer homologically thin, then CF K −(K) is a

direct sum of box and staircases complexes each of which consist only of arrows with all U

and V powers equal to 1.

3.3 The Family HF Kn(K)

In an effort to fill in the right side of Figure 3.1, Dowlin defined a family of Floer-theoretic

knot invariants meant to enjoy similar properties to the Khovanov-Rozansky homology the-

ories. Indeed, the forthcoming invariants, HF Kn(K), have many properties beyond invari-

ance. For a detailed description, see [Dow18], as many proofs will be omitted here. For

23

completeness, the original definition for these invariants for links in an arbitrary 3-manifold

is presented. A link in a 3-manifold Y is simply a knot of multiple components, i.e. an

isotopy class of embeddings L : ⊔S1 (cid:44)→ Y . The discussion will begin with links in mind, but

quickly be constrained to the case when the link has only one component, i.e. a knot.

3.3.1 Heegaard Diagrams

Once again, a modification of the Heegaard diagram Definition 2.2.1 is needed to include

the link information:

Definition 3.3.1. A (multi-pointed) Heegaard diagram H for a link L in a closed, oriented

3-manifold Y is the data (Σg, α, β, w, z) where

• Σg is a closed, oriented, genus g surface, called the Heegaard surface.

• α (respectively, β) is a set of g + k − 1 disjoint embedded circles {α1, ..., αg+k−1} (resp.

{β1, ..., βg+k−1}) in Σ such that the alpha curves intersect the beta curves transversely.

• α and β each span a g-dimensional subspace of H1(Σg; Z), and

• w and z are each sets of k basepoints {w1, ...wk} and {z1, ..., zk} such that each compo-

nent of Σg \ α and each component of Σg \ β contains a w basepoint and a z basepoint.

As before, the manifold Y is constructed from this data by thickening Σg to Σg × [0, 1]

and attaching thickened discs along the alpha curves at Σg × {0} and along the beta curves

at Σg × {1} and then filling in the resulting 2-sphere boundaries (now multiple) with 3-balls.

The link L in Y can be seen as follows. When thickening Σg, also thicken the w and z

basepoints to get arcs w ⊔ z × [0, 1] ⊂ Σg × [0, 1]. Since each component of Σg \ α has a w

and z basepoint, we connect those arcs in w ⊔ z × [0, 1] with an arc in the 3-ball attached to

that component’s S2 boundary. We do this similarly for the Σ × {1} boundaries.

Dowlin extended this notion to a punctured Heegaard diagram which adds to H the

following data:

• a basepoint p, called the puncture, and

24

• curves αg+k and βg+k which bound discs, and which separate p from the w and z

basepoints, such that α and β still intersect transversely and span a g-dimensional

subspace of H1(Σg; Z).

Puncturing the Heegaard diagram is like considering L ⊔ U (where U is the unknot) where

the unlinked unknot has only a single basepoint p. This is just a (0, 3)-stabilization in the

sense of [OS04b].

3.3.2 Complexes

The first construction needed is, in some sense, the widest one can think of to extrapolate

the full knot Floer complex in Section 2.2.2. In fact, the following is not even a chain complex

in general, without identifications in the ground ring or restrictions on the link. In addition,

we make all definitions henceforth in this chapter with the ground field the rationals, Q,

since Dowlin’s work requires it.

Definition 3.3.2. Given a multi-pointed Heegaard diagram H = (Σg, α, β, w, z) compatible

with L, the master knot Floer complex of the pair (Y, L) is

• the Q[U, V ]-module freely generated by all intersection points x ∈ Tα ∩ Tβ, denoted

CF KU,V (H) :=

(cid:77)

Q[U, V ],

x∈Tα∩Tβ

• and the boundary map given by counting the Maslov index 1 pseudo-holomorphic discs

from one intersection point to the others, denoted (using subscripts to distinguish it

from Definition 2.2.2)

∂U,V (x) =

(cid:88)

(cid:88)

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1

# (cid:99)M(ϕ)U nw1 (ϕ)

1

nwk (ϕ)
...U
k

V nz1 (ϕ)
1

nzk (ϕ)
...V
k

y.

In general, CF KU,V (H) is not a true chain complex, as mentioned, but a curved one. Let

wa(i) be the w basepoint connected to zi via an arc in Hα, and wb(i) be the arc connected to

zi via an arc in Hβ. Then the curved complex has potential

∂2
U,V =

k
(cid:88)

(Ua(i) − Ub(i))Vi.

i=1

25

When each component of L has exactly one w and z basepoint, CF KU,V (H) is a true

complex, since a(i) = b(i) for the singular value of i.

As with other algebraic variations on complexes involving two formal variables, Dowlin

makes a particular one to define a family of complexes:

Definition 3.3.3 ([Dow18]). Given a multi-pointed Heegaard diagram H for a link L in a

3-manifold Y , define the chain complexes CF Kn(H) to be the quotient

CF Kn(H) = CF KU,V (H)

Vi −

(cid:30)(cid:18)

a(i) − U n
U n
b(i)
Ua(i) − Ub(i)

(cid:19)

,

and the differential ∂n is the induced differential from ∂U,V . If there is only one basepoint w

and one basepoint z, define

CF Kn(H) = CF KU,V (H)

(cid:30)(cid:18)

V − nU n−1

(cid:19)

.

Together with the following theorems, HF Kn(L) = H∗(CF Kn(H)) where H is any punc-

tured Heegaard diagram for L, are invariants of the pair (Y, L).

Theorem 3.3.4 ([Dow18]). The map ∂n : CF Kn(H) → CF Kn(H) satisfies ∂2

n = 0.

Theorem 3.3.5 ([Dow18]). If H1 and H2 are two punctured Heegaard diagrams with k

curves for a null-homologous link L, then CF Kn(H1) and CF Kn(H2) are chain homotopy

equivalent as Q[U1, ..., Uk]-modules.

We perform some computations in Section 3.5. These examples illustrate that, for knots,

one has a simpler equivalent definition of HF Kn(K). In fact, in the case where L = K is

a knot, HF Kn(K) can be computed from CF KU,V (K) without a punctured Heegaard dia-

gram. Dowlin presents an alternate definition, which he proves is equivalent in the following

theorem.

26

Theorem 3.3.6 ([Dow18]). If H is an unpunctured diagram for a knot K in S3 with a single

pair of basepoints w and z, then

HF Kn(K) = H∗

(cid:18) CF KU,V (H)

(U V, V − nU n−1)

(cid:19)

.

3.4 Structure Theorem

Since Q[U ] is a principal ideal domain,

H∗

(cid:0)CF K −(K) ⊗
Q[U ]

(cid:20)

Q[U ]
(U n)

(cid:1) ∼=

H∗

(cid:0)CF K −(K); Q[U ](cid:1) ⊗
Q[U ]

(cid:21)

Q[U ]
(U n)

⊕ Tor

Q[U ]
1

(HF K −(K); Q[U ]).

This fact motivates Conjecture 3.0.3 as well as the idea for the following algebraic machinery.

Recall that Dowlin’s definition of the complex CF Kn(K) for knots in Theorem 3.3.6 is

the quasi-isomorphism

CF Kn(K) ∼=

CF KU,V (K)
(U V, V − nU n−1)

.

In the examples later in Section 3.5, this complex turns out to be isomorphic to

(cid:94)CF Kn(K) := CF K −(K) ⊗
Q[U ]

Q[U ]
(U n)

which we can compute using the universal coefficient theorem. This leads to another defini-

tion:

Definition 3.4.1. Given a chain complex of free Q[U, V ]-modules C = (cid:76) Q[U, V ], define

the Q[U ]-modules

Cn :=

C
(U V, V − nU n−1)

(cid:101)Cn :=

C
(V, U n)

.

Notice that

C
(U V, V − nU n−1)

∼= C ⊗

Q[U,V ]

Q[U, V ]
(U V, V − nU n−1)

,

27

x1
U x1
U 2x1
...
U n−1x1

x2
U x2
U 2x2
...
U n−1x2

x3
U x3
U 2x3
...
U n−1x3

Figure 3.4 The tower (vector space) structure of the complex CF Kn(T2,3) (the complex
shown in Figure 3.7).For example, ∂n(x2) = U x1 + nU n−1x3 and ∂n(U x2) = U 2x1.

and similarly that

C
(V, U n)
so CF Kn(K) and (cid:94)CF Kn(K) are as in Definition 3.4.1.

∼= C ⊗

Q[U, V ]
(V, U n)

Q[U,V ]

∼=

C
(V )

Q[U ]
(U n)

,

⊗
Q[U ]

Given a chain complex C over Q[U, V ], we often have Figure 3.4 in mind for the complexes

Cn and (cid:101)Cn. Each generator corresponding to a summand has a tower of chains, and the
differential is marked by arrows between them. Figure 3.4 shows CF Kn(T2,3). This structure

also clearly shows C as a Q-vector space.

In general, Cn and (cid:101)Cn are not isomorphic as Q[U, V ]-modules or quasi-isomorphic, while

they are isomorphic as Q[U ]-modules. In fact, we have the following counterexample:

Lemma 3.4.2. Cn and (cid:101)Cn do not always have the same homology.

Proof. Consider the complex C given by

Then Cn is

and (cid:101)Cn is

On the level of homology,

Q[U, V ]

V

Q[U, V ].

nU n−1

Q[U ]
(U n)

Q[U ]
(U n)

Q[U ]
(U n)

0

Q[U ]
(U n).

H∗(Cn) ∼=

H∗( (cid:101)Cn) ∼=

Q[U ]
(U n−1)
Q[U ]
(U n)

⊕

⊕

Q[U ]
(U n−1)

Q[U ]
(U n)

.

28

U k

x2

x1

nU n−1
U k

x4

−nU n−1

x3

U k

U k

x2

0
x4

x1

0
x3

Figure 3.5 On the left: Cn for l = 1. On the right: (cid:101)Cn for l = 1.

To prove the main theorem of this chapter, we employ the following algebraic lemmata.

Lemma 3.4.3 (Cancellation lemma, [HN13]). Let (C, ∂) be a chain complex of R-modules,

freely generated by chains {xi}, and suppose that d(xk, xl) is a unit in R, where

∂(xk) =

(cid:88)

m̸=l

amxm + d(xk, xl)xl.

Then we can define a complex (C ′, ∂′), freely generated by {xi|i ̸= k, i ̸= l}, which is chain

homotopy equivalent to (C, ∂).

Lemma 3.4.4. For any box complex C, H∗(Cn) ∼= H∗( (cid:101)Cn) is an isomorphism of Q[U ]-
modules.

Proof. Any box complex can be pictorially represented as in Definition 3.2.1. Note that after

the identifications V = nU n−1 and U V = 0 in the ground ring, if l > 1,

V l = V · V l−1

= nU n−1V l−1

= U V · nU n−2V l−2

= 0.

So we see that if l > 1, the vertical arrows become zero in both Cn and (cid:101)Cn, and Cn = (cid:101)Cn.
Thus, we restrict our attention to when l = 1. In this case, Cn and (cid:101)Cn are as in Figure 3.5.
By direct computation using a similar diagram to Figure 3.4,

H∗(Cn) ∼=

(cid:19)4

(cid:18) Q[U ]
(U k)

∼= H∗( (cid:101)Cn).

29

Lemma 3.4.5. For any staircase complex C, H∗(Cn) ∼= H∗( (cid:101)Cn) is an isomorphism of Q[U ]-
modules.

Proof. Any staircase complex can be pictorially represented as in Definition 3.1.1. We break

the proof into a case for each type of staircase.

Case 1: The staircase is like the left in Figure 3.3. Using a similar argument as in the

proof of Lemma 3.4.4, we can restrict our attention to the even i for which ki = 1, which

are the only places Cn and (cid:101)Cn differ. Regarding Cn and (cid:101)Cn as Q-vector spaces (and, as
such, Q-modules), we have that ∂nxi = U ki−1xi−1 + nU n−1xi+1. Since d(xi, U ki−1xi−1) = 1,

finite applications of Lemma 3.4.3 yield a chain homotopy equivalent complex of Q-modules,

C n, where the generators U ki−1xi−1 and xi are removed. A visual representation of the

cancellations are as in Figure 3.6. After cancelling the same arrows in (cid:101)Cn, the resulting
chain complex is equal to C n as in Figure 3.6. We can readily compute H∗(Cn) and H∗( (cid:101)Cn)
towers coming in pairs of the form
by computing H∗(C n). The complex C n is simply m−1
2

in Figure 3.6, with arrows given by multiplication by U ki−1 for even i, and one single tower

with no arrows for i = m. Thus,

H∗(C n) =

m−1

2(cid:77)

s=1

(cid:32) Q[U ]
(U k2s−1)

< x2s−1 > ⊕

Q[U ]
(U k2s−1)

(cid:33)

< U n−k2s−1x2s >

⊕

Q[U ]
(U n)

< xm > .

Case 2: The staircase is like the right in Figure 3.3. The argument is exactly the same as

in case 1, except we now restrict our attention to the odd i for which Ki = 1, and we apply

Lemma 3.4.3 on the generator pairs (U n−1xi+1, U n−ki+1−1xi+2) for such i. The new complex

C n is still m−1
2

towers coming in pairs of the form in Figure 3.6, but now with arrows given

by multiplication by U ki−1 for odd i, and the singular tower with no arrows is at i = 1.

In both cases, to see the isomorphism of Q[U ]-modules, we check how U acts on homology.

Indeed, the distinction in module structures arose from the V -action on C, but the U -action

remains unchanged throughout.

The main theorem of this chapter is the following structural theorem for the shape of

HF Kn(K):

30

xi−1

U xi−1
...
U ki−1xi−1
...
U n−1xi−1

xi

U xi
...
U ki−1xi
...
U n−1xi

xi+1

U xi+1
...
U ki−1xi+1
...
U n−1xi+1

xi−1

U xi−1
...

...
U n−1xi−1

U xi
...
U ki−1xi
...
U n−1xi

xi+1

U xi+1
...
U ki−1xi+1
...
U n−1xi+1

Figure 3.6 A local picture of a cancellation for some even i where ki = 1. On the left: the
cancelled arrow appears in red in the complex Cn. On the right: the new complex, Cn,
which is chain homotopy equivalent to Cn.

Theorem 3.4.6. If K ⊂ S3 is an L-space knot or a thin knot, then

HF Kn(K) ∼= (cid:0)HF K −(K) ⊗
Q[U ]

Q[U ]
(U n)

(cid:1) ⊕ Tor

Q[U ]
1

(HF K −(K); Q[U ]).

Proof. The argument is a chain of isomorphisms given by Lemmata 3.4.5, 3.4.4, and the

universal coefficient theorem. The universal coefficient theorem says

H∗

(cid:0)CF K −(K) ⊗
Q[U ]

(cid:20)

Q[U ]
(U n)

(cid:1) ∼=

H∗

(cid:0)CF K −(K); Q[U ](cid:1) ⊗
Q[U ]

(cid:21)

Q[U ]
(U n)

⊕ Tor

Q[U ]
1

(HF K −(K); Q[U ]).

But

HF K −(K) = H∗(CF K −(K))

and Lemmata 3.4.5 and 3.4.4 yield

HF Kn(K) ∼= H∗

(cid:0)CF K −(K) ⊗
Q[U ]

Q[U ]
(U n)

(cid:1).

Piecing it all together gives the result.

3.5 Examples

3.5.1 Trefoil

As mentioned above, CF KU,V (Tp,q) can be read off from its Alexander polynomial. For

the trefoil, it is well-known that ∆(T2,3) = t−1 − 1 + t. Since the spacing between the powers

of t are only one each, we have only a single power of U and V as the maps on the complex,

31

Q[U ]

U

Q[U ]

Q[U ]
(U n)

U

Q[U ]
(U n)

0

Q[U ]

nU n−1

Q[U ]
(U n)

Figure 3.7 On the left: the complex CF K −(T2,3). On the right: the complex CF Kn(T2,3).

yielding the usual staircase picture:

Q[U, V ]

U

Q[U, V ]

V

Q[U, V ]

The complex CF Kn(T2,3) looks the same with the appropriate quotients. Both CF K −(K)

and CF Kn(K) are pictured in Figure 3.7. On the level of homology, we see the following:

HF Kn(T2,3) ∼=

Q[U ]
(U n)

⊕ Q ⊕ Q

HF K −(T2,3) ∼= Q[U ] ⊕ Q.

Notice that since Tor

Q[U ]
1

(Q ⊕ Q[U ],

Q[U ]
(U n) ) ∼= Q, by the universal coefficient theorem,

HF Kn(T2,3) ∼= HF K −(T2,3) ⊗
Q[U ]

(cid:77)

Tor

Q[U ]
1

(HF K −(T2,3); Q[U ])

Q[U ]
(U n)
Q[U ]
(U n)

(cid:77) Q

∼= (Q[U ] ⊕ Q) ⊗
Q[U ]

∼=

Q[U ]
(U n)

⊕ Q ⊕ Q.

We see similar behavior for a knot whose master complex has boxes and is not a torus knot.

3.5.2 Figure Eight

We first look at the master complex for 41, and then its quotients, as shown in Figure

3.8.

The invariants for 41 are

HF Kn(41) ∼= Q ⊕ Q ⊕ Q ⊕ Q ⊕

Q[U ]
(U n)

HF K −(41) ∼= Q[U ] ⊕ Q ⊕ Q.

32

U

U

•

V

•

•

•

V

•

U

U

•

0

•

•

•

0

•

U

•

nU n−1
U
•

•

•

nU n−1
•

Figure 3.8 On the top: the complex CF KU,V (41), where each dot represents the module
Q[U, V ]. On the left: the complex CF K −(41), where each dot represents the module Q[U ].
On the right: the complex CF Kn(41), where each dot represents the module Q[U ]
(U n)

.

Again, the universal coefficient theorem gives us the following:

HF Kn(41) ∼= HF K −(41) ⊗
Q[U ]

Q[U ]
(U n)

(cid:77)

Tor

Q[U ]
1

(HF K −(41); Q[U ])

∼=

Q[U ]
(U n)

⊕ Q ⊕ Q ⊕ Q ⊕ Q.

33

CHAPTER 4

DEEPLY SLICE KNOTS

This chapter is dedicated to work stemming from a collaborative project with three

other doctoral candidates at Michigan State University: Rob McConkey, Christopher St.

Clair, and Chen Zhang. Theorems will be labeled accordingly, and many figures were also

collaborative efforts.

The primary conclusion of the project is the following result.

Theorem 4.0.1 (McConkey, St. Clair, W., Zhang). For the first 6-crossing knot, K = 61,

the Whitehead double of the dual knot to 1/n surgery along K, D+(µ1/n(K)), is not slice in

(K) × I for n ∈ Z, but is slice in a contractible 4-manifold with boundary S3

1/n(K).

S3

1
n

Our calculations motivate the following conjecture.

Conjecture 4.0.2. If a non-trivial knot K is slice in the 4-ball, then the Whitehead double

of the dual knot to 1/n surgery along K, D+(µ1/n(K)), is not slice in S3

1
n

(K) × I for n ∈ Z,

despite being slice in a contractible 4-manifold with boundary S3

1/n(K).

Conjecture 4.0.3. If a non-trivial knot K is slice in the 4-ball and has an acyclic summand

in its knot Floer complex over F[U, V ]/(U V ) with a vertically and horizontally simplified

basis, then the Whitehead double of the dual knot to 1/n surgery along K, D+(µ1/n(K)), is

not slice in S3

1
n

(K) × I for n ∈ Z.

4.1 Topological Preliminaries

So far, little discussion on some of the most active topics in knot theory (and low-

dimensional topology, for that matter) are present. We rectify that here, where we define

the integral components to Theorem 4.0.1.

4.1.1 Satellite Knots

There is an operation on knots called satellite operations, where one constructs a new

knot from two given knots. Given a knot P embedded in the solid torus S1 × D2 and K

34

Figure 4.1 A common projection of the first 6-crossing knot, 61.

an arbitrary knot in S3, gluing S1 × D2 along their common torus boundary to S3 \ ν(K)

yields a new knot in S3, called the satellite knot P (K), where the meridians of the tori are

identified and the longitude of S1 × D2 is identified with the Seifert longitude of K. We call

P ⊂ S1 × D2 the pattern knot and K ⊂ S3 the companion knot.

The pattern appearing in Theorem 4.0.1 is the Whitehead pattern, denoted D+. Here, the

Whitehead pattern is the positively (hence the “+” in the symbol) clasped unknot wrapped

around the S1 factor of the solid torus in which its embedded. A figure involving the right

handed trefoil as companion to the Whitehead pattern is shown in Figure 4.2.

4.1.2 Slice Knots

Given a smooth 4-manifold X whose boundary is Y , we say a knot K ⊂ Y is smoothly

slice, or simply slice in X, if there exists a smoothly embedded disc D (cid:44)→ X such that

K = ∂D ⊂ ∂X = Y. A schematic of a slice disc as well as a trivial example is shown in

Figure 4.3. Sliceness is an extremely active topic for study when considering knots in S3

thought of as the boundary of the standard smooth 4-ball, B4. When not specified, when

we say “K is slice,” we mean that K ⊂ S3 is slice in B4. A particular knot invariant

coming from knot Floer homology, the τ -invariant, is a concordance invariant, and hence

can obstruct sliceness of a knot. We discuss this in further detail in Section 4.3.2. For a nice

survey on knot concordance, see [Liv05].

Recall from Section 3.1.1 that surgery along K requires removing a solid torus in S3 and

35

Figure 4.2 Left: A diagram for the right handed trefoil in S3. Right: The Whitehead
pattern, D+, in S1 × D2. Bottom: A diagram for the knot D+(T2,3) in S3.

Figure 4.3 Left: A half-dimensional schematic of a slice disc for some knot K. Right: A
slice disc schematic for the unknot, pushed into B4.

36

Figure 4.4 A half-dimensional schematic if a 2-handle attachment.

gluing it back in. The core of the surgery solid torus S1 × D2 once glued in to produce the

surgered manifold is often called the dual knot to the surgery along K. In the case of 1/n

surgery, we denote the dual knot as µ1/n(K) ⊂ S3

1/n(K). Much information is known about
dual knots as well as various flavors of their knot Floer homology, including surgery formulae

in [HHSZ22].

There is a feature of the dual knots that needs highlighting. Performing integral (that is,

p/q ∈ Z) surgery along a knot K in a 3-manifold Y corresponds to attaching 4-dimensional

2-handles, D2 × D2, to the 4-manifold X whose boundary is Y (See Figure 4.4). Therefore,

the core of the surgery torus is also the isotopic to the boundary of the cocore (the second

D2 factor) of the attached 2-handle. Thus, the dual knot is slice in the 4-manifold (Y ×

I) ∪ (2 − handle), where the slice disc is the cocore of the 2-handle itself. For a reference on

handle decompositions of 4-manifolds and Kirby calculus, see [GS99]. Moreover,

Theorem 4.1.1 ([Gor75]). If K is slice, then there is a smooth, contractible 4-manifold W

with boundary S3

1/n(K).

There is a nice Kirby diagram for such a 4-manifold W obtained from the surgery diagram

for S3

1/n(K), show in Figure 4.6. We first note that the dual knot to 1/n surgery along K
looks like a pushoff of K in the surgery diagram in Figure 4.5. We then perform a Kirby

move known as the slam dunk to obtain a surgery diagram with 0-framed surgery on K and

37

Figure 4.5 A surgery diagram for 1/n surgery along K, keeping track of the dual knot to
the surgery, µ1/n(K), in blue. Also, a Kirby diagram for a 4-manifold whose boundary is
1/n(K).
S3

Figure 4.6 A slam dunk Kirby move applied to obtain a Kirby diagram for a new
4-manifold, W , whose boundary is homeomorphic to S3

1/n(K).

n-framed surgery on a new, unknotted component, denoted L. Finally, slide the dual knot

representative over K itself to obtain the desired surgery diagram for S3

1/n(K). Now we show
that all the Kirby diagrams in Figure 4.7 are Kirby diagrams for W , since each arrow only

describes isotopies of the µ1/n(K) in S3

1/n(K).

Proposition 4.1.2. Suppose K is slice. Then the Kirby diagram on the right hand side of

Figure 4.6 describes a smooth, contractible 4-manifold W whose boundary is S3

1/n(K).

Proof. First, we note that the Kirby diagram is a surgery diagram for S3

1/n(K), by Kirby’s
Theorem [Kir78]. This is because we used only Kirby moves to manipulate the diagram,

keeping track of the isotopy class of the dual knot, in red. Since K is slice in S3 = ∂B4, we

begin by removing a neighborhood of a slice disc for K. This neighborhood is diffeomorphic

38

Figure 4.7 The process (in reverse) of obtaining a nice Kirby diagram for W , featured in
the top left.

to D2 × D2, with the first factor thought of as the slice disc, and the second as the thickening

to a neighborhood. Hence, we are really removing a 2-handle from B4 by introducing instead

its canceling 1-handle, as in the “digging a ditch” analogy in [GS99]. With this perspective,

we can see that the fundamental group of B4 \ (D2 × D2) is generated by a meridian of

the boundary of the removed disc; that is, a meridian of K. Now, we attach the other

2-handle with framing n along L, which is also a meridian of K, killing that generator in

π1(B4 \ (D2 × D2), so W = (B4 \ (D2 × D2) ∪ (2 − handle) is contractible.

The particular feature of the dual knot to 1/n surgery of import is the following.

39

Lemma 4.1.3. The dual knot to 1/n surgery along a slice knot K ⊂ S3, µ1/n(K) ⊂ S3

1/n(K),

is slice in W as in Theorem 4.1.1.

Proof. This can readily be seen using the argument above, since the blue curve is the dual

knot to n surgery on L in Figure 4.6. Thus, it is the boundary of the core of the 2-handle

attached along L, which is a smooth disc in W .

A well known fact about Whitehead doubling in S3 is that the Whitehead double of a

smoothly slice knot is again smoothly slice. Analogously, we achieve the following corollary:

Corollary 4.1.4. The Whitehead double of the dual knot to 1/n surgery along a slice knot

K ⊂ S3, D+(µ1/n(K)), is slice in W as in Theorem 4.1.1.

Proof. By Lemma 4.1.3, the dual knot is itself slice. Given an annulus with one boundary

component the dual knot and one boundary the unknot, guaranteed by sliceness, then we

can “Whitehead double” the annulus in the same fashion as the dual knot itself. Since the

Whitehead double of the unknot is again the unknot, we have presented an annulus between

the Whitehead double of the dual knot and the unknot.

In [KR21], the concept of deeply slice knots is introduced in a proposed strategy to answer

a question on the Kirby list, attributed to Akbulut.

Definition 4.1.5 (Deeply Slice). A knot K ⊂ ∂X is deeply slice in X in X if it is slice in

X but K is not slice in ∂X × I.

That is to say that not only is the given slice disc not in ∂X × I but there is no slice

disc there. In essence, this means that the slice disc is “interesting” because it requires use of

the topology of the 4-manifold X rather than just what happens near the boundary. Then

Akbulut’s question in [Kir97] can be phrased as: Are there contractible, smooth 4-manifolds

with boundary an integral homology 3-sphere which contain deeply slice knots that are null-

homotopic in the boundary? With this definition, one can interpret Theorem 4.0.1 as saying

that the Whitehead double of the dual knot is deeply slice in W described above. If the

40

generalization in Conjecture 4.0.2 holds, there would be a good place to look to find such

knots.

4.2 Bordered Floer Theory

The first main tool used to obstruct the sliceness of the Whitehead double of the dual

knot in a collar neighborhood of the boundary of W , i.e. S3

1/n(K) × I, is a version of
Heegaard Floer homology associated to 3-manifolds with boundary, called bordered Heegaard

Floer homology. The most consolidated resource on the matter is [LOT18]. Since then,

bordered Heegaard Floer homology has led to significant results in 3-manifold topology and

knot theory.

In [HRW23, HRW22, Han23], Hanselman, Rasmussen, and Watson use the

framework of bordered Heegaard Floer homology to construct a very useful description of

knot Floer homology using immersed curves, which is the description we use here.

4.2.1

Immersed Heegaard Diagrams

As with the variants of Heegaard Floer theory discussed in Chapter 2, bordered theory

also begins with a Heegaard diagram. Immersed Heegaard diagrams arise as a “composition”

of two components, corresponding to pairing theorems in bordered Heegaard Floer homology.

In this case, immersed Heegaard diagrams are built from pairing an immersed multicurve in

a marked torus, thought of as the boundary of a bordered 3-manifold, and a pointed bordered

Heegaard diagram. This definition and further discussion of immersed Heegaard Floer theory

in full generality can also be found in [CH23].

Definition 4.2.1. An immersed doubly-pointed Heegaard diagram is a tuple H = (Σg, α, β, w, z)

where

• Σg is a closed oriented genus g surface,

• α = {α1, ..., αg−1, αg} is a collection of curves in Σg where {α1, ..., αg−1} are embed-

ded and disjoint, αg = {α1

local systems for which α1
g

g, ..., αn
has the trivial local system, {α1, ..., αg−1, α1

g } is a collection of immersed curves decorated with
g} are linearly
is trivial in H1(Σg; Z)/(α1, ..., αg−1) for i > 1,

independent in H1(Σg; Z), and each αi
g

41

• β = {β1, ..., βg} is a collection of embedded disjoint curves in Σg which are linearly

independent in H1(Σg; Z), and

• w and z are basepoints on Σg lying in the same component of Σg \ α and in the same

component of Σg \ β.

Often times, and in this discussion, the local systems involved are all trivial. Also, we

often denote αg by αim for “immersed alpha curves,” following conventions in [CH23]. As

usual, there are admissibility conditions on these kinds of Heegaard diagrams. We refer to

[CH23] for details.

4.2.2 Knot Floer Complexes

We now discuss another way to get the knot Floer chain complex, now from an immersed

Heegaard diagram.

Definition 4.2.2. Given an immersed doubly-pointed Heegaard diagram H = (Σg, α, β, w, z),

the “U V equals zero” knot Floer complex of H is freely generated over R = F[U, V ]/(U V )

by all intersection points x ∈ Tα ∩ Tβ, denoted

CF KR(H) :=

(cid:77)

R⟨x⟩,

x∈Tα∩Tβ

with boundary map defined on generators by

∂R(x) :=

(cid:88)

(cid:88)

# (cid:99)M(ϕ)U nw(ϕ)V nz(ϕ)y.

y∈Tα

(cid:84) Tβ

ϕ∈π2(x,y)
µ(ϕ)=1

Working over the ring R = F[U, V ]/(U V ), often called the “U V equals zero” ring, has

become increasingly popular in studying knot Floer homology, and is useful in eliminating

pesky arrows in the chain complex. As is the case for the full knot Floer complex in Section

2.2.2, CF KR(H) supports a bigrading by (grU , grV ) or by (grU , A) defined the same way. As
is obligatory, the following theorems guarantee an invariant chain complex.

Theorem 4.2.3 ([CH23]). The complex (CF KR(H), ∂R) is a chain complex, i.e. ∂2

R = 0.

42

Theorem 4.2.4 ([CH23]). The bigraded chain homotopy type of (CF KR(H), ∂R) is invari-

ant under isotopies of the α curves and β curves, handeslides, and (de)stabilization of the

Heegaard diagram, H.

Later, in the discussion of pairing theorems, we will see that this complex is chain homo-

topy equivalent to the full knot Floer complex of the compatible knot with the identification

U V = 0 in the ground ring. To see the relationship between a knot-3-manifold pair and the

knot Floer complex, we turn our attention to the two pieces paired to create an immersed

Heegaard diagram.

4.2.3 Doubly-pointed Bordered Heegaard Diagrams

A doubly-pointed bordered Heegaard diagram for a pair (Y, K) is to a bordered Heegaard

diagram for Y as a doubly-pointed Heegaard diagram for a pair (Y, K) is to a Heegaard

diagram for Y . That is, we first start with the following information which encodes a bordered

3-manifold (Y, Z, ϕ), where Z and ϕ are some auxiliary data specifying a parametrization

of the boundary of Y .

Definition 4.2.5 (Bordered Heegaard Diagram). A bordered Heegaard diagram for a smooth

3-manifold Y with boundary is a tuple (Σg, α, β, w) where

• Σg is a genus g surface with a single boundary component,

• β is a collection of g pairwise disjoint properly embedded simple closed curves in the

interior of Σg which are linearly independent in H1(Σg; Z),

• α is a collection of g − k pairwise properly disjoint embedded simple closed curves

αc := {αc

1, ..., αc

g−k} in the interior of Σg and 2k pairwise disjoint properly embedded

arcs αa := {αa

1, ..., αa

2k} in Σg with transverse intersection with ∂Σg, and

• w is a point on ∂Σg \ (α ∩ ∂Σg).

Given a bordered Heegaard diagram, reconstruction of the bordered 3-manifold is very

similar to that of an ordinary Heegaard diagram. We outline the notable difference here.

43

Since there are now arcs αa in the diagram, we must complete them to circles to attach

handles along them. Roughly speaking, this is done using the data of a point matched circle

that encodes the parametrization of the boundary of the bordered 3-manifold. For further

details, see [LOT18].

As we have come to expect, we have the following theorem.

Theorem 4.2.6 ([LOT18]). Any bordered 3-manifold can be represented by some bordered

Heegaard diagram.

To complete the analogy, we now incorporate the information of a knot present in a

bordered manifold.

Definition 4.2.7 (Doubly-pointed bordered Heegaard diagram). A doubly-pointed bordered

Heegaard diagram for Y compatible with a knot K (cid:44)→ Y is a tuple (Σg, α, β, w, z) where

• (Σg, α, β, w) is a bordered Heegaard diagram for Y as in Definition 4.2.5 and

• z is a basepoint along with w in Σg \ (αc ∪ β) such that K can be recovered from w

and z.

The recovery of the knot is exactly the same as it is for Definition 2.2.1. As usual,

every knot in a bordered 3-manifold can be realized by a doubly-pointed bordered Heegaard

diagram. These diagrams are the first pieces in constructing an immersed Heegaard diagram.

4.2.4

Immersed Curves

The second ingredient in an immersed Heegaard diagram is the immersed multicurve in

the marked boundary of a knot compliment, the marked torus. The marked torus is simply

the standard torus, thought of as R2/Z2 in the plane, where the x-axis is the preferred

longitude and the y-axis is the preferred meridian, and a basepoint z located at (1 − ϵ, 1 − ϵ)

for as small ϵ > 0 as we like. An immersed multicurve is a set of immersed curves in the

marked torus away from z decorated with local systems, which we suppress, since our result

44

does not concern them directly. We often simply say immersed curve to mean an immersed

multicurve.

Immersed curves conveniently package the information of CF K −(Y, K) in the form of the

bordered invariant, sometimes denoted (cid:100)HF (Y \ ν(K)), which is an invariant of a bordered 3-
manifold arising from a bordered Heegaard diagram. For discussions on the immersed curve

formulation of bordered Heegaard Floer homology, see [HRW23], [HRW22], and [Han23].

Technically, to carry out this packaging, a particular basis is required for CF K −(Y, K).

Recalling that CF K −(Y, K) comes with two filtrations, one by the action of multiplying by

U and the other by the Alexander filtration, as in [HRW22], we may choose a representative of

the chain homotopy type of CF K −(Y, K) for which the boundary map ∂− strictly decreases

one of these filtrations. A filtered basis for CF K −(K) is {vi} such that the equivalence classes
{[vi]} in the associated graded complex gCF K −(Y, K) = (cid:76) FA(x)/FA(x)−1 are a basis.

Definition 4.2.8 ([HRW22]). A filtered basis {vi} is vertically simplified if for each vi,

either ∂vi ∈ U · CF K −(K) or ∂vi ∼ vj + x where x ∈ U · CF K −(K). The filtered basis is

horizontally simplified if for each vi with Alexander filtration level A(vi) = l, either A(∂vi) < l

or A(∂vi) = U kvj + x where A(U kvj) = l and A(x) < l.

Being vertically simplified can be thought of as requiring that each basis generator only

has one vertical arrow pointing to or away from it, and likewise for horizontally simplified.

The complex in Figure 2.8 is both vertically and horizontally simplified. For any knot,

CF K −(Y, K) always admits a vertically simplified basis and always admits a horizontally

simplified basis. As a warning, it may not be the case that CF K −(Y, K) admits a basis

which is simultaneously vertically and horizontally simplified. Nevertheless, we restrict our

attention to when there is a basis which is both.

We now describe an algorithm for obtaining an immersed curve from CF K −(Y, K),

following the method in [HRW22]. It is most convenient to present this curve as a lift in the

cover of the marked torus, [−1/2, 1/2] × R, where the interval corresponds to the preferred

longitude λ and {−1/2} × R and {1/2} × R are identified and is a lift of the preferred

45

meridian, µ. We then center the lifts of the basepoint z at {0} × {n + 1/2} for n ∈ Z. This

setup, along with a curve obtained from the following procedure, can be seen in Figure 4.8.

Proposition 4.2.9 ([HRW22]). Given a horizontally and vertically simplified basis for

CF K −(K), a lift of the immersed multicurve αim = (cid:100)HF (Y \ ν(K)) in the infinite strip can
be obtained by the following procedure:

1. For each basis element vi of CF K −(K), place a short horizontal segment at

[−1/4, 1/4] × {t} where t = A(vi).

2. If CF K −(K) contains a vertical arrow from vi to vj, then connect the left endpoints

of the horizontal segments corresponding to vi and vj by a vertical arc.

3. If CF K −(K) contains a horizontal arrow from vi to vj, then connect the right endpoints

of the horizontal segments corresponding to vi and vj by a vertical arc.

4. Connect the unique horizontal segment with an unattached left endpoint to {−1/2}×{0}

and the unique horizontal line segment with an unattached right endpoint to {1/2}×{0}

each with an arc.

4.2.5 Pairing Theorems

Ordinarily, bordered Heegaard Floer homology is presented using type D modules and

type A modules over differential graded algebras associated to the boundaries of the manifolds

involved. Then a special model of the derived tensor product, the box tensor product, is used

to obtain a representative of the chain homotopy class of the Heegaard Floer complex or knot

Floer complex from Section 2.1. The genius and beauty of the immersed curves perspective

is that we can now package all of the complicated algebraic information into pictures of

curves. Many important invariants can then be read directly from the immersed Heegaard

diagrams coming from pairing immersed curves with bordered Heegaard diagrams.

From a doubly-pointed bordered Heegaard diagram H as in Definition 4.2.5 and an

immersed curve αim as described in Section 4.2.4, there is a way to pair them by gluing to

46

a

b

c

U a

U 2a

U 2b

U 3a

U b

U c

U 2c

(a)

Figure 4.8 (a) CF K −(T2,3). (b) Each step of the construction in Proposition 4.2.9, with a
projection to the marked torus on the bottom left.

(b)

47

obtain a doubly-pointed immersed Heegaard diagram H(αim) as in Definition 4.2.1. While

a more detailed description can be found in [CH23], essentially, the two diagrams are glued

along their common boundary, thought of as “filling in” the bordered Heegaard diagram with

the immersed curve. After some isotopies, the resulting doubly-pointed immersed Heegaard

diagram looks like a superposition of the bordered Heegaard diagram and the immersed

curve. Figure 4.9 illustrates this process. One can remove many immersed points by looking

at the curves in various lifts of the torus, such as the infinite strip. To see why this is useful,

it is best to introduce the necessary pairing theorem, as it is one of the primary theorems

applied to prove Theorem 4.0.1.

Theorem 4.2.10 ([CH23]). Let H be a doubly-pointed bordered Heegaard diagram for a

pattern knot P ⊂ S1 × D2, and let αK be the immersed multicurve associated to a companion

knot K. Let H(αK) be the immersed doubly-pointed Heegaard diagram obtained by pairing

H with αK. Then the CF KR(H(αK)) is bigraded chain homotopy equivalent to the knot

Floer complex of the satellite knot P (K) over R, where R = F[U, V ]/(U V ).

Theorem 4.2.10 is actually a generalization of an earlier theorem of Chen in [Che23],

presented below. In [CH23], they remark that Theorem 4.2.10 is particularly useful when

the pattern knot is a (1,1) pattern, which means that it has a genus 1 doubly-pointed bordered

Heegaard diagram. This is because the resulting immersed Heegaard diagram is also genus

1, so it is easy to extract CF KR(H(αim)) even when the curves self-intersect. In fact, the

process is entirely combinatorial. We will see a classic example in Section 4.4. The earlier

theorem, while less general, is still useful to present here as its proof more carefully specifies

the homeomorphism of the pairing of the knot compliment and the pattern torus:

Theorem 4.2.11 ([Che23]). Let P ⊂ S1 × D2 be a (1, 1)-pattern knot and K in S3 a

companion. Let αK ⊂ ∂S3 \ ν(K) be the immersed curve for K, and let H be a genus 1

bordered Heegaard diagram for P , thought of as curves and basepoints in ∂S1 × D2. Let

h : ∂(S3 \ ν(K)) → ∂(S1 × D2) be an orientation preserving homeomorphism such that

48

Figure 4.9 Top left: A bordered Heegaard diagram. Top right: An immersed curve in the
marked torus. Bottom: The immersed Heegaard diagram obtained by gluing.

49

• h identifies the meridian and Seifert longitude of K with µ and λ respectively;

• h maps the z basepoint for αK to the z basepoint for H;

• there is a regular neighborhood U ⊂ ∂(S1 × D2) of z such that U ∩ (λ ∪ µ) = ∅ and

U ∪ h(αK) = ∅.

Suppose αK is connected. Then there is a grading-preserving isomorphism of chain complexes

(cid:92)CF K(H(αK)) ∼= (cid:92)CF K(S3, P (K)).

The goal is to adapt this theorem to manifolds other than S3, and extend the chain

homotopy equivalence to the U V = 0 complex CF KR rather than simply (cid:92)CF K.

Theorem 4.2.12. Let H be a doubly-pointed bordered Heegaard diagram for a pattern knot

P ⊂ S1 × D2, and let αK be the immersed multicurve associated to a companion knot K.

Let H(αK) be the immersed doubly-pointed Heegaard diagram obtained by pairing H with

αK using a framing change in accordance with 1/n surgery on K. Then the knot Floer

complex CF KR(H(αK)) is bigraded chain homotopy equivalent to the knot Floer complex of

the satellite knot P (µ1/n(K)) in S3

1/n(K) over R, where R = F[U, V ]/(U V ).

Proof. The proof of Theorem 4.2.10 passes through an arced bordered Heegaard diagram,

which is a version of a bordered Heegaard diagram for a manifold with two boundary com-

ponents. The manifold in question is S1 × D2 \ ν(P ), the compliment of the pattern knot

in the solid torus. In their proof, the parametrization of the outer boundary is the usual

meridian-longitude parametrization and the inner boundary is parametrized by the meridian

of P and a longitude of P . Here, we will parametrize the outer boundary instead with a fram-

ing change given by the 1/n surgery we will perform along K. That is, a homeomorphism
(cid:0)(S1 × D2) \ ν(P )(cid:1) → ∂(S3 \ ν(K)), given by how it acts on homology,

Φn : ∂outer





Φ∗

n =




1

0

n −1


 .

50

Now, when pairing H with αK, we simply take this map into account by adding Dehn twists

in the torus for H to skew the diagram to slope −1/n (really, we can skew either diagram

using Dehn twists, but we prefer to look at covering spaces which maintain the basis already

in place for αK rather than H = (Σg, α, β, w, z), so the linear map on H1(Σg) =< µ, λ > is

the inverse of Φ∗
n

). The remainder of the proof is unchanged from that of Theorem 4.2.10.

4.3

Invariants

All of the aforementioned diagrams grant access to a slew of numerical knot invariants,

especially concordance invariants. While we forgo details on concordance and omit definitions

of the invariants not involved in the proof of Theorem 4.0.1, we mention their existence.

4.3.1 Knot genus, τ , and ϵ

Given an immersed curve αK representing the bordered invariant of S3 \ ν(K), the

construction in Proposition 4.2.9 makes it easy to see two popular numerical invariants of

K. To do this, it is easiest (while not required) to pull the immersed curve tight, to create a

so-called pegboard diagram. For a knot, the immersed curve pulled tight will just be a vertical

strand in the neighborhood of {0} × R, where the “pegs” are located, and one homologically

horizontal strand which wraps around the infinite strip.

The genus of the knot, g(K), is simply the difference between the maximum height

achieved by αK and the minimum height achieved, rounded to the nearest integer when

pulled tight. This is the same as checking how many pegs are encompassed by the curve

and dividing by 2. The tau invariant of Ozsváth and Szabó, τ (K), can be see by starting

anywhere on the horizontal strand and tracing to the right (in the positive interval direction)

until hitting the vertical strand. The nearest integer height where they meet is τ (K). Finally,

ϵ(K), defined by Hom, is shown by the behavior of the horizontal line segment after crossing

the vertical portion. If the curve has an upward slope, ϵ(K) = 1. If downward, ϵ(K) = −1. If

it continues straight, which is only possible if τ (K) = 0 by Proposition 4.2.9, then ϵ(K) = 0

as well.

51

For example, the curve for the right handed trefoil shown in Figure 4.8 has the following

invariants: τ (T2,3) = 1, since the horizontal curve first intersects above the higher basepoint

when traveling to the right (along the green arrow indicating λ). Then, since when leaving

the vertical strand, the curve again has positive slope, µ, ϵ(T2,3) = 1 as well.

4.3.2 τα(Y, K)

From an immersed Heegaard diagram of genus 1, Chen gives a nice calculus for deter-

mining τ (K) along with the Alexander gradings of other generators in [Che23]. The calculus

introduces to the diagram A-buoys attached to the β curves. Essentially, these A-buoys are

small arrows which record the change in Alexander filtration level between two generators

as we isotope away discs only crossing the z basepoint. Isotoping discs away is akin to can-

celing components of the differential on the filtered complex. The difference in Alexander

filtration level between two intersection points indicates the lengths of differentials that we

cancel, which corresponds to “turning the page” on the spectral sequence converging to the

Heegaard Floer homology of the underlying 3-manifold, as described in Section ??.

So, in practice, we perform isotopies to cancel differentials of filtration length one until

we no longer can, recording the filtration change using A-buoys. We can then iterate this

process for length two, or three, should we like, to see further pages in the spectral sequence.

For a knot in S3, there will eventually be only one remaining intersection point, whose

Alexander grading is τ (K). However, for a knot in a 3-manifold other than S3, such as

a 3-manifold obtained by surgery, more than one intersection point may remain, since the

3-manifold might have rk(cid:100)HF (Y ) > 1.

In [HR23], Hedden and Raoux introduce the invariants τα(Y, K), which are assignments

to each Heegaard Floer homology class α ∈ (cid:100)HF (Y ) and knot K ⊂ Y a number which
records the Alexander filtration level of α. Algebraically, τα(Y, K) is the Alexander grading
of the surviving generator of (cid:92)CF K(Y, K) in the spectral sequence to (cid:100)HF (Y ). Geometrically,
τα(Y, K) gives a lower bound for the genus of surfaces with boundary K in 4-manifolds with

boundary. The theorem is as follows.

52

Proposition 4.3.1 ([HR23]). Let K be a null-homologous knot in Y . If Σ ⊂ Y × I is a

smoothly embedded oriented surface with boundary K ⊂ Y × {1}, then

τα(Y, K) ≤ g(Σ).

Corollary 4.3.2. If K ⊂ Y is slice in Y × I, then τα(Y, K) = 0 for all α ∈ (cid:100)HF (Y ).

Proof. Recall that if K is slice, then it bounds a smooth disc in Y × I. Equivalently, it

means that K cobounds a smooth annulus with the unknot. If there exists some α ∈ (cid:100)HF (Y )
with τα(K) ̸= 0, Proposition 4.3.1 implies that the genus of any such surface is nonzero, a

contradiction.

Clearly, this is a generalization of an already well known fact that the τ -invariant ob-

structs sliceness in B4. Corollary 4.3.2 also implies that we can see obstructions to sliceness

in immersed Heegaard diagrams of genus 1 using Chen’s A-buoy calculus. In fact, the same

way one detects the τ -invariant for the knot using A-buoys, one can detect the other τα(Y, K)

by checking the Alexander gradings of any surviving generators in the spectral sequence from

knot Floer homology to (cid:100)HF (Y ) (yielded by considering the filtration induced by the pres-
ence of a knot). In Figure 4.12, the satellite knot still lives in S3, which has a unique Spinc

structure, and thus only one τα, which is just the usual τ -invariant for the satellite knot. we

treat an extended example in the next subsection.

4.4 Examples

Now that the meat of the theory is introduced, we move to illuminating examples. We

turn our attention to the (1, 1) pattern knot of focus, the Whitehead double, D+, depicted

in Figure 4.2. A doubly-pointed bordered Heegaard diagram for D+, denoted simply by H

in this section, can be seen in Figure 4.10.

4.4.1 D+(T2,3)

Now, we have a plethora of immersed curves with which we can check consistency with

what we already know of knot Floer homology of satellites. We begin with the right handed

53

Figure 4.10 A genus 1 Heegaard diagram for the Whitehead pattern, D+. We think of the
two arcs {αa
1, αa
intersect ∂Σg as required.

2} as µ and λ, the two sides of Σg, the punctured torus (g = 1). They

U a

b

V c

Figure 4.11 Left: A shorthand representation of the complex CF KR(T2,3), the right
handed trefoil. Right: The immersed curve arising from the complex on the left.

trefoil, since it is a torus knot.

Its knot Floer complex over R is in Figure 2.8, which,

for simplicity, we often draw only one copy, as in Figure 4.11. Following the procedure in

Proposition 4.2.9, we obtain the immersed curve for the trefoil shown also in Figure 4.11.

From Figure 4.11, we see that τ (T2,3) = 1 and ϵ(T2,3) = 1. To establish τ (D+(T2,3)),

we pass to an immersed Heegaard diagram by paring αT2,3
determine τ (D+(T2,3)) from this immersed Heegaard diagram, we employ two steps. First,

and H, as in Figure 4.12. To

the Alexander grading of each intersection point in the diagram can be determined by looking

at their relative Alexander gradings given by Whitney discs between them. Second, we

54

Figure 4.12 A lift of an immersed Heegaard diagram for the Whitehead double of the right
handed trefoil in S3.

symmetrize the gradings so that the top-most and bottom-most are just opposite in sign.

This establishes the absolute Alexander grading of each generator of CF KR(D+(T2,3)) in

this diagram. The gradings are as follows:

x

4, 7, 11, 14

1, 3, 6, 8, 10, 13, 15

2, 5, 9, 12

A(x)

1

0

−1

Following Chen’s A-buoy calculus, we begin canceling differentials of length one, then length

two, and so on, until a single generator remains, corresponding to the single generator of

(cid:100)HF (S3). Then τ (D+(T2,3)) is the Alexander grading of the remaining generator, . The
entire manipulation of the immersed Heegaard diagram is carried out in Figures 4.13 and

4.14. We see that the only generator surviving after pulling β straight is x7, which has

A(x7) = 1. Thus, the singular τα corresponding to the only generator α ∈ (cid:100)HF (S3) is just
τ (D+(T2,3)) = 1. Consequently, D+(T2,3) is not slice in S3.

55

Figure 4.13 Cancellation of length one differentials. βd+
the z basepoints.

is essentially pulled tight across

Figure 4.14 The immersed Heegaard diagram for S3 arising from canceling all the
differentials.

56

4.4.2 D+(µ1/2(41))

We now turn our attention to an example computation for the Whitehead double of the

dual knot to 1/n surgery. Since the surgered 3-manifold is no longer S3, the expectation is

that there will be more than one generator α ∈ (cid:100)HF (S3
The immersed curve α41
pattern as seen in Figure 4.10 using the map Φ∗
2

is shown in Figure 4.15. Combining α41

1/n(K)) with which to compute τα.
with H for the Whitehead

as in the proof of Theorem 4.2.12, we arrive

at the immersed Heegaard diagram in Figure 4.16. In Figure 4.16, a convenient lift to a suit-

able covering space was chosen to remove as many immersed points as possible. This way, it

is significantly easier to execute the combinatorics of counting discs and to see the genera-

tors. From Figure 4.16, we can compute the complex CF KR(S3

1/n(K), µ1/n(K)) for K = 41
directly, or simply apply the A-buoy calculus to see the Alexander gradings of the surviving

generators after pulling the β curve straight, allowing isotopy over z basepoints. In this case,

only length one differentials need to be cancelled before arriving at Figure 4.17, where no

more useful isotopies can be made. It is clear that only generators 7, 8, 9, 10, and 11 survive.

From the A-buoys, we see that A(x7) ̸= A(x8), so τx7(D+(µ1/n(K)) ̸= τx8(D+(µ1/n(K)),
and, in particular, one of them is nonzero. By Corollary 4.3.2, D+(µ1/n(K)) cannot be

slice in S3

1/n(K) × I. Remarkably, if 41 were slice in S3 (it is not), then we could conclude
that D+(µ1/n(K)) is deeply slice in the 4-manifold described in Section 4.1.2, by applying

Corollary 4.1.4.

4.5 Proof of Main Theorem

A noteworthy feature of the computation in Example 4.4.2 is that the distinct τα-

invariants came from generators on the closed component of the immersed curve for the

knot. As discussed later, it is suspected this is usually the case. For Theorem 4.0.1, we make

use of the fact that its immersed curve has a curve corresponding to a box complex which

manifest as a some kind of “8”-looking shape in the immersed curve. Since Theorem 4.0.1 is

only for the knot 61 as stated, the proof is identical to the computation in Section 4.4.

Proof of Theorem 4.0.1. First, note that 61 ⊂ S3 is an alternating, slice knot in B4. There-

57

U a

e

b

U V d

V c

Figure 4.15 Left: A shorthand representation of the complex CF KR(41), the figure eight
knot. Right: The immersed curve arising from the complex on the left.

Figure 4.16 A lift of the immersed Heegaard diagram for the pairing of the Whitehead
double pattern with the dual knot to 1/2 surgery along the figure eight knot, 41. Notice the
change of framing for the Whitehead pattern corresponding to the 1/2 surgery on 41,
giving the β curve a −1/2 slope.

58

Figure 4.17 The immersed Heegaard diagram arising from canceling all the length one
differentials.

fore, (cid:92)HF K(61) is determined by its Alexander polynomial, ∆61(t) = −2t−1 + 5 − 2t, and its
signature, σ(61) = 0. By [OS03, theorem 3.1],

(cid:92)HF K(S3, 61, A) = F2

−1 ⊕ F5

0 ⊕ F2
1,

where A is the Alexander grading. Using the spectral sequence from (cid:92)HF K(61) to (cid:100)HF (S3), we
can reconstruct the vertical arrows present in the U V = 0 knot Floer complex, as in Figure

4.18. Now, using the symmetry granted by swapping the roles of U and V , we reconstruct

horizontal arrows in CF KR(61) and plot some more of the generators in the (grU , grV ) plane,
as in Figure 4.19.

Now that we have the full information CF KR(61) and in a horizontally and vertically

simplified basis, we can construct its immersed curve using the method in Proposition 4.2.9,

shown in Figure 4.20. Notably, the curve strikingly resembles the curve for 41 as in Figure

4.15, with another closed component overlapping the first. Since in Example 4.4.2 only

the closed component is necessary to see differing τα-invariants, we will only keep track of

one of these “8” shapes in the pairing diagram with the bordered Heegaard diagram for the

Whitehead double, H. In this case, the local picture near generators x7 and x8 in Figure

59

a

c

f

h

e

b

d

g

j

a

c
f

h

b

de
g

j

Figure 4.18 Left: (cid:92)HF K(61) arranged by Alexander grading. Right: the location of the
vertical arrows in CF KR(61) if there is to only be one generator for (cid:100)HF (S3).

U a

U c

U b

U d

f

h

e
g

j

Figure 4.19 Left: adding horizontal arrows in the plane using symmetry along grU = grV
Right: a short hand representation of the complex CF KR(61).

.

Figure 4.20 The immersed curve for 61, α61

.

60

4.16 looks identical (after isotoping away the other curve components) to the local picture

for the pairing of H and α61
. The same two generators, then, still have τx7(D+(µ1/n(K))) ̸=
τx8(D+(µ1/n(K))) for K = 61, so one is nonzero. By Corollary 4.3.2, D+(µ1/n(K)) cannot be
1/n(K)×I. However, since 61 is itself slice, Corollary 4.1.4 implies that D+(µ1/n(K))
slice in S3

is deeply slice in W , the contractible 4-manifold of Proposition 4.1.2.

61

BIBLIOGRAPHY

[CH23] Wenzhao Chen and Jonathan Hanselman, Satellite knots and immersed heegaard

floer homology, arXiv preprint arXiv:2309.12297 (2023).

[Che23] Wenzhao Chen, Knot floer homology of satellite knots with (1, 1) patterns, Selecta

Mathematica 29 (2023), no. 4, 53.

[Dow18] Nathan Dowlin, A family of sln-like invariants in knot floer homology, arXiv

preprint arXiv:1804.03165 (2018).

[Dow24]

, A spectral sequence from khovanov homology to knot floer homology, Jour-

nal of the American Mathematical Society (2024).

[Gor75]

[GS99]

C McA Gordon, Knots, homology spheres, and contractible 4-manifolds, Topology
14 (1975), no. 2, 151–172.

Robert E Gompf and András Stipsicz, 4-manifolds and kirby calculus, no. 20,
American Mathematical Soc., 1999.

[Han23]

Jonathan Hanselman, Knot floer homology as immersed curves, arXiv preprint
arXiv:2305.16271 (2023).

[HHSZ22] Kristen Hendricks, Jennifer Hom, Matthew Stoffregen, and Ian Zemke, An invo-

lutive dual knot surgery formula, arXiv preprint arXiv:2205.12798 (2022).

[HN13] Matthew Hedden and Yi Ni, Khovanov module and the detection of unlinks, Ge-

ometry & Topology 17 (2013), no. 5, 3027–3076.

[HR23] Matthew Hedden and Katherine Raoux, Knot floer homology and relative adjunc-

tion inequalities, Selecta Mathematica 29 (2023), no. 1, 7.

[HRW22] Jonathan Hanselman, Jacob Rasmussen, and Liam Watson, Heegaard floer ho-
mology for manifolds with torus boundary: properties and examples, Proceedings
of the London Mathematical Society 125 (2022), no. 4, 879–967.

[HRW23]

, Bordered floer homology for manifolds with torus boundary via immersed

curves, Journal of the American Mathematical Society (2023).

[Hut02] Michael Hutchings, Lecture notes on morse homology (with an eye.

[Kho00] Mikhail Khovanov, A categorification of the jones polynomial.

[Kir78]

[Kir97]

Robion Kirby, A calculus for framed links in s 3, Inventiones mathematicae 45
(1978), no. 1, 35–56.

Rob Kirby, Problems in low-dimensional topology, Proceedings 1993 Georgia In-
ternational Topology Confence, Geometric Topology 2 (1997).

[KR08] Mikhail Khovanov and Lev Rozansky, Matrix factorizations and link homology ii,

Geometry & Topology 12 (2008), no. 3, 1387–1425.

62

[KR21] Michael Klug and Benjamin Ruppik, Deep and shallow slice knots in 4-manifolds,
Proceedings of the American Mathematical Society, Series B 8 (2021), no. 17,
204–218.

[Liv05]

Charles Livingston, A survey of classical knot concordance, Handbook of knot
theory, Elsevier, 2005, pp. 319–347.

[LOT14] Robert Lipshitz, Peter S Ozsváth, and Dylan P Thurston, Computing hf by fac-

toring mapping classes, Geometry & Topology 18 (2014), no. 5, 2547–2681.

[LOT18] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston, Bordered heegaard floer

homology, vol. 254, American Mathematical Society, 2018.

[Mil63]

John Willard Milnor, Morse theory, no. 51, Princeton university press, 1963.

[ML04]

Khovanov Mikhail and Rozansky Lev, Matrix factorizations and link homology.

[MOT09] Ciprian Manolescu, Peter Ozsváth, and Dylan Thurston, Grid diagrams and hee-

gaard floer invariants, arXiv preprint arXiv:0910.0078 (2009).

[OS03]

Peter Ozsváth and Zoltán Szabó, Heegaard floer homology and alternating knots,
Geometry & Topology 7 (2003), no. 1, 225–254.

[OS04a]

, Heegaard diagrams and holomorphic disks, Different faces of geometry,

Springer, 2004, pp. 301–348.

[OS04b]

, Holomorphic disks and knot invariants, Advances in Mathematics 186

(2004), no. 1, 58–116.

[OS04c]

, Holomorphic disks and three-manifold invariants: properties and appli-

cations, Annals of Mathematics (2004), 1159–1245.

[OS04d]

, Holomorphic disks and topological invariants for closed three-manifolds,

Annals of Mathematics (2004), 1027–1158.

[OS05a]

, On knot floer homology and lens space surgeries, Topology 44 (2005),

no. 6, 1281–1300.

[OS05b]

, On the heegaard floer homology of branched double-covers, Advances in

Mathematics 194 (2005), no. 1, 1–33.

[Pet13]

[Ras03]

[Ras05]

Ina Petkova, Cables of thin knots and bordered heegaard floer homology, Quantum
Topology 4 (2013), no. 4, 377–409.

Jacob Andrew Rasmussen, Floer homology and knot complements, Harvard Uni-
versity, 2003.

Jacob Rasmussen, Knot polynomials and knot homologies, arXiv preprint
math/0504045 (2005).

63

[Sav11]

[SW10]

[Tur97]

Nikolai Saveliev, Lectures on the topology of 3-manifolds: an introduction to the
casson invariant, Walter de Gruyter, 2011.

Sucharit Sarkar and Jiajun Wang, An algorithm for computing some heegaard
floer homologies, Annals of mathematics (2010), 1213–1236.

Vladimir Turaev, Torsion invariants of spinΘc-structures on 3-manifolds, Mathe-
matical Research Letters 4 (1997), no. 5, 679–695.

64