HEEGAARD FLOER D-INVARIANTS AND ITS APPLICATIONS

By

Chen Zhang

A DISSERTATION

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

Mathematics—Doctor of Philosophy

2024

ABSTRACT

This thesis studies Heegaard Floer d-invariants and its applications. The first result is applying

d-invariants and linking form obstruction to prove an algebraically slice linear combination of L-

space knots is not smoothly slice. The second result is the computation of d-invariants of splicing

of circle bundles of higher genus surface.

Copyright by
CHEN ZHANG
2024

ACKNOWLEDGEMENTS

Throughout my six years at MSU, I received tremendous help from many people.

I would like to express my deepest gratitude to my advisor, Matt Hedden, who has always been

patient and encouraging during my six years at MSU. Under his guidance, I have learned various

branches of mathematics. Additionally, he has taught me invaluable lessons in facing failure,

living a better life, and becoming a better person. I would also like to acknowledge the members

of my committee, Effie Kalfagianni, Thomas Parker, and Matt Stroffregen, for their guidance in

mathematics and their service on my committee. I am thankful to Wenzhao Chen, Peter Johnson,

and Abhishek Mallick, with whom I have had many helpful discussions. Furthermore, I extend

my gratitude to my peers, Chris St. Clair and Tristan Wells, as well as everyone in the MSU

topology group, who are my reasons for going to the office every day. Lastly, I am grateful to

Chuck Livingston for his assistance during my job application.

I would like to thank my friends for being helpful and supportive, and for the wonderful times

I have spent with them playing games, sharing food, and traveling.

Finally, I would like to thank my parents, Wei Zhang and Yongxia Ji, and my girlfriend, Xiaofu

Dai, for being my safe harbor no matter what difficulties I encounter.

iv

TABLE OF CONTENTS

CHAPTER 1

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

CHAPTER 2

BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

5

CHAPTER 3

NONSLICENESS OF ALGEBRAICALLY SLICE KNOTS . . . . . . . 19

CHAPTER 4

D-INVARIANTS OF SPLICING OF CIRCLE BUNDLES OF HIGHER
GENUS SURFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 37

BIBLIOGRAPHY .

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APPENDIX .

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. 66

v

CHAPTER 1

INTRODUCTION

In the early 2000s, Ozsváth and Szabó introduced a collection of invariants for 3- and 4-

manifolds called Heegaard Floer homology [OS04b][OS04a]. Given a based 3-manifold (𝑌 , 𝑧)
with a Spin𝑐 structure 𝔰 ∈ Spin𝑐 (𝑌 ), Heegaard Floer homology associates F2 [𝑈]-modules with

different flavors

𝐻𝐹−(𝑌 , 𝑧, 𝔰),

𝐻𝐹+(𝑌 , 𝑧, 𝔰),

𝐻𝐹∞(𝑌 , 𝑧, 𝔰), (cid:100)𝐻𝐹 (𝑌 , 𝑧, 𝔰).

Heegaard Floer theory has been generalized in various senses and provides numerous invariants

for the study of 3- and 4-dimensional manifolds, as well as contact and symplectic topology and

knots. In this thesis, we will explore the Heegaard Floer d-invariants, which are invariants that

encode the grading information of 3- and 4-dimensional manifolds and have various applications.

For example, they have been used to reprove Donaldson’s diagonalizable theorem in [OS03a,

Theorem 9.1] and to study cosmetic surgery in [NW15].

We will talk about several computations of the d-invariants on certain manifolds and their

applications.

1.1 Nonsliceness of algebraically slice knots

In [Rud76], Rudolph asks whether the set of algebraic knots is linearly independent in the

knot concordance group C. An algebraic knot is, by definition, the connected link of an isolated

singularity of a polynomial map 𝑓

: C2 → C.

It can also be defined as an iterated torus knot

𝑇𝑝1,𝑞1;··· ;𝑝𝑛,𝑞𝑛 with indices satisfying 𝑝𝑖, 𝑞𝑖 > 0 and 𝑞𝑖+1 > 𝑝𝑖𝑞𝑖 𝑝𝑖+1.

A knot is algebraically slice if it is in the kernel of Levine’s classifying homomorphism [Lev69].

Livingston and Melvin [LM83] observed that, for any knot 𝐾,

𝐾𝑝,𝑞1#𝑇𝑝,𝑞2# − 𝐾𝑝,𝑞2# − 𝑇𝑝,𝑞1

is an algebraically slice knot. Here, 𝐾𝑝,𝑞 is the ( 𝑝, 𝑞)-cable of 𝐾.

When 𝐾 is an algebraic knot, all the components in the above connected sum, up to mirror

images, are algebraic knots provided the 𝑞𝑖’s are large enough. Since the sliceness of a knot implies

1

that it is algebraically slice, it is interesting to ask when the knot above is slice.

In [KHL12],

Hedden, Kirk and Livingston used Casson-Gordon invariants [CG86] to show that

Theorem 1.1 ([KHL12]). For appropriately chosen integers 𝑞𝑖,

𝑇2,3;2,𝑞𝑛#𝑇2,𝑞1# − 𝑇2,3;2,𝑞1# − 𝑇2,𝑞𝑛

are not slice.

Using the metabelian Blanchfield pairing, introduced by Miller and Powell [MP18], this result

is generalized by Conway, Kim, and Politarczyk [CKP23] to the following theorem.

Theorem 1.2 ([CKP23]). Fix a prime power p. Let S𝑝 be the set of iterated torus knots

𝑇 ( 𝑝, 𝑞1; 𝑝, 𝑞2; · · · ; 𝑝, 𝑞𝑙), where the sequences (𝑞1, 𝑞2, · · · , 𝑞𝑙) of positive integers that are co-

prime to p satisfy

1) 𝑞𝑙 is a prime;

2) for 𝑖 = 1, · · · , 𝑙 − 1, the integer 𝑞𝑖 is coprime to 𝑞𝑙 when 𝑙 > 1;

The set S𝑝 is linearly independent in the topological knot concordance group 𝐶𝑡𝑜 𝑝.

In this thesis, we use the d-invariants obstruction from [HLR12] to show that for any L space

knot 𝐾,

Theorem 1.3 ([Zha23a]). 𝐾2,𝑘1#𝑇2,𝑘2# − 𝐾2,𝑘2# − 𝑇2,𝑘1 has infinite order in the knot concordance
group C when 𝑘1 and 𝑘2 are any pair of distinct prime numbers greater than 3.

As a corollary, we obtain a generalization of Theorem 1.1:

Corollary 1.3.1 ([Zha23a]). 𝐾2,𝑘1#𝑇2,𝑘2# − 𝐾2,𝑘2# − 𝑇2,𝑘1 has infinite order in the knot concordance
group C when 𝐾2,𝑘1 and 𝐾2,𝑘2 are algebraic knots.

Note that all results about the nonsliceness of 𝐾2,𝑘1#𝑇2,𝑘2# − 𝐾2,𝑘2# − 𝑇2,𝑘1 have used Casson-

Gordon invariants and our result is the first to use the Heegaard Floer homology. Moreover, the

2

result of Conway, Kim and Politarczyk have the same 𝑝 in the iterated cabling, whereas our results

can cover cases when we use different 𝑝 values in the iterated cabling.

1.2 d-invariants of splicing of circle bundles on higher genus surface

We compute the d-invariants of the splicing of circle bundles on higher genus surface in

the pursuit of a Heegaard Floer proof of the 10/8 theorem.

In 1982, Matusumoto conjectured

that if 𝑀 is a closed spin manifold, then 𝑏2(𝑀) ≥ (11/8)|𝜎(𝑀)|. This is known as the 11/8

conjecture. Here 𝑏2(𝑀) represents the second Betti number and 𝜎(𝑀) denotes the signature.

Furuta proved the 10/8 theorem by studying the Pin(2) Seiberg-Witten theory [Fur01], which states

that 𝑏2(𝑀) ≥ (10/8)|𝜎(𝑀)| + 2.

Using Heegaard-Floer homology, in collaboration with Hedden, we relate the proof of the 10/8

theorem to the computation of d-invariants of certain manifolds illustrated in Figure 1.1, which is

the boundary of a 4 manifold with intersection form

𝐻 =

0 1

1 0









.









Figure 1.1 is the Kirby diagram of the manifolds 𝐻𝑔1,𝑔2. Here, 𝑔𝑖 denotes the genus of each

surface and this graph denotes the splicing of two circle bundles with Euler number 0.

Figure 1.1 𝐻𝑔1,𝑔2

We compute d invariants using the link surgery formula from [MO10, Theorem 1.1].

Theorem 1.4 ([Zha23b]). 𝑑top(𝐻𝑔1,𝑔2) = |𝑔1 − 𝑔2| − 2, when min(𝑔1, 𝑔2) ≥ 1.

3

More specifically, we decompose 𝐻𝑔1,𝑔2 into three components, using Zemke’s bordered link

surgery description along with the connected sum formula, to derive the link surgery complex for

the framed link in Figure 1.1.

1.3 Outline

The thesis is organized as follows. In Chapter 2, we review Heegaard Floer homology, including

the definition of d-invariants and the statement of link surgery formula. In Chapter 3, we give the

proof of Theorem 1.3, which includes a discussion of the topology of branched covers and the

computation of d-invariants.

In Chapter 4, we relate the 10/8 theorem to d-invariants, review

Zemke’s general surgery formula, and use it to compute the d-invariants. In Appendix ??, we apply

lattice homology to prove the special case of Theorem 1.3, where 𝐾 = 𝑇2,3.

4

CHAPTER 2

BACKGROUND

2.1 Preliminaries on Heegaard Floer homology and correction term

2.1.1 Heegaard Floer homology

Given a closed, oriented, based 3-manifold 𝑌 with a Spin𝑐 structure 𝔰, Ozsváth and Szabó defined

an invariant known as Heegaard Floer homology in [OS04b]. These invariants are F2 [𝑈]-modules,
which come in different flavors, (cid:100)𝐻𝐹, 𝐻𝐹−, 𝐻𝐹∞ and 𝐻𝐹+.

In fact, these groups are related by functorially associated long exact sequences:

· · · → (cid:100)𝐻𝐹 (𝑌 , 𝔰)

ˆ𝜄
−→ 𝐻𝐹+(𝑌 , 𝔰)

𝑈
−→ 𝐻𝐹+(𝑌 , 𝔰) → · · ·

and

· · · → 𝐻𝐹−(𝑌 , 𝔰)

𝜄
−→ 𝐻𝐹∞(𝑌 , 𝔰)

𝜋
−→ 𝐻𝐹+(𝑌 , 𝔰) → · · ·

Later in this thesis, we will use 𝐻𝐹◦ to denote the Heegaard Floer homology when we do not

specify the flavor. We will define 𝐻𝐹◦ in Section 2.2, where we define the link Floer homology, and

treat the Heegaard Floer homology of closed 3-manifolds as a special case of link Floer homology,

such that there is only one basepoint on the Heegaard diagram.

When 𝑊 is a smooth cobordism from a three-manifold 𝑌1 to 𝑌2, equipped with a Spin𝑐 structure

𝔰 whose restrictions to the two boundary components are 𝔰1 and 𝔰2, respectively, then there are

induced maps between the Heegaard Floer homology:

𝐻𝐹◦(𝑌1, 𝔰1)

𝐹◦

𝑊 ,𝔰−−−→ 𝐻𝐹◦(𝑌1, 𝔰2).

Note that in the case where 𝔰 is a torsion Spin𝑐 structure on 𝑌 , 𝐻𝐹◦(𝑌 , 𝔰) can be endowed with

a relative Z grading. In Theorem 7.1 of [OS06], it has been shown that 𝐻𝐹◦(𝑌 , 𝔰) can be given

an absolute Q grading ˜gr which lifts the relative Z grading. It is uniquely characterized by the

following properties:

1) ˆ𝜄, 𝜄 and 𝜋 above preserve the absolute grading

5

2) (cid:100)𝐻𝐹 (𝑆3) is supported in absolute grading zero

3) If 𝑊 is a cobordism from 𝑌1 to 𝑌2, and 𝜉 ∈ 𝐻𝐹∞(𝑌1, 𝔰1), then

˜gr(𝐹𝑊,𝔰 (𝜉)) − ˜gr(𝜉) =

𝑐1 (𝔰)2−2𝜒(𝑊)−3𝜎(𝑊)
4

,

where 𝔰𝑖 = 𝔰|𝑌𝑖 for 𝑖 = 1, 2.

2.1.2 Correction terms

The definition of the Heegaard Floer correction terms depends on the structure of 𝐻𝐹∞(𝑌 , 𝔰).

When 𝑏1(𝑌 ) = 0, 𝐻𝐹∞(𝑌 , 𝔰) = F[𝑈, 𝑈−1] [OS04a, Theorem 10.1]. With the absolute grading,

Ozsváth and Szabó defined a numerical invariant called the correction term, denoted by 𝑑 (𝑌 , 𝔰)

[OS03a, Definition 4.1]:

Definition 2.1. 𝑑 (𝑌 , 𝔰) = min𝛼≠0∈𝐻𝐹+ (𝑌 ,𝔰) { ˜gr(𝛼)|𝛼 ∈ Im𝑈 𝑘 , for all 𝑘 ≥ 0}.

The above definition is equivalent to the original definition provided in [OS03a], as both are the

grading of the bottom element of the unique non-torsion tower. The terminology correction term

reflects that 𝑑 (𝑌 , 𝔰) is the correction term in the formula [OS03a, Theorem 1.3], which relates the

Euler characteristic of the reduced Heegaard Floer homology and the Casson invariant.

The d-invariants satisfy the following two properties,

1) (Additivity) 𝑑 (𝑌 #𝑌 ′, 𝔰#𝔰′) = 𝑑 (𝑌 , 𝔰) + 𝑑 (𝑌 ′, 𝔰′): that is, 𝑑 is additive under connected sums.

2) (Vanishing) Suppose (𝑌 , 𝔰) = 𝜕 (𝑊, 𝔱), where 𝑊 is a Q-homology ball and 𝔱 is a Spin𝑐 structure

on 𝑊 that restricts to 𝔰 on 𝑌 . Then 𝑑 (𝑌 , 𝔰) = 0.

When 𝑏1(𝑌 ) > 0, 𝐻𝐹◦(𝑌 , 𝔰) is acted upon by the exterior algebra Λ∗(𝐻1(𝑌 ; F)/Tors. This

action is natural with respect to the cobordism map in the following sense:

if elements 𝛾𝑖 ∈

𝐻1(𝑌𝑖)/Tors for 𝑖 = 1, 2, are homologous in 𝑊, then

𝐹𝑊,𝔰 (𝛾1 · 𝜉) = 𝛾2 · 𝐹𝑊,𝔰 (𝜉).

We say that 𝐻𝐹∞(𝑌 ) is standard if for each torsion Spin𝑐 structure 𝔰0,

6

𝐻𝐹∞(𝑌 , 𝔰0) (cid:27) (Λ𝑏𝐻1(𝑌 ; 𝐹))⊗FF[𝑈, 𝑈−1]

as Λ𝑏𝐻1(𝑌 ; 𝐹)⊗FF[𝑈, 𝑈−1]-modules, where 𝑏 = 𝑏1(𝑌 ). The Λ𝑏𝐻1(𝑌 ; 𝐹)⊗FF[𝑈, 𝑈−1] is induced
by the interior product between 𝐻1(𝑌 ) and 𝐻1(𝑌 ).

Osváth and Szabó [OS04a, Theorem 10.1] proved that 𝐻𝐹∞(𝑌 ) is standard when 𝑏1(𝑌 ) ≤ 2.

When 𝑏1(𝑌 ) ≥ 3, it depends on the triple cup product structure by the results of Lidman [Lid10].

When 𝐻𝐹∞(𝑌 ) is standard, we can specify two generators using 𝐻1(𝑌 ) action: a "bottom-most"

generator which is in the kernel of the action by 𝐻1(𝑌 ) and a "top-most" generator which is acted
on non-trivially by any non-zero element in Λ𝑏𝐻1(𝑌 , 𝐹). Then the corresponding d-invariants are

defined similarly to Definition 2.1, which uses the grading of the bottom elements of the image of

these two towers under the map 𝜋.

Definition 2.2. Let 𝑌 be a three-manifold with standard 𝐻𝐹∞, equipped with a torsion Spin𝑐

structure 𝔰. 𝑑𝑏 (𝑌 , 𝔰) is the correction term that corresponds to the "bottom-most" generator, and

𝑑𝑡 (𝑌 , 𝔰) is the correction term corresponds to the "top-most" generator.

2.2 Preliminaries on surgery formula

Given a three-manifold 𝑌 with a surgery description, we can compute 𝐻𝐹◦(𝑌 ) with the surgery

formula. This formula requires the link Floer complex and the flip map. The Knot surgery formula

was proved by Osváth and Szabó in [OS08] [OS10]. It has been generalized by Manolescu and

Ozsváth to the case of surgery on an integral framed null-homologous link in [MO10]. Zemke

provided a bordered interpretation of the link surgery formula in [Zem21a] and generalized it to a

general surgery formula for arbitrary links in closed 3-manifolds in [Zem23].

2.2.1 Link Floer complex

In this section, we review the definition of generalized Heegaard Floer complexes for links

following the convention in [MO10] and we treat the knot Floer complex as a special case of the

link Floer complex, where the link has one connected component.

Definition 2.3. A multi-pointed Heegaard diagram consists of H = (Σ, 𝜶, 𝜷, w, z), where:

7

• Σ is a closed, oriented surface of genus 𝑔;

• 𝜶 = {𝛼1, . . . , 𝛼𝑔+𝑘−1} is a collection of disjoint, simple closed curves on Σ which span

a 𝑔-dimensional lattice of 𝐻1(Σ; Z), hence specify a handlebody 𝑈𝛼; the same goes for

𝜷 = {𝛽1, . . . , 𝛽𝑔+𝑘−1}, which specify a handlebody 𝑈𝛽;

following property. Let {𝐴𝑖}𝑘

• w = {𝑤1, . . . , 𝑤 𝑘 } and z = {𝑧1, . . . , 𝑧𝑚} (with 𝑘 ≥ 𝑚) are collections of points on Σ with the
𝑖=1 be the connected components of Σ − 𝛼1 − · · · − 𝛼𝑔+𝑘−1 and
𝑖=1 be the connected components of Σ − 𝛽1 − · · · − 𝛽𝑔+𝑘−1. Then there is a permutation
𝜎 of {1, . . . , 𝑚} such that 𝑤𝑖 ∈ 𝐴𝑖 ∩ 𝐵𝑖 for 𝑖 = 1, . . . , 𝑘, and 𝑧𝑖 ∈ 𝐴𝑖 ∩ 𝐵𝜎(𝑖) for 𝑖 = 1, . . . , 𝑚.

{𝐵𝑖}𝑘

A Heegaard diagram H describes a closed, connected, oriented 3-manifold 𝑌 = 𝑈𝛼 ∪Σ 𝑈𝛽, and

an oriented link (cid:174)𝐿 ⊂ 𝑌 (with ℓ ≤ 𝑚 components), obtained as follows. For 𝑖 = 1, . . . , 𝑚, we join 𝑤𝑖

to 𝑧𝑖 inside 𝐴𝑖 by an arc which we then push by an isotopy into the handlebody 𝑈𝛼; then we join 𝑧𝑖

to 𝑤𝜎(𝑖) inside 𝐵𝑖 by an arc which we then push into 𝑈𝛽. The union of these arcs (with the induced

orientation) is the link (cid:174)𝐿. We then say that H is a multi-pointed Heegaard diagram representing

(cid:174)𝐿 ⊂ 𝑌 .

Remark 2.1. In cases 𝑘 = 1 and 𝑚 = 0, the Heegaard diagram represents an empty link, which is

simply a Heegaard diagram for the a pointed 3-manifold.

We require that the Heegaard multi-diagram satisfies the following admissibility condition to

ensure that there are only finitely many disks counting when defining the differential:

Definition 2.4. Let H = (Σ, 𝜶, 𝜷, w, z) be a multi-pointed Heegaard diagram.

(𝑎) A region in H is the closure of a connected component of Σ − (𝛼1 ∪ · · · ∪ 𝛼𝑔+𝑘−1 ∪ 𝛽1 ∪

· · · ∪ 𝛽𝑔+𝑘−1);

(𝑏) A periodic domain in H is a two-chain 𝜙 on Σ obtained as a linear combination of regions

(with integer coefficients), such that the boundary of 𝜙 is a linear combination of 𝛼 and 𝛽 curves,

and the local multiplicity of 𝜙 at every 𝑤𝑖 ∈ w is zero.

8

(𝑐) The diagram H is called admissible if every non-trivial periodic domain has some positive

local multiplicities and some negative local multiplicities.

From now on, we will assume that all the Heegaard diagrams in this paper are admissible.

Moreover, we will use a more restrictive class of Heegaard diagrams.

Definition 2.5. A Heegaard diagram (Σ, 𝜶, 𝜷, w, z) is called link-minimal if 𝑚 = ℓ; that is, each

link component has only two basepoints.

Definition 2.6. A Heegaard diagram (Σ, 𝜶, 𝜷, w, z) for a nonempty link is called minimally-pointed

if 𝑘 = 𝑚 = ℓ; that is, each link component has only two basepoints, and there are no free basepoints.

Definition 2.7. A Heegaard diagram (Σ, 𝜶, 𝜷, w, z) is called basic if it is minimally-pointed and,

further, for each 𝑖 = 1, . . . , ℓ, the basepoints 𝑤𝑖 and 𝑧𝑖 (which determine one of the link components)

lie on each side of a beta curve 𝛽𝑖, and are not separated by any alpha curves.

Remark 2.2. Under the condition in Definition 2.7, 𝛽𝑖 is a meridian for 𝐿𝑖.

Given a link Heegaard diagram H = (Σ, 𝜶, 𝜷, w, z), which describes an 𝑙-components link in

an integral homology sphere (cid:174)𝐿 ⊂ 𝑌 , the Heegaard diagram determines tori

T𝛼 = 𝛼1 × · · · × 𝛼𝑔+𝑘−1, T𝛽 = 𝛽1 × · · · × 𝛽𝑔+𝑘−1 ⊂ Sym𝑔+𝑘−1(Σ).

We define the generators of the link Floer complex CFL−(H ) as the intersection point x ∈ T𝛼 ∩ T𝛽.
Each intersection point x ∈ T𝛼∩T𝛽 is assigned a relative Spin𝑐 structure on (𝑌 , 𝐿), via a construction

of non-vanishing vector fields, which we denote it by 𝔰(𝑥).

For x, y ∈ T𝛼 ∩ T𝛽, we let 𝜋2(x, y) be the set of homotopy classes of Whitney disks from x to y

relative to T𝛼 and T𝛽, as in [OS04b]. For each homotopy class of disks 𝜙 ∈ 𝜋2(x, y), we denote by

𝑛𝑤 𝑗 (𝜙) and 𝑛𝑧 𝑗 (𝜙) ∈ Z the multiplicity of 𝑤 𝑗 (resp. 𝑧 𝑗 ) in the domain of 𝜙. Furthermore, we let

𝜇(𝜙) be the Maslov index of 𝜙.

Each generator x of the link Floer complex is bigraded by the Maslov grading 𝑀 (x) ∈ Z and

an Alexander multi-grading, which takes value in the following set:

9

H(𝐿)𝑖 =

lk(𝐿𝑖, 𝐿 − 𝐿𝑖)
2

+ Z ⊂ Q, H(𝐿) =

H(𝐿)𝑖,

ℓ(cid:63)

where lk denotes linking number. Let us also set

𝑖=1

ℓ(cid:63)

H(𝐿)𝑖 = H(𝐿)𝑖 ∪ {−∞, +∞}, H(𝐿) =

H(𝐿)𝑖,

𝑖=1

such that

𝐴𝑖 (x) ∈ H(𝐿)𝑖, 𝑖 ∈ {1, . . . , ℓ}.

Let W𝑖 and Z𝑖 be the set of indices for the 𝑤’s (resp. 𝑧’s) belonging to the 𝑖th component of the

link. We then have

𝐴𝑖 (x) − 𝐴𝑖 (y) =

𝑛𝑧 𝑗 (𝜙) −

∑︁

𝑗 ∈Z𝑖

∑︁

𝑗 ∈W𝑖

𝑛𝑤 𝑗 (𝜙),

where 𝜙 is any class in 𝜋2(x, y).

Following [MO10], we define the completed link Floer complex CFL−(H ) as follows. We let

CFL−(H ) be the free module over R (H ) = F[[𝑈1, . . . , 𝑈𝑘 ]] generated by T𝛼 ∩ T𝛽, and equipped

with the differential:

∑︁

∑︁

𝜕x =

y∈T𝛼∩T𝛽

𝜙∈𝜋2 (x,y)
𝜇(𝜙)=1

#(M (𝜙)/R) · 𝑈

(𝜙)

𝑛𝑤
1
1

· · · 𝑈

𝑛𝑤𝑘 (𝜙)
𝑘

y.

(2.1)

Here, M (𝜙) is the moduli space of pseudo-holomorphic curves (solutions to Floer’s equation)

in the class 𝜙, and R acts on M (𝜙) by translations. Note that M (𝜙) depends on the choice of

a suitable path of almost complex structures on the symmetric product. We suppress the almost

complex structures from notation for simplicity.

The Maslov grading 𝑀 produces the homological grading on CFL−(H ), with each 𝑈𝑖 decreas-

ing 𝑀 by two. Furthermore, each Alexander grading 𝐴𝑖 defines a filtration on CFL−(H ), with 𝑈𝑖

decreasing the filtration level 𝐴𝑖 by one, and leaving 𝐴 𝑗 constant for 𝑗 ≠ 𝑖.

Remark 2.3. We use the 𝐻𝐹+ version for the knot surgery formula later in this paper, and we define

𝐻𝐹+(𝑌 , 𝐾) as a special case of CFL−(H ) here.

10

The Heegaard diagram H for (𝑌 , 𝐾) is H = (Σ, 𝜶, 𝜷, w, z), where we only have a pair of base

points (𝑤, 𝑧).

For each generator x ∈ T𝛼 ∩ T𝛽, the Alexander grading takes a value in H(𝐾) = Z, which is

given by evaluating 𝔰(𝑥) on a Seifert Surface for 𝐾. For simplicity, we will also use 𝔰(𝑥) to denote

the Z we obtain from the evaluation.

We define CFK∞ as the free module over R (H ) = F[[𝑈, 𝑈−1] generated by 𝑈𝑖𝑥, s.t. 𝑥 ∈ T𝛼∩T𝛽,

and equipped with the differential:

∑︁

∑︁

𝜕x =

#(M (𝜙)/R) · 𝑈𝑛𝑤 (𝜙)y.

(2.2)

y∈T𝛼∩T𝛽

𝜙∈𝜋2 (x,y)
𝜇(𝜙)=1

CFK− is the subcomplex of CFK∞ generated by 𝑈𝑖𝑥, s.t. 𝑖 ≤ 0 and 𝐶𝐹𝐾 + is defined as the

quotient-complex CFK∞/ CFK−.

Moreover, we can identify 𝑈𝑖𝑥 with [𝑥, 𝑖, 𝑗], s.t 𝑗 = 𝔰(𝑥) + 𝑖. Then CFK− is identified with the

subcomplex [𝑥, 𝑖, 𝑗], s.t. 𝑖 ≤ 0. From now on, we will use this version of the knot Floer complex

when we talk about the knot surgery formula.

We also define the subcomplex that will be used in the surgery formula here. Given s =

(𝑠1, . . . , 𝑠ℓ) ∈ H(𝐿), we define the generalized Heegaard Floer complex

A−(H , s) = A−(H , 𝑠1, . . . , 𝑠ℓ) = A−(T𝛼, T𝛽, s)

to be the subcomplex of CFL−(H ) generated by elements x ∈ T𝛼 ∩ T𝛽 with 𝐴𝑖 (x) ≤ 𝑠𝑖 for all

𝑖 = 1, . . . , ℓ.

Note that for the knot Floer complex, the generalized Floer complex A−(H , 𝑠) is the completion

of the subcomplex 𝐴−

𝑠 = 𝐶{max(𝑖, 𝑗 − 𝑠) ≤ 0}. The subcomplex used in the 𝐻𝐹+ version of the

knot surgery formula is 𝐴+

𝑠 = 𝐶{max(𝑖, 𝑗 − 𝑠) ≥ 0} and 𝐵+

𝑠 = 𝐶{𝑖 ≥ 0}.

2.2.2 Knot and link surgery formula

In this section, we review the link surgery formula from [MO10]. We also describe the knot

surgery formula for the plus flavor in Chapter 3 and Zemke’s bordered surgery formula in Chapter

4.

11

We now describe the algebraic structure for the link surgery formula, which is called a hypercube

of chain complexes in [MO10].

Define

E𝑛 = {0, 1}𝑛

as the set of vertices of the 𝑛-dimensional unit hypercube. If 𝜀, 𝜀′ ∈ 𝐸𝑛, we write 𝜀 ≤ 𝜀′

if the

inequality holds for each coordinate of 𝜀 and 𝜀′

.

Definition 2.8. An 𝑛-dimensional hypercube of chain complexes consists of a collection of Z-

graded vector spaces

(𝐶𝜀)𝜀∈E𝑛, 𝐶𝜀 =

𝐶𝜀
∗ ,

(cid:202)

∗∈Z

together with a collection of linear maps for each pair of indices 𝜀, 𝜀′ ∈ 𝐸𝑛 such that 𝜀 ≤ 𝜀′

𝐷𝜀,𝜀′ : 𝐶𝜀 → 𝐶𝜀′

,

The maps are required to satisfy the relations whenever 𝜀 and 𝜀′′

∑︁

𝜀≤𝜀′ ≤𝜀′′

𝐷𝜀′ ,𝜀′′ ◦ 𝐷𝜀,𝜀′ = 0.

(2.3)

The input data for the link surgery complex is called a complete system of hyperboxes H for

the link (cid:174)𝐿, which is defined in Section 8 of [MO10]. We will focus only on the basic system.

Let (cid:174)𝐿 be a 𝑛-component link and denote its components by 𝐿1, 𝐿2, · · · , 𝐿𝑛. Fix a framing Λ
for the link (cid:174)𝐿. For a component 𝐿𝑖 of 𝐿, we let Λ𝑖 be its induced framing, thought of as an element
in 𝐻1(𝑌 − 𝐿). The latter group can be identified with Z𝑛 via the basis of oriented meridians for (cid:174)𝐿.
Given a sublink 𝑀 ⊆ 𝐿, we let Ω(𝑀) be the set of all possible orientations on 𝑀. For (cid:174)𝑀 ∈ Ω(𝑀),

we let 𝐼−( (cid:174)𝐿, (cid:174)𝑀) denote the set of indices 𝑖 such that the component 𝐿𝑖 is in 𝑀 and its orientation

induced from (cid:174)𝑀 is opposite to the one induced from (cid:174)𝐿. Set

Λ (cid:174)𝐿, (cid:174)𝑀 =

∑︁

Λ𝑖 ∈ 𝐻1(𝑌 − 𝐿) (cid:27) Z𝑛.

𝑖∈𝐼− ( (cid:174)𝐿, (cid:174)𝑀)

Let 𝑌Λ(𝐿) be the three-manifold obtained from 𝑌 by surgery on the framed link (𝐿, Λ).

12

Given a basic Heegaard diagram H 𝐿 for L, the other diagrams appearing in a basic complete

system H are the reductions

H 𝐿−𝑀 := 𝑟 𝑀 (H 𝐿),

obtained from H 𝐿 by deleting the 𝑧 basepoints on the sublink 𝑀. Note that all the diagrams H 𝐿−𝑀

are link-minimal. Let us denote the remaining components by 𝑁 := 𝐿 − 𝑀.

To the diagrams H 𝐿−𝑀 we associate generalized Floer complexes 𝔄−(H 𝐿−𝑀, s). These are

modules over the ground ring

R := R (H 𝐿) = F[[𝑈1, . . . , 𝑈𝑛]].

There is a corresponding reduction on the Spin𝑐 structures

𝜓 𝑀 : ¯H(𝐿) → ¯H(𝑁).

For each remaining component in 𝐿𝑖 ⊂ 𝑁, we denote it by 𝑗𝑖. The map 𝜓 𝑀 is defined on each

component by

𝜓 𝑀
𝑖

: ¯H(𝐿)𝑖 → ¯H(𝑁) 𝑗𝑖 , 𝑠𝑖 → 𝑠𝑖 −

𝑙 𝑘 (𝐿𝑖, 𝑀
2

).

The surgery complex is the infinite direct product

𝐶−(H , Λ) =

(cid:202)

(cid:214)

𝑀 ⊆𝐿

s∈H(𝐿)

𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)).

To simplify the notation somewhat, we denote a typical term in the chain complex by

s = 𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)),
𝐶𝜀

where 𝜀 = 𝜀(𝑀) = (𝜀1, · · · , 𝜀𝑛) ∈ {0, 1}𝑛 is such that 𝐿𝑖 ⊆ 𝑀 if and only if 𝜀𝑖 = 1.

Furthermore, the differential on the complex C−(H , Λ) is given by

𝐷−(s, x) =

∑︁

∑︁

𝑁 ⊆𝐿−𝑀

(cid:174)𝑁 ∈Ω(𝑁)

(s + Λ (cid:174)𝐿, (cid:174)𝑁 , Φ

(cid:174)𝑁
𝜓 𝑀 (s) (x)),

for s ∈ H(𝐿) and x ∈ 𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)). Note that in this formula, the maps

Φ

(cid:174)𝑁

𝜓 𝑀 (s) : 𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)) → 𝔄−(H 𝐿−𝑀−𝑁 , 𝜓 𝑀∪𝑁 (s))

13

(2.4)

(2.5)

are constructed from polygon maps of the type considered in [MO10], depending on the choice of

orientation. We omit their precise definition in this thesis and only give the definition for the knot
case. Note that when 𝑁 = ∅, the map Φ (cid:174)𝑁
𝜓 𝑀 (s) is just the usual differential on 𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)),

counting holomorphic disks.

It has been shown in [MO10] that the above construction forms a hypercube of chain complexes

and its homology is isomorphic to the Heegaard Floer homology of the surgery manifold 𝑌Λ(𝐿).

Theorem 2.1 ([MO10]). Fix a complete system of hyperboxes H for an oriented, ℓ-component link

(cid:174)𝐿 in an integral homology three-sphere 𝑌 , and fix a framing Λ of 𝐿. There is an isomorphism of

homology groups:

𝐻∗(C−(H , Λ)) (cid:27) HF−

∗ (𝑌Λ(𝐿)),

(2.6)

where HF− is the completed version of Heegaard Floer homology over the power series ring

F[[𝑈]].

Remark 2.4. In Theorem 2.1, the link surgery complex involves direct product, which is different

from the knot surgery formula in [OS08], where a directed sum is used. Using directed sum only

is only appropriate for computing the plus and hat flavor using an analogous surgery formula. For

the minus flavor, the isomorphism does not hold (see the unknot computation in Example 2.2).

To achieve the isomorphism for the minus flavor, we need to complete the direct sum, which

gives us the direct product. In Section 4.2, there is a completion in the bordered surgery formula

by the same reason and we will provide a more detailed discussion there.

We describe the knot surgery formula as an example of the link surgery formula.

Example 2.1. When (cid:174)𝐿 has only one component (cid:174)𝐾, the hypercube is 1 dimensional, which forms a

mapping cone. The mapping cone complex is a F[[𝑈]]-module

𝐶 =

(cid:214)

𝐶0

s ⊕

(cid:214)

𝑠 .
𝐶1

𝑠∈Z
𝑠 is the generalized Floer complex 𝔄−(H 𝐾, 𝑠), which is isomorphic to

𝑠∈Z

Here, 𝐶0

𝐴−

𝑠 = 𝐶{max(𝑖, 𝑗 − 𝑠) ≤ 0}

14

and each 𝐶1

𝑠 is a copy of the complex

𝔄−(H ∅, 0) = CF−(H ∅),

whose homology is HF−(𝑌 ). Let us denote it by 𝐵−

𝑠 . For the 𝑚-surgery on 𝐾, the differentials

on 𝐶 consist of three parts. The first one is the self differential on 𝐴−

𝑠 and 𝐵−

𝑠 . The second is the

differential which corresponds to the map that has the same orientation of (cid:174)𝐾

𝑠 = Φ𝐾
𝑣−
𝑠

: 𝐴−

𝑠 → 𝐵−
𝑠 ,

which is the inclusion. The third map corresponds to the opposite orientation of (cid:174)𝐾

𝑠 = Φ−𝐾
ℎ−
𝑠

: 𝐴−

𝑠 → 𝐵−

𝑠+𝑚,

which is the composition of the inclusion 𝐴−

𝑠 to 𝐶{ 𝑗 ≤ 𝑠} and the isomorphism between 𝐶{ 𝑗 ≤ 𝑠}

and 𝐵−

𝑠 given by the flip map.

Thus, the complex C can be viewed as the mapping cone of the map

𝐴−

𝑠 →

(cid:214)

𝑠∈Z

𝐵−
𝑠 ,

(cid:214)

𝑠∈Z

(𝑠, x) ↦→ (𝑠, 𝑣−

𝑠 ) + (𝑠 + 𝑚, ℎ−

𝑠 ).

(2.7)

Example 2.2. Using the above description, we can compute the +1 surgery on the unknot (cid:174)𝑈 in 𝑆3.

The knot Floer complex of 𝑈 is generated by one element over F[[𝑈]], hence we have

𝑠 (cid:27) 𝐵−
𝐴−

𝑠 (cid:27) F[[𝑈]],

we denote generators by 𝑎𝑠 and 𝑏𝑠.

The flip map in this case is the identity, and the maps are

𝑣−
𝑠 =





1

if 𝑠 ≥ 0

𝑈−𝑠

if 𝑠 ≤ 0,

ℎ−

𝑠 =

𝑈 𝑠

if 𝑠 ≥ 0

1

if 𝑠 ≤ 0.





(2.8)

The homology of the complex C is then isomorphic to F[[𝑈]], being freely generated by the element

in the kernel

𝑈 |𝑠|(|𝑠|−1)/2𝑎𝑠.

∑︁

𝑠∈Z

15

The +1-surgery on the unknot is 𝑆3, which has 𝐻𝐹−(𝑆3) (cid:27) F[[𝑈]]. Hence, the above compu-

tation gives the right answer.

However, if we use the direct sum, the element (cid:205)𝑠∈Z 𝑈 |𝑠|(|𝑠|−1)/2𝑎𝑠 which generates the homology

no longer exists. Instead, we would have nontrivial cokernel in the new C = ⊕C1

𝑠 . The cokernel

would be generated as a F[[𝑈]]-module by classes [𝑏𝑖], 𝑖 ∈ Z, subjects to the relations:

[𝑏0] = [𝑏1] = 𝑈 [𝑏−1] = 𝑈 [𝑏2] = 𝑈3 [𝑏−2] = 𝑈3 [𝑏−3] = · · ·

Remark 2.5. We also need another version of link Floer homology over the ring

F[𝒰1, · · · , 𝒰𝑛, 𝒱1, · · · , 𝒱𝑛],

which is used in Zemke’s bordered reinterpretation of link surgery formula. Let us denote it by

CF L−(H ). CF L−(H ) is the free module over F[𝒰1, · · · , 𝒰𝑛, 𝒱1, · · · , 𝒱𝑛] generated by T𝛼 ∩ T𝛽.

The differential is defined similar to 2.1 except that we also count the z basepoints, and we weight

a holomorphic curve by

𝑛
(cid:214)

𝑖=1

𝑛𝑤𝑖 ( 𝜙)
𝒰
𝑖

𝒱
𝑖

𝑛𝑧𝑖 ( 𝜙)

.

We have the following identification of subcomplexes:

Proposition 2.1 ([Zem21a]). Suppose that H is a link minimal Heegaard diagram for a link 𝐿 in

𝑆3, which has no free basepoints. Let 𝑀 be a sublink of 𝐿. Write 𝑆𝑀 ⊂ F[𝒰1, . . . , 𝒰ℓ, 𝒱1, . . . , 𝒱ℓ]

for the multiplicatively closed subset generated by 𝒱𝑖 for 𝑖 such that 𝐾𝑖 ⊂ 𝑀. Then there is an

F[𝑈1, . . . , 𝑈ℓ]-equivariant chain isomorphism

(cid:202)

s∈H(𝐿)

𝐴−(H 𝐿−𝑀, 𝜓 𝑀 (s)) (cid:27) 𝑆−1

𝑀 · CF L (H ),

(2.9)

where we view 𝑈𝑖 as acting by 𝒰𝑖𝒱𝑖 on the right-hand side. Furthermore, if s ∈ H (𝐿), this

isomorphism intertwines the summand 𝐴−(H 𝐿−𝑀, 𝜓 𝑀 (s)) with the subspace of 𝑆−1

𝑀 · CF L (H ) in

Alexander multi-grading s.

We omit the proof of this proposition here. Note that the isomorphism in 2.9 is constructed by

the map

𝐴−(H 𝐿−𝑀, 𝜓 𝑀 (s)) → 𝑆−1

𝑀 · CF L (H )

16

via the formula

𝑈𝑖1
1

· · · 𝑈𝑖ℓ

𝑖1
ℓ · x ↦→ 𝒰
1

· · · 𝒰

𝑠1−𝐴𝐿
𝑖ℓ
ℓ 𝒱
1
1

(x)+𝑖1

· · · 𝒱
ℓ

𝑠ℓ −𝐴𝐿

ℓ (x)+𝑖ℓ

· x.

We can also rephrase the link surgery formula in this setting. We only talk about the knot

surgery formula here. For the general case, one may refer to Chapter.7 of [Zem21a].

Given a knot surgery complex X𝜆 (𝐾) = Cone((cid:206)𝑠∈Z 𝐴−
𝑠

𝑣 −
𝑠 +ℎ−
𝑠−−−−−→ (cid:206)𝑠∈Z 𝐵−

𝑠 ), (cid:206)𝑠∈Z 𝐴−

𝑠 is identi-

fied with a completion of CF K (𝐾) and (cid:206)𝑠∈Z 𝐵−

𝑠 is identified with a completion of 𝒱−CF K (𝐾)

by the above equivalence.

Let x1, · · · , x2 be a free basis of CF K (𝐾) over F[𝒰, 𝒱]. Then the differential in the mapping

cone is determined by the map on this basis together with the homomorphism of the coefficient ring:

via the formula

𝑇 : F[𝒰, 𝒱] → F[𝒰, 𝒱, 𝒱−1]

𝑇 (𝒰) = 𝒱−1 and 𝑇 (𝒱) = 𝒰𝒱2,

and 𝐼 : F[𝒰, 𝒱] → F[𝒰, 𝒱, 𝒱−1], which is the canonical inclusion.

Using the above equivalence, we can compute the three summands of differentials on the

generators in the mapping cone to this setting:

(i) The internal differential on each summand, let us denote it by 𝜕.

(ii) The map corresponds to the inclusion; let us denote it by 𝑣.

(iii) The map corresponds to the opposite orientation; let us denote it by ℎ𝜆.

Together with the coefficient ring homomorphism, given a ∈ F[𝒰, 𝒱], the differential of the

mapping cone is given by:

(i) 𝑣(a · x𝑖) = 𝐼 (a) · x

(ii) ℎ𝜆 (a · x𝑖) = 𝑇 (a) · ℎ𝜆 (x𝑖)

17

Using the above description, the complex of 𝜆 surgery on unknot O is the following.

CF K (O) is generated by one generator. Hence the mapping cone is

F[[𝒰, 𝒱]]

𝑣+ℎ𝜆−−−→ F[[𝒰, 𝒱, 𝒱−1],

where

𝑣(𝒰𝑖𝒱 𝑗 ) = 𝑣(𝒰𝑖𝒱 𝑗 ) and ℎ𝜆 (𝒰𝑖𝒱 𝑗 ) = 𝒰 𝑗 𝒱2 𝑗−𝑖+𝜆.

18

CHAPTER 3

NONSLICENESS OF ALGEBRAICALLY SLICE KNOTS

In this chapter, we give the proof of Theorem 1.3. In Section 3.1, we talk about the obstruction

from the linking form and d-invariants. In Section 3.2, we describe the topology of 2-fold branched

covers from different aspects. In Section 3.3, we compute the d-invariants using knot surgery for-

mula and give the proof of Theorem 1.3. We also include the computation with lattice cohomology

in the Appendix ??.

3.1 Linking form and d-invariants obstruction

In this section, we review an obstruction for a knot to be slice from [HLR12]. Let 𝐾 be a

knot in 𝑆3 and let Σ(𝐾) be the 2-fold branched cover of 𝐾. Suppose 𝐾 is slice, which means 𝐾

bounds a smooth disk in 𝐷4. Then Σ(𝐾) bounds a Z/2Z-homology 4-ball 𝑊. Hence, to show the

nonsliceness of 𝐾, it is enough to show the nonexistence of 𝑊.

The linking form on Σ(𝐾) provides an initial constraint on the pair (𝑊, Σ(𝐾)). To state it,

recall that a subgroup 𝑀 ⊂ 𝐻1(Σ(𝐾)) is called a 𝑚𝑒𝑡𝑎𝑏𝑜𝑙𝑖𝑧𝑒𝑟 if

• |𝑀 |2 = |𝑇1(𝑌 )|, where 𝑇1 denotes the torsion subgroup of 𝐻1(𝑌 ), and

• The Q/Z-valued linking form on 𝐻1(𝑌 ) is identically zero on 𝑀.

If 𝐾 is slice, then the Z/2Z-homology ball bounded by Σ(𝐾) gives rise, via the kernel of the

inclusion induced map, 𝑖∗ : 𝐻1(Σ(𝐾)) → 𝐻1(𝑊), to a metabolizer of 𝐻1(Σ(𝐾))[CG86]. This

linking form can often be used to obstruct sliceness. Note, however, that when 𝐾 is algebraically

slice, this obstruction vanishes.

With Heegaard Floer homology, we get additional structures on the 𝑚𝑒𝑡𝑎𝑏𝑜𝑙𝑖𝑧𝑒𝑟. Recall that,
for a rational homology 3-sphere 𝑌 , the Heegaard Floer homology of 𝑌 splits with respect to Spin𝑐

structures over 𝑌 ,

𝐻𝐹+(𝑌 ) = (cid:201)

𝑡∈Spin𝑐 (𝑌 )

𝐻𝐹+(𝑌 , 𝔰),

and we can define the d-invariants 𝑑 (𝑌 , 𝔰) for each 𝔰 ∈ Spin𝑐 (𝑌 ), 𝐻𝐹+(𝑌 , 𝔱).

The obstruction is defined as a difference of correction terms.

19

Definition 3.1. For 𝑌 a Z/2Z-homology sphere, define the relative d-invariants as ¯𝑑 (𝑌 , 𝔰) =

𝑑 (𝑌 , 𝔰) − 𝑑 (𝑌 , 𝔰0), where 𝔰0 is the unique spin structure on 𝑌 .

When 𝐻1(𝑌 ; Z/2Z) = 0, Poincare duality and the Chern class provide a bijection Spin𝑐 ←→
𝐻1(𝑌 ). Combining with the d-invariants property 2.1.2 above, we see that the d-invariants vanish

on a metabolizer. One can package this using the following[HLR12],

Theorem 3.1. Let 𝑃 be a finite set of (distinct) odd primes. Suppose that 𝑊 is a Z/2Z-homology

4-ball and 𝜕𝑊 = #𝑝∈𝑃𝑌𝑝#𝑌1, where

(i) 𝑝𝑘 𝐻1(𝑌𝑝) = 0 for each 𝑝 ∈ 𝑃 and some 𝑘 ≥ 0.

(ii) 𝑌1 is a Z-homology 3-sphere.

Then for each 𝑝 ∈ 𝑃, there is a 𝑚𝑒𝑡𝑎𝑏𝑜𝑙𝑖𝑧𝑒𝑟 𝑀𝑝 ⊂ 𝐻1(𝑌𝑝) for which ¯𝑑 (𝑌𝑝, 𝔰𝑚 𝑝 ) = 0 for all

𝑚 𝑝 ∈ 𝑀𝑝.

By using branched covers, the theorem yields the desired concordance obstruction.

Corollary 3.1.1. Let 𝐾 = #𝑝∈𝑃𝐾𝑝#𝐾1 be a connected sum of knots satisfying
• 𝑝𝑘 𝐻1(Σ(𝐾𝑝)) = 0 for each 𝑝 in a set of primes, 𝑃, and some 𝑘,
• 𝐻1(Σ(𝐾1)) = 0.

Suppose 𝐾 is slice.Then for each 𝑝 ∈ 𝑃, there is a 𝑚𝑒𝑡𝑎𝑏𝑜𝑙𝑖𝑧𝑒𝑟 𝑀𝑝 ⊂ 𝐻1(Σ(𝐾𝑝)) for which

¯𝑑 (Σ(𝐾𝑝), 𝔰𝑚 𝑝 ) = 0 for all 𝑚 𝑝 ∈ 𝑀𝑝.

Note that Corollary 3.1.1 shows more; normally the linear combination (cid:205) 𝐾𝑝 isn’t concordant

to any knot with det(𝐾) = 1.

3.2 Topology of the 2-fold branched covers

3.2.1 Surgery description by rational unknotting number one patterns

In this section, we use the algorithm from [DHMS22] to give a knot surgery description of

Σ2(𝐾2,𝑝).

20

We first review the notion of 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑢𝑛𝑘𝑛𝑜𝑡𝑡𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑛𝑒 𝑝𝑎𝑡𝑡𝑒𝑟𝑛𝑠. For the definition

of rational tangle and the bijection between the rational tangles in a fixed 3-ball 𝐵3 and Q ∪ {∞},

one can refer to section 2.1 in [DHMS22].

Definition 3.2. Let 𝑃 ⊆ 𝑆1 × 𝐷2 be a pattern. We say that P has a rational unknotting number one

if there exists a rational tangle T in P such that replacing T with another rational tangle 𝑇 ′

gives a

knot which is unknotted in the solid torus. We say that P has proper rational unknotting number

one if 𝑇 ′

can be taken to be a proper tangle replacement: that is, connecting the same two pairs of

marked points as T.

For a rational unknotting number one pattern P, we have

Σ2(𝑃(𝑈)) (cid:27) 𝑆3

𝑝/𝑞 (𝐽)

for some strongly invertible knot 𝐽 and surgery coefficient 𝑝/𝑞. The claim is immediate from the

Montesinos trick: since 𝑃′

is an unknot, the branched double cover over 𝑃′

is 𝑆3. The 3-ball 𝐵3

containing 𝑇 ′ lifts to a solid torus in 𝑆3, and replacing 𝑇 ′ with 𝑇 corresponds to doing surgery on

the core of this solid torus. Moreover, we can explicitly produce J and the surgery coefficient 𝑝/𝑞.

Here, we will use 𝑇2,𝑘 to illustrate the procedure, which is given in Figure 3.1. For the general case,

one can refer to [DHMS22].

(i) Let 𝛾 be a reference arc in 𝐵3 which has one endpoint on each component of 𝑇 ′, displayed

in panel (2). When taking 2-fold branched cover of 𝐵3 along 𝑇 ′, we first cut 𝐵3 at the disk

bounds by each arc of 𝑇 ′, which gives us a cylinder 𝐷2 × 𝐼 and 𝛾 is isotopic to 0 × 𝐼. Gluing

two copies of cylinders gives a solid torus and the lift of 𝛾 in the solid torus is the core of

this solid torus, i.e. 𝐽.

(ii) We also have a concrete description for the knot 𝐽. Let 𝐹𝑡 be an isotopy of the solid torus

moving 𝑃′

into a local unknot in 𝑆1 × 𝐷2. We then cut along the disk bounded by the unknot

and glue two copies of the disk complement to get the 2-fold branched cover. We also keep

21

Figure 3.1 Montesinos trick

track of 𝛾 along 𝐹𝑡 and lift it to the 2-fold branched cover. This gives the desired strongly

invertible knot 𝐽, which in this case is just an unknot, displayed in panel (4).

(iii) To compute the surgery coefficient 𝑝/𝑞, we must find the unique rational tangle 𝑆 in 𝐵3

which lifts to a pair of 𝜏-equivariant Seifert framings of 𝐽, which can be done by running

𝐹𝑡 backwards. We first find a 𝜏-invariant Seifert framing of 𝐽 in the 2-fold branched cover,

then quotient it by 𝜏 and reverse the isotopy 𝐹𝑡 to draw 𝑆 in the original 3-ball 𝐵3. Since 𝐽 is

a 𝜏-equivariant unknot, we can pick a parallel copy of 𝐽 to be Seifert framing. Quotienting

this pair by 𝜏 gives us a pair of arcs. Keeping track of 𝐹𝑡 backwards, it adds 𝑘 − 1 negative

22

crossings to the pair of arcs. Hence, the rational tangle with 𝑘 − 1 negative crossings is the

desired rational tangle in 𝑆 in 𝐵3. By the Montesinos trick, the surgery coefficient 𝑝/𝑞 is

then precisely the rational number identified with the original tangle 𝑇 relative to the choice

of reference tangles 𝑇∞ = 𝑇 ′

and 𝑇0 = 𝑆. In our example, the surgery coefficient is 𝑘. From

the discussion above, we have

Σ2(𝑇2,𝑘 ) (cid:27) 𝑆3

𝑘 (𝑈).

Let 𝐾 be an oriented knot in 𝑆3. We can now extend the discussion above to the branched cover

of a cable knot 𝐾2,𝑘 . Recall that 𝐾2,𝑘 can be constructed by taking the image of 𝑇2,𝑘 inside the

gluing

𝑆3 (cid:27) (𝑆3 − 𝑁 (𝜇))∪𝜕𝑁 (𝜇) (𝑆3 − 𝑁 (𝐾))

formed by a boundary identification which maps a meridian 𝜇 of 𝑇2,𝑘 to a Seifert framing of 𝐾 and

a longitude of 𝑇2,𝑘 to a meridian of 𝐾.

Taking the 2-fold branched cover lifts 𝑇2,𝑘 to an unknot and the meridian 𝜇 to ˜𝜇∩𝜏 ˜𝜇. Combining

the discussion of the satellite operation above, we have

Σ2(𝑃(𝐾)) (cid:27) (𝑆3

𝑘 (𝐽) − 𝑁 ( ˜𝜇 − 𝑁 (𝜏 ˜𝜇))∪𝜕𝑁 ˜(𝜇) (𝑆3 − 𝑁 (𝐾))∪𝜕𝑁 (𝜏 ˜𝜇) (𝑆3 − 𝑁 (𝐾)),

which is illustrated in Figure 3.1. Since 𝐽 is unknot, we have

Σ2(𝑃(𝐾)) (cid:27) 𝑆3

𝑘 (𝐾#𝐾𝑟)

Figure 3.2 knot surgery description

Note that, since it is a 𝑘 surgery on a knot in 𝑆3, we have |𝐻1(Σ2(𝑃(𝐾))| = 𝑘.

23

3.2.2 Topology of p-fold branched cover from complex polynomial

In this section, we study the topology of Σ𝑝 (𝐾 ( 𝑝, 𝑞)). Using the definition of torus knot with

complex curve intersection, we give a link surgery description of the p-fold branched cover. Note

that we take p-fold branched cover instead of 2-fold branched cover to make the manifold having
nontrivial 𝑏1, such that we have enough Spin𝑐 structures to apply the d-invariants obstruction.

Using the same strategy as in Section 3.2, we first study the topology of Σ𝑝 (𝑇𝑝,𝑞), with the

description of the lifting of the meridian in Σ𝑝 (𝑇𝑝,𝑞). Then we splice the knot complement to each

lift to get the link surgery description of Σ𝑝 (𝐾 ( 𝑝, 𝑞)).

𝑇𝑝,𝑞 can be defined as the intersection of complex curves. Let

and

𝐶𝑝,𝑞 = {(𝑧2, 𝑧3) ∈ 𝐶2 | 𝑧2

𝑝 + 𝑧3

𝑞 = 0}

𝜖 = {(𝑧2, 𝑧3) ∈ 𝐶2 | |𝑧2|2 + |𝑧3|2 = 𝜖 }
𝑆2

be the two complex curves which intersect transversely. Define

𝑇𝑝,𝑞 = 𝐶𝑝,𝑞 ∩| 𝑆2
𝜖 .

The p-fold branched cover is the intersection of

𝐶𝑝,𝑝,𝑞 = {(𝑧1, 𝑧2, 𝑧3) ∈ 𝐶3 | 𝑧1

𝑝 + 𝑧2

𝑝 + 𝑧3

𝑞 = 0}

with

𝜖 = {(𝑧1, 𝑧2, 𝑧3) ∈ 𝐶3 | |𝑧1|2 + |𝑧2|2 + |𝑧3|2 = 𝜖 },
𝑆5

which is denoted by Σ( 𝑝, 𝑝, 𝑞) in [JN83, Chapter. 7]. Applying Theorem 7.2 of [JN83], Σ( 𝑝, 𝑝, 𝑞)

is a Seifert manifold 𝑀 (0; (1, 𝑟), 𝑝(𝑞, 𝑠)), where 𝑟, 𝑠 are a pair of numbers such that 𝑞𝑟 + 𝑝𝑠 = 1.

𝑀 (0; (1, 𝑟), 𝑝(𝑞, 𝑠)) can be represented by the plumbing diagram in Figure 3.3.

Note that, the Seifert manifold representation is only unique up to a set of operations ([JN83,

Theorem 1.5]):

(i) Add or delete any Seifert pair (𝛼, 𝛽) = (1, 0)

24

Figure 3.3 𝑀 (0; (1, 𝑟), 𝑝(𝑞, 𝑠))

(ii) Replace any (0, ±1) by (0, ∓1)

(iii) Replace each (𝛼𝑖, 𝛽𝑖) by (𝛼𝑖, 𝛽𝑖 + 𝐾𝑖𝛼𝑖) provided (cid:205) 𝐾𝑖 = 0

Applying continuous fraction to each

𝑞
𝑠 vertex, we can represent it by the following plumbing

diagram with only integer weights.

Figure 3.4 Plumbing diagram with integer weights

Embedding 𝑇𝑝,𝑞 into the solid torus, one can check that the meridian of the solid torus is isotopic

to {𝑧3 = 0} in 𝑆3. Hence, the lift of the meridian is also isotopic to {𝑧3 = 0} in 𝑀 (0; (1, 𝑟), 𝑝(𝑞, 𝑠)),

which corresponds to the core of 𝑝 singular fibers with coefficient

𝑞
𝑠 . We use an arrow in Figure

3.4 to denote the singular fiber and 𝑎0 to denote its framing. We can compute the framing of the

core in the plumbing diagram in the following way.

Let us denote the meridian and longitude of the singular fiber complement by 𝜇 and 𝜆 and the

meridian and longitude of the boundary of the singular fiber by 𝜇′

and 𝜆′

. The singular fiber is

25

          𝑞

𝑠

glued back with matrix (cid:169)
(cid:173)
(cid:173)
−𝑝 𝑟
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

, which maps the 𝜆′

to −𝑝𝜇 + 𝑟𝜆. Hence, to specify the framing in

the plumbing diagram, we need to make sure the framing has −𝑝𝜇 + 𝑟𝜆 as its image.

The meridian corresponds to the vertex with weight 𝑎𝑖 by 𝑚𝑖. We have 𝑚1 = 𝜇 and 𝑚0 = 𝜇′

.

These following continuous fractions

𝑞
𝑠 = 𝑎𝑛 −

1

𝑎𝑛−1 − · · · −

1
𝑎1

𝑞
𝑠′ = 𝑎1 −

1

𝑎2 − · · · −

1
𝑎𝑛

satisfy 𝑠𝑠′ ≡ 1 (mod q). Hence 𝑠′ = 𝑝 + 𝑘𝑞, for some 𝑘 ∈ 𝑍. One can compute that, in the plumbing

diagram, we have 𝑚0 = 𝑞𝑚𝑛 and 𝑚1 = 𝛼𝑚𝑛, where

𝛼
𝛽

= 𝑎2 −

1

𝑎3 − · · · −

,

1
𝑎𝑛

𝑎1𝛼−𝛽
𝛼

. In particular, we have 𝛼 = 𝑠′ = 𝑝 + 𝑘𝑞. Then for the framing

𝑞
𝑠′ = 𝑎1 − 1

𝛼/𝛽 =
and we have
𝑎0, we have the image of 𝜆′

is

−𝑚1 + 𝑎0𝑚0 = (−𝛼 + 𝑎0𝑞)𝑚𝑛.

Hence, we should choose 𝑘 as the framing.

Figure 3.5 Σ𝑝 (𝐾𝑝,𝑞)

26

     Splicing 𝐾 to the core of singular fiber has the following description in Figure 3.5. We label

the vertices with (𝐾, 𝑠) indicate that it is the knot 𝐾 with its Seifert framing.

Denote the meridian of each copy of 𝐾 by 𝑚𝑖, we have

𝐻1(Σ𝑝,𝑞 (𝑇𝑝,𝑞)) (cid:27) (cid:202)
𝑝−1

𝑍/𝑝,

which is generated by 𝑚1 − 𝑚𝑖, 𝑖 = 1, · · · , 𝑝 − 1.

When 𝑝 = 2, using the Seifert manifold description above and applying the operators in 3.2.2,

we have the following two equivalent surgery descriptions of Σ2(𝑇𝑝,𝑞) in Figure 3.6.

Figure 3.6 Seifert manifold of 2-fold branched cover

Using the description in Figure 3.6 (b), it is equivalent to gluing two solid torus via the matrix

𝑞

1
(cid:170)
(cid:174)
(cid:174)
(cid:172)

, which is equivalent to the 0 surgery on the unknot. The two cores of the singular fiber are

(cid:169)
(cid:173)
(cid:173)
−1 0
(cid:171)
homotopy to the meridians of the unknot with opposite orientations. Splicing the two meridians

with the knot 𝐾, we get the same description as in section 3.2.1.

3.3 Computations with knot surgery formula

3.3.1 Knot Floer complex

We will give a description of 𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟) in this subsection. For any knot 𝐾, we have a

filtered chain homotopy equivalence

𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟) (cid:27) 𝐶𝐹𝐾 ∞(𝐾) ⊗ 𝐶𝐹𝐾 ∞(𝐾𝑟).

27

from the connected sum formula [OS04b].

Since 𝐶𝐹𝐾 ∞(𝐾𝑟) (cid:27) 𝐶𝐹𝐾 ∞(𝐾), we have that 𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟) (cid:27) 𝐶𝐹𝐾 ∞(𝐾) ⊗ 𝐶𝐹𝐾 ∞(𝐾).

When 𝐾 is a L-space knot, the knot Floer complex is in a relatively simple form

𝐶𝐹𝐾 ∞(𝐾) (cid:27) St(𝐾) ⊗ Z2 [𝑈, 𝑈−1].

Here, St(𝐾) is the staircase complex associated to 𝐾. We have an example of St(𝑇3,4) in Figure

3.7, where each dot represents a generator and the arrows represent differentials in the complex.

The other complex in Figure 3.7 is the tensor complex, we omit some differentials induced from

the second components for simplicity.

Following the notation from section 4.1 of [BL14], we can also denote this staircase complex

by an array St(1,2,2,1). Each integer here denotes the length of the segments starting at the top left

and moving to the bottom right in alternating right and downward steps. For a L-space knot 𝐾, the

Alexander polynomial is in the form of Δ𝐾 (𝑡) = Σ2𝑚
𝑖=0

(−1)𝑖𝑡𝑛𝑖 . We can get the staircase complex

from the Alexander polynomial by St(𝐾) = St(𝑛𝑖+1 − 𝑛𝑖), where 𝑖 runs from 0 to 2𝑚 − 1.

The absolute grading of the generator gives us a filtration on the staircase complex. The

generator which does not have arrows pointing to other generators has grading 0 and we call these

generators type A. Starting from top left, we denote these generators by 𝑎1,𝑎2, . . . ,𝑎𝑚+1. Similarly,

we call the other generators which have nontrivial differentials type B and denote them by 𝑏1,𝑏2,

. . . , 𝑏𝑚.

In the tensor product 𝐶𝐹𝐾 ∞(𝐾) ⊗ 𝐶𝐹𝐾 ∞(𝐾), we have a subcomplex 𝐶 ′

, which is generated

by the concatenation of 𝑎1 ⊗ St(𝐾) and St(𝐾) ⊗ 𝑎𝑛. We call the concatenation staircase the double

of original staircase and denote it by D(St(𝐾)). As an example, D(St(𝑇3,4)) is the red staircase in

Figure 3.7.

Let ˜𝐶 := 𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟)/𝐶 ′

be the quotient complex.

Proposition 3.1. 𝐻 ( ˜𝐶) (cid:27) 0.

Proof. We prove it by inductively quotienting the sub square complex from ˜𝐶. At each generator

𝑏𝑖𝑏 𝑗 , we have the following square complex, which is demonstrated in 3.8, as a subcomplex of

28

Figure 3.7 knot Floer complex

𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟).

Figure 3.8 square complex

In ˜𝐶, when 𝑖 = 1 or 𝑗 = 𝑚, the square complexes become the ones in Figure 3.9.

Figure 3.9 square complex in ˜𝐶

Quotienting these square complexes and performing a change of basis, the quotient complex

29

we get is the same as deleting the square complex in the bottom of Figure 3.9 with 𝑖 = 1 or 𝑗 = 𝑚.

Let us denote the new quotient complex by ˜𝐶1.

Suppose we have quotiented 𝑘 times and got the quotient complex ˜𝐶𝑘 . By the same argument

above, we can quotient the sub square complex with 𝑖 = 𝑘 + 1 or 𝑗 = 𝑚 − 𝑘, which is the same as

deleting the subcomplex in Figure 3.9 with 𝑖 = 𝑘 + 1 or 𝑗 = 𝑚 − 𝑘 and get the new quotient complex

˜𝐶𝑘+1. Moreover, we have ˜𝐶𝑚 = 0 since we have deleted all the generators.

Hence, we have ˜𝐶 (cid:27) ˜𝐶𝑚 (cid:27) 0.

□

Combining Proposition 3.1 and the exact sequence for the pair (𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟), D(St(𝐾)) ⊗

Z2 [𝑈, 𝑈−1]), we have the following:

Proposition 3.2. 𝐻 (𝐶𝐹𝐾 ∞(𝐾#𝐾𝑟)) (cid:27) 𝐻 (D(St(𝐾)) ⊗ Z2 [𝑈, 𝑈−1]).

The middle generator 𝑎1𝑎𝑛+1 of the double staircase is on the diagonal. This is easy to be shown

since the staircase of any L-space knot is symmetric along the diagonal. The 𝑖-th filtration level of

𝑎1𝑎𝑛+1 is the distance of 𝑎1𝑎𝑛+1 to the 𝑗 axis, which is equal to sum of length of horizontal arrows

in the staircase: Σ𝑖∈2𝑍+1𝑛𝑖. Since 𝑎1𝑎𝑛+1 is on the diagonal, we have

Proposition 3.3. The bigrading of 𝑎1𝑎𝑛+1 is (Σ𝑖∈2𝑍+1𝑛𝑖, Σ𝑖∈2𝑍+1𝑛𝑖).

3.3.2 Computation of d-invariants

We will use the argument from [NW15] to compute the d-invariants.

In their paper, they used the plus version of the integer surgery formula from [OS08]. Given

a knot 𝐾 in 𝑆3, let 𝐶 = 𝐶𝐹𝐾 ∞(𝐾) be the knot Floer complex associated to it. The plus integer

surgery formula involves the following two subcomplex:

𝐴+

𝑠 = 𝐶{max(𝑖, 𝑗 − 𝑠) ≥ 0}

𝐵+ = 𝐶{𝑖 ≥ 0}.

There are two canonical chain maps 𝑣+

𝑠 : 𝐴+

𝑠 → 𝐵+ and ℎ+

𝑠 : 𝐴+

𝑠 → 𝐵+ as in [OS08]. We only

need 𝑣+

𝑠 in this paper, which is the projection from 𝐴+

𝑠 onto 𝐶{𝑖 ≥ 0}.

Let A = ⊗𝑠∈Z 𝐴+

𝑠 and B = ⊗𝑠∈Z𝐵+

𝑠 and let 𝐷+

𝑛 : A+ → B+ be the map

30

𝐷+

𝑛 ({𝑎𝑠}𝑠∈Z) = {𝑏𝑠}𝑠∈Z,

where here

𝑏𝑠 = ℎ+

𝑠−𝑘 (𝑎𝑠−𝑛 + 𝑣+

𝑠 (𝑎𝑠).

Let X+(𝑘) denote the mapping cone of 𝐷+
𝑘 .

Theorem 3.2 ([OS08]). For any non-zero integer 𝑘, the homology of the mapping cone X+

𝑘 of

𝐷+

𝑘 : A+ → B+

is isomorphic to 𝐻𝐹+(𝑆3

𝑘 (𝐾)).

In [NW15], Ni and Wu gave an efficient way to compute the d-invariants from the integer

surgery formula. We first recall the notation from their paper.

Let

𝔄+

𝑠 = 𝐻∗( 𝐴+

𝑠 ), 𝔅+ = 𝐻∗(𝐵+).

Indeed, 𝐵+ = 𝐶{𝑖 ≥ 0} is identified with 𝐶𝐹+(𝑆3) and 𝔅+ (cid:27) T +. Here T + (cid:27) Z2 [𝑈, 𝑈−1]/Z2 [𝑈].

Let

𝑠 , 𝔟+
𝔳+

𝑠 : 𝔄+

𝑠 → 𝔅+

be the map induced on homology.

Let 𝔄𝑇

𝑠 = 𝑈𝑛𝔄+

𝑠 for 𝑛 ≫ 0, we have 𝔄𝑇

𝑠 (cid:27) T +. Since each 𝔞+

sufficiently high grading and is 𝑈-equivariant, 𝔞+

𝑠 | 𝔄𝑇

𝑠 is a graded isomorphism at
𝑠 is modeled on multiplication by 𝑈𝑉𝑠 . Note

that the number 𝑉𝑠 is an invariant. Also, by Proposition 3.2, we can use D(St(𝐾)) ⊗ Z2 [𝑈, 𝑈−1] to

compute 𝑉𝑠. We have a useful property of 𝑉𝑠.

Proposition 3.4 ([NW15][Ras04]). 𝑉𝑠 ≥ 𝑉𝑠+1.

The formula given in [NW15] computes d-invariants of 3-manifold constructed from a rational

surgery in 𝑆3. In our case, we just need the formula in the integer surgery case.

31

Proposition 3.5 ([NW15]). Suppose 𝑘 > 0 and fix 0 ≤ 𝑖 ≤ 𝑘 − 1. Then

𝑑 (𝑆3

𝑘 (𝐾), 𝑖) = 𝑑 (𝐿 (𝑘, 1), 𝑖) − 2max{𝑉𝑖, 𝑉𝑘−𝑖}.

Combining this and the Proposition above, together with the symmetry of the d-invariants for

lens space, we have

𝑑 (𝑆3

𝑘 , 𝑖) = 𝑑 (𝑆3

𝑘 , 𝑘 − 𝑖) = 𝑑 (𝐿 (𝑘, 1), 𝑖) − 2𝑉𝑖 , when 0 ≤ 𝑖 ≤ (𝑘 − 1)/2.

Lemma 3.3.1. For a staircase St ⊆ 𝐶 and subcomplexes 𝐴+

𝑠 and 𝐵+, let us denote the restriction of

St to the subcomplexes by 𝑟 (St). 𝐻∗(𝑟 (St)) is nontrivial iff St is fully included in the subcomplex.

Proof. Each staircase in the subcomplex is truncated by a horizontal line and a vertical line.

Suppose it is not fully included in the subcomplex, since the staircase starts horizontally and ends

vertically, each connected component of the remaining part is a staircase with an even number of

generators. Hence, the homology will be trivial on these staircases. Below in Figure 3.10, we have

3 (D(St(𝑇 (3, 4))) ⊗ Z2 [𝑈, 𝑈−1]) as an example.
𝐴+

□

Figure 3.10 𝐴+

3 (D(St(𝑇 (3, 4))) ⊗ Z2 [𝑈, 𝑈−1])

Let us first study 𝔅+. By Lemma 3.3.1, the bottom generator is represented by the staircase

whose left top corner is on the 𝑗-axis, since it is the first staircase which is fully included in 𝐵+.

Let us denote it by 𝑆𝑡 𝐵.

32

The first staircase included in 𝔄+

0 is the one that has the middle generator at (0, 0), let us denote
it by 𝑆𝑡0. 𝑉0 is the 𝑈-distance between these two staircases, i.e. the 𝑈 power in 𝑈𝑉0 𝑆𝑡 𝐵 = 𝑆𝑡0. By
Proposition 3.3, the bigrading of the middle term in St𝐵 is (Σ𝑖∈2𝑍+1𝑛𝑖, Σ𝑖∈2𝑍+1𝑛𝑖). Since the middle
generator of 𝑆𝑡0 is at (0, 0), we get 𝑉0 = Σ𝑖∈2𝑍+1𝑛𝑖.

Let us denote the gap 𝑉𝑠 − 𝑉0 by ¯𝑉𝑠. Note that when 𝑠 ≥ 2Σ𝑖∈2𝑍+1𝑛𝑖, 𝑉𝑠 = 𝑉0 and ¯𝑉𝑠 = 0. On the

j axis, we denote the overlap with all the staircases by 𝑂𝑖=0 and the restriction of 𝑂𝑖=0 to 0 ≤ 𝑗 ≤ 𝑠
by 𝑂 𝑠

𝑖=0. We also denote the length of 𝑂 𝑠

𝑖=0 by 𝐿𝑠

𝑖=0.

Proposition 3.6. When 0 ≤ 𝑠 ≤ 2Σ𝑖∈2𝑍+1𝑛𝑖, ¯𝑉𝑠 = 𝑠 − 𝐿𝑠

𝑖=0.

Proof. Let us look at the gap of 𝑉𝑠 − 𝑉𝑠+1. Suppose 𝑂 𝑠+1
𝑖=0
most staircase of 𝐴+

𝑠 remains fully included in 𝐴+
𝑖=0 is empty, then the bottom most staircase of 𝐴+

𝑖=0 is nonempty, then the bottom
𝑠+1. Hence, 𝑉𝑠 = 𝑉𝑠+1 and 𝑉𝑠 − 𝑉𝑠+1 = 0. Suppose
𝑠+1 is the one which is once above the

𝑂 𝑠+1
𝑖=0

\ 𝑂 𝑠

\ 𝑂 𝑠

bottom most staircase of 𝐴+

𝑠 . Hence 𝑉𝑠 − 𝑉𝑠+1 = 1.

Sum all of these gaps up, we get the conclusion.

Combining the propositions above, we have

□

Corollary 3.2.1.

¯𝑑 (𝑆3

𝑘 (𝐾, 𝑠) = ¯𝑑 (𝐿 (𝑘, 1), 𝑠) − 𝑠 + 𝐿𝑠

𝑖=0, when 0 ≤ 𝑠 ≤ (𝑘 − 1)/2.

3.3.3 Main theorem

In this subsection, we give the statement and proof of the main theorem.

Let us denote Max{𝑠|𝑠 − 𝐿𝑠

𝑖=0 = 0} by 𝑚(𝐾). From the discussion of the staircase complex in
Section 3.3.1, 𝑚(𝐾) is equal to the difference of degrees of the highest degree generator and the
second top degree generator of Δ𝐾 (𝑡). For any polynomial 𝑃(𝑡) = (cid:205)𝑛
𝑎𝑖𝑡 𝑑𝑖 such that 𝑑𝑖 < 𝑑𝑖+1,
𝑖=1

let 𝑚(𝑃(𝑡)) = 𝑑𝑛 − 𝑑𝑛−1. Then 𝑚(𝐾) = 𝑚(Δ𝐾 (𝑡)).

It is been shown in [HW18] that, for any L space knot 𝐾,

Δ𝐾 (𝑡) = 𝑡𝑔 − 𝑡𝑔−1 · · · − 𝑡1−𝑔 + 𝑡−𝑔,

where 𝑔 denotes the Seifert genus of 𝐾. Hence, we have 𝑚(𝐾) = 1 for any L-space knot 𝐾.

33

Theorem 3.3. For any L space knot 𝐾, 𝐾2,𝑘1# − 𝑇2,𝑘1# − 𝐾2,𝑘2#𝑇2,𝑘2 has infinite order in the smooth
knot concordance group C when 𝑘1 and 𝑘2 are a pair of distinct prime numbers such that 𝑘1 > 3

and 𝑘2 > 3.

Note that, since 𝑚(𝐾) = 1, the assumption in Theorem 3.3 is the same as 𝑘𝑖 > 2𝑚(𝐾) + 1.

In the special case, where 𝐾2,𝑘1 and 𝐾2,𝑘2 are algebraic knots, 𝑘𝑖 > 2𝑝𝑞 > 3, for some 𝑝, 𝑞

which are co-prime. We have the following corollary:

Corollary 3.3.1. 𝐾2,𝑘1# − 𝑇2,𝑘1# − 𝐾2,𝑘2#𝑇2,𝑘2 has infinite order in the smooth knot concordance
group 𝐶 when 𝐾2,𝑘1 and 𝐾2,𝑘2 are 2 distinct algebraic knots.

Before proving the main theorem, let us study the linking form on the manifold 𝑀 = Σ2(𝐾2,𝑘 )#−

Σ2(𝑇2,𝑘 ) first. From Section 3.2, we have 𝑀 (cid:27) 𝑆3

𝑘 (𝑈), here 𝑈 denotes the unknot.
𝐻1(𝑀) (cid:27) Z/𝑘Z ⊕ Z/𝑘Z, which is generated by the meridians of the surgery knots. Let us denote
the meridian of 𝐾#𝐾𝑟 by 𝛼 and the meridian of 𝑈 by 𝛽. Then the linking form evaluating on these

𝑘 (𝐾#𝐾𝑟)# − 𝑆3

generators gives:

𝜆(𝛼, 𝛼) ≡ 1/𝑘, 𝜆(𝛽, 𝛽) ≡ −1/𝑘, 𝜆(𝛼, 𝛽) ≡ 0. (mod 𝑍)

Let us denote the generators of 𝐻1(𝑛𝑀) by 𝛼𝑖 and 𝛽𝑖, 1 ≤ 𝑖 ≤ 𝑛. Here, 𝛼𝑖 are the meridians

of each 𝐾#𝐾𝑟 and 𝛽𝑖 are the meridians of each 𝑈. We also use 𝑀 1

𝑖 and 𝑀 2
𝑖

to denote the

corresponding summands of Σ2(𝐾2,𝑘 ) and Σ2(𝑇2,𝑘 ). Via a change of basis, we can use 𝛼𝑖 and

𝛼𝑖 + 𝛽𝑖 as the generators for 𝐻1(𝑛𝑀).

For a knot 𝐾 satisfies the assumption in Theorem 3.3, we have

Lemma 3.3.2. For any 𝑖, on the Spin𝑐 structures correspond to the subgroup 𝐺𝑖 generated by
𝛼𝑖 + 𝛽𝑖, there exists at least one Spin𝑐 structure 𝔰, such that ¯𝑑 (𝑛𝑀, 𝔰) ≠ 0.

Proof. We prove it by contradiction. 𝐺𝑖 = {𝑙 (𝛼𝑖 + 𝛽𝑖)|𝑙 ∈ Z/𝑘Z}. Suppose for each 𝑙, ¯𝑑 (𝑛𝑀, 𝑙 (𝛼𝑖 +

𝛽𝑖)) = 0. Then

(cid:205)𝑘

𝑙=1

¯𝑑 (𝑛𝑀, 𝑙 (𝛼𝑖 + 𝛽𝑖)) = 0.

34

Using additivity of the relative d-invariants, we can rewrite it as

(cid:205)𝑘

𝑙=1

¯𝑑 (𝑀 1

𝑖 , 𝑙𝛼𝑖) = (cid:205)𝑘
𝑙=1

¯𝑑 (𝑀 2

𝑖 , 𝑙 𝛽𝑖).

By Corollary 3.2.1, ¯𝑑 (𝑀 1

𝑖 , 𝑙𝛼𝑖) = ¯𝑑 (𝑀 2

𝑖 , 𝑙 𝛽𝑖) − 𝑙 + 𝐿𝑙

𝑖=0, when 0 ≤ 𝑙 ≤ (𝑘 − 1)/2.

Note that −𝑙 + 𝐿𝑙

𝑖=0

≤ 0 for any 𝑙. When 𝑘 > 2𝑚(𝐾) + 1, −(𝑘 − 1)/2 + 𝐿 (𝑘−1)/2

𝑖=0

< 0 by the

assumption. Hence, we have

(cid:205)𝑘

𝑙=1

¯𝑑 (𝑀 1

𝑖 , 𝑙𝛼𝑖) < (cid:205)𝑘
𝑙=1

¯𝑑 (𝑀 2

𝑖 , 𝑙 𝛽𝑖),

which contradicts the equation above.

□

Let us denote the subgroup generated by 𝛼𝑖 + 𝛽𝑖 by ˜𝐺.

Lemma 3.3.3. For any metabolizer 𝐺 of 𝐻1(𝑛𝑀) with vanishing relative d-invariants, 𝐺 ∩ ˜𝐺 ≠ ∅.

Proof. For any metabolizer 𝐺, we have |𝐺 |2 = 𝑘 2𝑛. Hence, |𝐺 | = 𝑘 𝑛 and 𝐺 is generated by 𝑛
linearly independent elements {𝑔 𝑗 = (cid:205)𝑛
𝑖=1
Suppose 𝐺 ∩ ˜𝐺 = ∅, then 𝐺 (cid:27) 𝐺/𝐺 ∩ ˜𝐺, which is generated by 𝑔′

𝑎𝑖 𝑗 𝛼𝑖 + 𝑏𝑖 𝑗 (𝛼𝑖 + 𝛽𝑖)| 𝑗 = 1, 2, · · · , 𝑛}.

𝑎𝑖 𝑗 𝛼𝑖. Since

𝑗 = (cid:205)𝑛
𝑖=1

|𝐺 | = 𝑘 𝑛, {𝑔′
𝑒 = 𝛼1 + (cid:205)𝑛
𝑖=1

𝑗 } are linearly independent. Hence 𝛼1 ∈ 𝐺/𝐺 ∩ ˜𝐺, which implies there exists
𝑐𝑖 (𝛼𝑖 + 𝛽𝑖) ∈ 𝐺 for some 𝑐𝑖 ∈ Z/𝑘Z.
Suppose 𝑐1 ≠ 0, 𝑒 = (1 + 𝑐1)𝛼1 + 𝑐1𝛽1 + (cid:205)𝑖=2

𝑛(𝛼𝑖 + 𝛽𝑖). Use the same argument from Lemma

3.3.2,

(cid:205)𝑘

𝑙=1

¯𝑑 (𝑛𝑀, 𝑙𝑒) ≠ 0,

which contradicts the assumption.
Suppose 𝑐1 = 0, 𝑒 = 𝛼1 + (cid:205)𝑖=2

𝑛(𝛼𝑖 + 𝛽𝑖).

¯𝑑 (𝑛𝑀, 𝑙𝑒) = ¯𝑑 (Σ2(𝐾2,𝑘 ), 𝑙) + (cid:205)𝑛
𝑖=2

¯𝑑 (𝑀𝑖, 𝑙𝑐𝑖 (𝛼𝑖 + 𝛽𝑖)) (cid:46) 0(mod𝑍),

which also contradicts the assumption. Hence 𝐺 ∩ ˜𝐺 ≠ ∅.

□

Proof of Theorem 3.3:

35

Proof. By Lemma 3.3.3, any metablizer 𝐺 of 𝐻1(𝑛𝑀) contains an element 𝑒 ∈ ˜𝐺. Then by Lemma
3.3.2 and additivity of relative d-invariants, there exists at least one Spin𝑐 structures 𝔰, such that

¯𝑑 (𝑛𝑀, 𝔰) ≠ 0. This shows the nonexistence of a metabolizer for which the relative d-invariants

vanish. Then by Corollary 3.1.1, it proves the nonsliceness of 𝑛(𝐾2,𝑘1# − 𝑇2,𝑘1# − 𝐾2,𝑘2#𝑇2,𝑘2). □

Using the jump function for the Levine-Tristram signature of a knot, we can show the linearly

independent of a set of knots.

For the knot 𝐾, let us use 𝑟 (𝐾) = {𝜃𝑙 } to denote the set of numbers such that when evaluated

at 𝜔 = 𝑒2𝜋𝑖𝜃𝑙 , the jump function is non-zero.

Corollary 3.3.2. Let 𝑘𝑖 be a set of distinct prime numbers such that, 𝑘𝑖 > 2𝑚(𝐾) + 1 and

1/2𝑘𝑖 ∉ 𝑟 (𝐾). Then the set of knots {𝑇2,𝑘𝑖 , 𝐾2,𝑘𝑖 } are linearly independent in the concordance

group C.

Proof. Consider a linear combination

𝐽 = (cid:205)𝑁
𝑖=1

𝑛𝑖𝑇2,𝑘𝑖 + 𝑚𝑖𝐾2,𝑘𝑖 .

Suppose 𝐽 is slice. Fix 𝑙, when we evaluate the jump function at 𝜔 = 𝑒2𝜋𝑖/2𝑘𝑙 . By the assumption,

the only knots have non-zero jump function are 𝑇2,𝑘𝑙 and 𝐾2,𝑘𝑙 , both having jump equal to -1. Hence,

𝑛𝑙 = −𝑚𝑙.

𝐽 = (cid:205)𝑁
𝑖=1

𝑚𝑖 (𝐾2,𝑘𝑖 − 𝑇2,𝑘𝑖 ).

Using the same argument above, 𝐽 is nonslice.

□

36

CHAPTER 4

D-INVARIANTS OF SPLICING OF CIRCLE BUNDLES OF HIGHER GENUS SURFACE

4.1 D=invariants obstruction to existence of certain intersection form

Given a smooth closed four-manifold 𝑋 4 with the intersection form 𝑄 𝑋 = 𝑚𝐸8 ⊕ 𝑛𝐻. The

d-invariants give a constraint on the pair (𝑚, 𝑛) in the following way.

Consider a basis 𝛽 for 𝐻2(𝑋) and let 𝑁 (Σ𝛽) := 𝑁 be a neighborhood of a collection of smooth,

closed surfaces representing the elements in 𝛽. Note that, by adding tubes, we can arrange the

surfaces to be embedded, with geometric intersection number equal to the algebraic intersection

number, at the expense of increasing the genus. Let 𝐶 be the complement of 𝑁 in 𝑋, then it gives

us a splitting of 𝑋 as 𝑋 = 𝑁 ∪𝑌 𝐶, which is illustrated in Figure 4.1. The intersection form 𝑄𝐶 of

𝐶 is trivial.

Figure 4.1 X

Given a four-manifold with boundary (𝑊, 𝜕𝑊), the d-invariants of 𝜕𝑊 give constraints on the

topology of 𝑊 by the following Theorem from [OS03a].

Theorem 4.1 ([OS03a]). Let (𝑊, 𝜕𝑊) be a smooth four-manifold with boundary, such that 𝑊 is

negative semi-definite and the restriction map 𝐻1(𝑊; Z) → 𝐻1(𝑌 ; Z) is trivial. Suppose 𝜕𝑊 has
standard 𝐻𝐹∞ and is equipped with a torsion Spin𝑐 structure 𝔱, then we have the inequality:

𝑐1(𝔰)2 + 𝑏−

2 (𝑊) ≤ 4𝑑𝑏 (𝜕𝑊, 𝔱) + 2𝑏1(𝜕𝑊)

for all Spin𝑐 structures 𝔰 over 𝑊 whose restriction to 𝜕𝑊 is 𝔱.

When 𝐻1(𝑊; Z) → 𝐻1(𝑌 ; Z) is surjective, we have a similar inequality for 𝑑𝑡 (𝜕𝑊, 𝔱):

𝑐1(𝔰)2 + 𝑏−

2 (𝑊) ≤ 4𝑑𝑡 (𝜕𝑊, 𝔱) − 2𝑏1(𝜕𝑊).

37

(4.1)

(4.2)

We first apply Theorem 4.1 to obstruct 𝑚𝐸8 as the intersection form for a smooth, closed

four-manifold as an example.

Figure 4.2 𝐸8

Example 4.1. Figure 4.2 is a plumbing diagram for 𝐸8, where each vertex represents a circle

bundle over a genus 𝑔𝑖 surface with Euler number −2. Denote the manifold corresponding to the

𝐸8 plumbing by (𝑊 (𝐸8), 𝑌 (𝐸8).

Since det(𝐸8) = 1, H1(𝑌 (𝐸8)) is generated by H1(Σ𝑔𝑖 ). Therefore, we have |H1(𝑌 (𝐸8))| =

2 (cid:205) 𝑔𝑖 and 𝐻1(𝑊 (𝐸8)) → 𝐻1(𝑌 (𝐸8)) is surjective.

Given a four-manifold 𝑋 with 𝑄 𝑋 = 𝑚𝐸8, we apply the splitting in Figure 4.1. Then by the

discussion above, 𝐻1(𝑁) → 𝐻1(𝑌 ) is surjective and |H1(𝑌 )| = 2 (cid:205)𝑚,𝑖 𝑔𝑚𝑖 .

There is a unique torsion Spin𝑐 structure on 𝑌 , let us denote it by 𝔱. Since 𝑚𝐸8 is an even
intersection form, (cid:174)0 is a characteristic element. Let us use the corresponding Spin𝑐 structure on

𝑊 (𝐸8). Then by the inequality 4.2, we have

8𝑚 ≤ 4𝑑𝑡 (𝑌 , 𝔱) − 2(2

∑︁

𝑚,𝑖

𝑔𝑚𝑖 )

(4.3)

On the other hand, since 𝐻1(𝐶) = 0, 𝐻1(𝐶) → 𝐻1(−𝑌 ) is trivial. Taking the trivial Spin𝑐

structure on 𝐶 and applying the inequality 4.1, we have:

0 ≤ 4𝑑𝑏 (−𝑌 , 𝔱) + 2(2

∑︁

𝑚,𝑖

𝑔𝑚𝑖 )

(4.4)

38

Combining the property that, 𝑑𝑏 (−𝑌 , 𝔱) = −𝑑𝑡 (𝑌 , 𝔱), we have

∑︁

𝑚,𝑖

𝑔𝑚𝑖 ≥ 𝑑𝑡 (𝑌 , 𝔱) ≥ 2𝑚 +

∑︁

𝑚,𝑖

𝑔𝑚𝑖 ,

which shows 𝑚 = 0.

Given the intersection form 𝑄 𝑋 = 𝑚𝐸8 ⊕ 𝑛𝐻, let us denote the submanifold corresponding to

𝑚𝐸8 by ˜𝑌𝑚 and the submanifold corresponds to 𝑛𝐻 by 𝑌𝑛. Let 𝑔( ˜𝑌𝑚) be the sum of the genera of

all surfaces in ˜𝑌𝑚 and 𝑔(𝑌𝑛) be the sum of the genera of all surfaces in 𝑌𝑛. By the similar argument

as in Example 4.1, using the four-manifold 𝐶, we have

𝑑𝑡 ( ˜𝑌𝑚#𝑌𝑛) = 𝑑𝑡 ( ˜𝑌𝑚) + 𝑑𝑡 (𝑌𝑛) ≤ 𝑔𝑚 + 𝑔𝑛.

Combining with the inequality 4.3, we have

2𝑚 + 𝑔𝑚 + 𝑑𝑡 (𝑌𝑛) ≤ 𝑑𝑡 (𝑌𝑚) + 𝑑𝑡 (𝑌𝑛) ≤ 𝑔𝑚 + 𝑔𝑛,

hence

2𝑚 + 𝑑𝑡 (𝑌𝑛) ≤ 𝑔𝑛

(4.5)

Suppose we have the d-invariants 𝑑𝑡 (𝑌 (𝐻𝑔1,𝑔2)) ≥ 𝑔1 + 𝑔2 − 𝑘, then we can rewrite equation 4.5

as

2𝑚 + 𝑔𝑛 − 𝑛𝑘 ≤ 𝑔𝑛

2𝑚 ≤ 𝑛𝑘

(4.6)

Note that, if 𝑘 = 2, we recover the 10/8 theorem.

However, we have

when min(𝑔1, 𝑔2) ≥ 1.

𝑑top(𝐻𝑔1,𝑔2) = |𝑔1 − 𝑔2| − 2,

We expect to improve the above argument to involutive Heegaard Floer setting and reprove the

10/8 theorem for any genus. One could ask if we could improve the bound to 𝑘 = 3/2 and prove

the 11/8 conjecture. However, by the properties of d-invariants, 𝑑𝑡 (𝑌 (𝐻𝑔1,𝑔2) ∈ Z. Hence, the best

we can do by this argument is the proof of 10/8 theorem.

39

4.2 Zemke’s general surgery formula

In this section, we review the bordered link surgery formula from [Zem21a] [Zem23]. We will

use the version of link surgery formula from Remark 2.5.

4.2.1 Type-D module

In [Zem21a], Zemke reinterprets the link surgery formula in [MO10] as a type-𝐷 module. The

type-𝐷 module is defined as following,

Definition 4.1. If A is an associative algebra over k, a right type-𝐷 module 𝑁 A is a right k-module

𝑁, equipped with a k-linear structure map

𝛿1 : 𝑁 → 𝑁 ⊗k A,

which satisfies

(id𝑁 ⊗𝜇2) ◦ (𝛿1 ⊗ idA) ◦ 𝛿1 = 0.

(4.7)

Remark 4.1. Given a type-𝐷 module (𝑁 A, 𝛿1), the associated pair (𝑁 ⊗k A, 𝛿1 ⊗ idA) is a chain

complex. The type-D relation 4.7 is equivalent to 𝛿1 ⊗ idA being a differential on 𝑁 ⊗k A.

We first talk about the type-D module for knots.

Definition 4.2. The knot surgery algebra K is an algebra over the idempotent ring

I = I0 ⊕ I1,

where each of I𝑖 is rank 1 over F2. Let 𝑖0 and 𝑖1 be the generators of I0 and I1, respectively. We set

I0 · K · I0 = F[𝒰, 𝒱]

and

I1 · K · I1 = F[𝒰, 𝒱, 𝒱−1].

Also

I0 · K · I1 = 0.

There are two special algebra elements, 𝜎, 𝜏 ∈ I1 · K · I0. These algebra elements satisfy the

relation

𝜎 · 𝑓 = 𝐼 ( 𝑓 ) · 𝜎 and

𝜏 · 𝑓 = 𝑇 ( 𝑓 ) · 𝜏,

40

for 𝑓 ∈ F[𝒰, 𝒱] = I0 · K · I0. Here 𝑇 : F[𝒰, 𝒱] → F[𝒰, 𝒱, 𝒱−1] is the algebra homomorphism

satisfying 𝑇 (𝒰) = 𝒱−1 and 𝑇 (𝒱) = 𝒰𝒱2. The map 𝐼 : F[𝒰, 𝒱] → F[𝒰, 𝒱, 𝒱−1] is the

canonical inclusion.

In particular, I1 · K · I0 is generated by two special algebra elements 𝜎 and 𝜏, together with the

left action of F[𝒰, 𝒱, 𝒱−1], which satisfy the relations

𝜎𝒰 = 𝒰𝜎 and 𝜎𝒱 = 𝒱𝜎

𝜏𝒰 = 𝒱−1𝜏

and

𝜏𝒱 = 𝒰𝒱2𝜏.

Let x1, · · · , x2 be a free basis of CF K (𝐾) over F[𝒰, 𝒱]. The type-𝐷 module X𝜆 (𝐾)K

associated to X𝜆 (𝐾) is defined as following:

(i) The underlying I-module of X𝜆 (𝐾)K is

X𝜆 (𝐾)K = SpanF(x1, · · · , x𝑛) ⊗F I.

Over F each x𝑖 contributes one generator in each idempotent. One can think of the generators
in I0 correspond to (cid:206)𝑠∈Z 𝐴−

𝑠 and the generators in I1 correspond to (cid:206)𝑠∈Z 𝐵−

𝑠 . We denote

these generators by x0

𝑖 and x1
𝑖 .

(ii) The map 𝛿1 on X𝜆 (𝐾) contains three summands:

1) Internal 𝛿1 summands from the differential on CF K (𝐾). If 𝜕 (x) contains a summand of

y · 𝒰𝑛𝒱𝑚, then 𝛿1(x𝜖 ) contains a summand of y𝜖 ⊗ 𝒰𝑛𝒱𝑚, for 𝜖 ∈ {0, 1}.

2) The summand corresponds to the inclusion map 𝑣, such that 𝛿1(x0) contains a summand

of the form x1 ⊗ 𝜎.

3) The summand corresponds to ℎ𝜆, such that if y · 𝒰𝑖𝒱 𝑗 is a summand of ℎ𝜆 (x) then we

define 𝛿1(x𝜖 ) to have a summand of the form

y1 ⊗ 𝒰𝑖𝒱 𝑗 𝜏.

It is been proved in [Zem21a] that X𝜆 (𝐾)K is a type-𝐷 module.

41

Example 4.2. The type-𝐷 module associated to 𝜆 surgery on the unknot O is the following (com-

pared to the knot surgery complex in Remark 2.5). The generators are We set

DK

𝜆 · I0 = ⟨x0⟩

and DK

𝜆 · I1 = ⟨x1⟩,

where ⟨x𝜀⟩ = F, spanned by x𝜀. The structure map is given by the formula

𝛿1(x0) = x1 ⊗ (𝜎 + 𝒱𝑛𝜏).

When the link has more than one component, we need to use the type-𝐷 modules over hyper-

cubes. We begin with the 1-dimensional cube algebra C1. The algebra C1 is an algebra over the

idempotent ring I0 ⊕ I1, where I𝜀 (cid:27) F. We set I𝜀 · C1 · I𝜀 = I𝜀, and we set

I1 · C1 · I0 = ⟨𝜃⟩,

i.e. we set I1 · C1 · I0 (cid:27) F, generated by an element 𝜃.

Next, we introduce the cube algebra C𝑛. The definition is

C𝑛 = C1 ⊗F · · · ⊗F C1 = ⊗𝑛

FC1.

Note that this is an algebra over the idempotent ring E𝑛 = ⊗𝑛

FI. We define the type-𝐷 module over

hypercubes as the type-𝐷 module over C𝑛.

Using the identification in Remark 4.7, we have the following proposition:

Proposition 4.1 ([Zem21a]). The category of 𝑛-dimensional hypercubes of chain complexes is

equivalent to the category of type-𝐷 modules over C𝑛.

Definition 4.3. The link algebra Lℓ is defined as

Lℓ := K ⊗F · · · ⊗F K.

We often write just L, when ℓ is determined by context. We view Lℓ as being an algebra over the

idempotent ring

Eℓ := I ⊗F · · · ⊗F I.

42

The type-𝐷 module for links depends on a choice of auxiliary data, which is a system of arcs

𝒜. Since the computation in this thesis is independent of the choice of the system of arcs, we omit

the definition here. One can refer to Chapter.9 in [Zem21a] for more details.

Similar to the construction of the type-𝐷 module of the framed knot, the type-𝐷 module of

framed link can be derived from the link surgery complex. Note that, There is an identification of

Lℓ-idempotents with points of the cube Eℓ. If 𝑀 ⊂ 𝐿, write 𝜀(𝑀) ∈ Eℓ for the coordinate such

that 𝜀(𝑀)𝑖 = 0 if 𝐿𝑖 ∉ 𝑀 and 𝜀(𝑀)𝑖 = 1 if 𝐿𝑖 ∈ 𝑀.

Let x1, . . . , x𝑛 be a free basis of CF L (𝐿) over F[𝒰1, . . . , 𝒰ℓ, 𝒱1, . . . , 𝒱ℓ], then the correspond-

ing type-D module is defined as the following:

(i) The generators of XΛ(𝐿, 𝒜)Lℓ are

SpanF(x1, . . . , x𝑛) ⊗F Eℓ

In particular, if x is a basis element of CF L (𝐿), we have a generator x𝜀 for each 𝜀 ∈ Eℓ. By

Remark 2.5, we may view x𝜀(𝑀) as an element of the group (cid:206)

s∈H(𝐿) 𝔄−(H 𝐿−𝑀, 𝜓 𝑀 (s)).

(ii) Suppose also that (cid:174)𝑁 is an oriented sublink of 𝐿 − 𝑀, and that Φ (cid:174)𝑁 (x𝜀(𝑀)) has a summand of

y𝜀(𝑀∪𝑁) · 𝑓 , where we view 𝑓 as an element of the 2ℓ-variable polynomial ring, localized at

the variables 𝒱𝑖 such that 𝐿𝑖 ⊂ 𝑁. We may naturally view 𝑓 as being an element of

E𝜀(𝑀∪𝑁) · Lℓ · E𝜀(𝑀∪𝑁).

There is an algebra element 𝑡 (cid:174)𝑁

𝜀(𝑀),𝜀(𝑀∪𝑁) ∈ E𝜀(𝑀∪𝑁) · Lℓ · E𝜀(𝑀) which is the tensor of 𝜎𝑖
for 𝑖 such that 𝐿𝑖 ⊂ (cid:174)𝑁 and 𝐿𝑖 is oriented the same as 𝐿, 𝜏𝑖 for 𝑖 such that 𝐿𝑖 ⊂ (cid:174)𝑁 and 𝐿𝑖

is oriented oppositely from 𝐿, and 1 for 𝑖 such that 𝐿𝑖 ∉ 𝑁. With this notation, we declare

𝛿1(x𝜀) to have the summand

y𝜀(𝑀∪𝑁) ⊗ 𝑓 · 𝑡 (cid:174)𝑁

𝜀(𝑀),𝜀(𝑀∪𝑁)

.

4.2.2 Type-A module of solid torus

One can also interpret the type-𝐷 module as a bordered invariant of the link complement.

Similar to the construction in [LOT18], one can also associate a type-𝐴 structure to the link

43

complement. The box tensor of the type-𝐷 structure of the link complement and the type-𝐴

structure of the solid torus recovers link surgery formula.

Definition 4.4. Suppose A is an associative algebra. A left 𝐴∞-module A 𝑀 over A is a left

k-module 𝑀 equipped with k-module map

𝑚 𝑗+1 : A ⊗ 𝑗 ⊗k 𝑀 → 𝑀

for each 𝑗 ≥ 0, such that if 𝑎𝑛, . . . , 𝑎1 ∈ A and x ∈ 𝑀, then

𝑛
∑︁

𝑗=0

𝑚𝑛− 𝑗+1(𝑎𝑛, . . . , 𝑎 𝑗+1, 𝑚 𝑗+1(𝑎 𝑗 , . . . , 𝑎1, x)) +

𝑛−1
∑︁

𝑘=1

𝑚𝑛 (𝑎𝑛, . . . , 𝑎𝑘+1𝑎𝑘 , . . . , 𝑎1, x) = 0.

Given a type-𝐷 module 𝑁 A and a type-𝐴 module A 𝑀, the box tensor product of Lipshitz,

Ozsváth and Thurston [LOT18] construct a chain complex.

The box tensor product 𝑁 A ⊠ A 𝑀 is the chain complex 𝑁 ⊗k 𝑀, with differential 𝜕⊠ as follows.
Denote that 𝛿 𝑗 : 𝑁 → 𝑁 ⊗ A ⊗ 𝑗 is the map given inductively by 𝛿 𝑗 = (idA ⊗ 𝑗 −1 ⊗𝛿1) ◦ 𝛿 𝑗−1 and
𝛿0 = id. Then the differential has the formula

𝜕⊠(x ⊗ y) =

∞
∑︁

𝑗=0

(id𝑁 ⊗𝑚 𝑗+1) (𝛿 𝑗 (x), y).

The differential is usually depicted via the following diagram:

x

𝛿

𝜕⊠(x ⊗ y) =

y

𝑚

(4.8)

According to [LOT18], the map 𝜕⊠ is a differential whenever one of 𝑀 or 𝑁 satisfies a boundedness

assumption.

Suppose that 𝜆 is an integer, the type-𝐴 module K D𝜆 for a solid torus is defined as the following:

Definition 4.5. We set

I0 · D𝜆 = F[𝒰, 𝒱]

and

I1 · D𝜆 = F[𝒰, 𝒱, 𝒱−1].

44

We define the type-𝐴 structure map 𝑚 𝑗 on D𝜆 to be 0 unless 𝑗 = 2. We define 𝑚2 on D𝜆 as follows.

If 𝑓 ∈ I𝑖 · K · I𝑖 and 𝑥 ∈ I 𝑗 · D 𝑗 , then we define 𝑚2( 𝑓 , 𝑥) to be 𝑓 · 𝑥 (ordinary multiplication of

polynomials) if 𝑖 = 𝑗 and to be 0 otherwise. If 𝑥 ∈ I0 · D𝜆, we define

𝑚2(𝜎, 𝑥) = 𝐼 (𝑥) ∈ F[𝒰, 𝒱, 𝒱−1] = I1 · D𝜆.

Similarly, we define

𝑚2(𝜏, 𝑥) = 𝒱𝜆 · 𝑇 (𝑥) ∈ F[𝒰, 𝒱, 𝒱−1],

where · denotes ordinary multiplication of polynomials.

Recall that in the definition of link surgery formula, it involves direct product instead of direct

sum, which can be viewed as a completion with respect to the cofinite topology.

Definition 4.6. If (𝑋𝑖)𝑖∈𝐼 is a family of groups, define the cofinite subspace topology on X = ⊕𝑖∈𝐼 𝑋𝑖

to be

X𝑐𝑜(𝑆); =

𝑋𝑖

(cid:202)

𝑖∈𝐼\𝑆

ranging over finite sets 𝑆 ⊂ 𝐼. When 𝑋𝑖 (cid:27) 𝐹 for all 𝑖, we will refer to this topology as the cofinite

basis topology.

It is not hard to see that the completion of the direct sum with this topology is direct product.

Similarly, we need to complete the type-𝐴 module K D𝜆. We view K D𝜆 as the direct sum of a

copy of F over each Alexander and Maslov grading supported by the module, and complete it with

respect to the cofinite basis topology. The completion is given by

I0 · D𝜆 = F[[𝒰, 𝒱]],

and I1 · D𝜆 = F[[𝒰, 𝒱, 𝒱−1]].

It is not hard to see that the box tensor of type-𝐷 module of 𝐾 with the 0-framed type-𝐴 module

of the solid torus recovers the knot surgery formula:

X𝜆 (𝐾) (cid:27) X𝜆 (𝐾)K ⊠ K [D0].

(4.9)

45

Note that, when taking the box tensor in this case, we use 𝑋𝑖 ⊗ 𝑌𝑗 as the open subspace, where

𝑋𝑖 and 𝑌𝑗 are open subspaces of X𝜆 (𝐾)K and K [D0] respectively. Withe this topology, the two

completions coincide with each other.

Remark 4.2. There is a similar construction for the link surgery formula, One can refer to Chapter.8

of [Zem21a] for more details.

4.2.3 Type-DA bimodule and connected sum formula

We also need type-𝐷 𝐴 bimodules in the definition of the connected sum formula.

Definition 4.7. Suppose that A and B are algebras over j and k, respectively. A type-𝐷 𝐴 bimodule,

denoted A 𝑀 B, consists of a (j,k)-module 𝑀, equipped with (j, k)-linear structure morphisms

𝑗+1 : A ⊗ 𝑗 ⊗j 𝑀 → 𝑀 ⊗k B,
𝛿1

which satisfy the following structure relation:

𝑛
∑︁

𝑗=0
𝑛−1
∑︁

𝑘=1

+

((id𝑀 ⊗𝜇2) ◦ (𝛿1

𝑛− 𝑗+1 ⊗ idB)) (𝑎𝑛, . . . , 𝑎 𝑗+1, 𝛿1

𝑗+1(𝑎 𝑗 , . . . , 𝑎1, x))

𝛿1
𝑛 (𝑎𝑛, . . . , 𝑎𝑘+1𝑎𝑘 , . . . , 𝑎1, x) = 0.

The box tensor of a type-𝐷 𝐴 bimodule and type-𝐷 module gives a type-𝐷 module, and the

differential is given by

𝛿𝐷⊠𝐴𝐷 =

x

𝛿

y

𝛿1

.

The bimodule used in the connected sum formula is the merge bimodule ⊗2K 𝑀2

K.

(i) The underlying space is a (⊗2I, I) bimodule. As a vector space, 𝑀2 is I = I0 + I1. The left

action of ⊗2I is given by

(𝑖1 ⊗ 𝑖2) · 𝑖 = 𝑖1 · 𝑖2 · 𝑖.

The right I -action is the standard action.

46

(ii) There are two non-trivial structure maps on 𝑀2, namely 𝛿1

2 and 𝛿1

𝑛+1.

1) The map 𝛿1

2 is as follows. Suppose that 𝑎1 ⊗ 𝑎2 is an elementary tensor in either

(⊗2I0 · ⊗2K · ⊗2I0)

or

(⊗2I1 · ⊗2K · ⊗2I1).

In this case, we set

2(𝑎1 ⊗ 𝑎2 ⊗ 𝑖) = 𝑖 ⊗ 𝑎1𝑎2.
𝛿1

On any other elementary tensor, we set 𝛿1

2 to vanish.

2) We define 𝛿1

3 as follows. We set

3((1 ⊗ 𝜎) ⊗ (𝜎 ⊗ 1) ⊗ 𝑖0) = 𝑖1 ⊗ 𝜎,
𝛿1

and

3((1 ⊗ 𝜏) ⊗ (𝜏 ⊗ 1) ⊗ 𝑖0) = 𝑖1 ⊗ 𝜏.
𝛿1

More generally, if 𝑎, 𝑏, 𝑐, 𝑑 are monomials concentrated in single idempotents, then we

set

3((𝑎 ⊗ 𝑏𝜎) ⊗ (𝑐𝜎 ⊗ 𝑑) ⊗ 𝑖0) = 𝑖0 ⊗ 𝑎𝑏𝑐𝜎𝑑
𝛿1

and similarly for the 𝜏 terms. Note that we set

3((𝜎 ⊗ 1) ⊗ (1 ⊗ 𝜎) ⊗ 𝑖0) = 0,
𝛿1

and similarly if 𝜏 replaces 𝜎.

Given a pair of framed links (𝑌1, 𝐿1, Λ1) and (𝑌2, 𝐿2, Λ2). Taking a pair of distinguished

components 𝐾1 and 𝐾2, let (𝑌1#𝑌2, 𝐿1#𝐿2, Λ1 + Λ2) denote the connected sum of 𝐿1 and 𝐿2 at

the distinguished components. Here, Λ1 + Λ2 is 𝜆1 + 𝜆2 at the connected sum of the distinguished

components and remains the same for other components. We can compute the type-𝐷 module of

the (𝐿1#𝐿2, Λ1 + Λ2) by the connected sum formula:

Theorem 4.2 ([Zem21a]).

XΛ1+Λ2 (𝑌1#𝑌2, 𝐿1#𝐿2)Lℓ

1+ℓ

2 −1 ≃

(cid:16)

XΛ1 (𝑌1, 𝐿1)Lℓ

1 ⊗F XΛ2 (𝑌2, 𝐿2)Lℓ

2

(cid:17) ⊠ K2 𝑀 K .

47

We can also gluing the solid torus to the distinguished component to get a one dimensional less

hypercube, XΛ1+Λ2 (𝑌1#𝑌2, 𝐿1#𝐿2)Lℓ

1+ℓ

2 −1 ⊠K D0:

(cid:16)

XΛ1 (𝑌1, 𝐿1)Lℓ

1 ⊗F XΛ2 (𝑌2, 𝐿2)Lℓ

2

(cid:17) ⊠ K2 𝑀 K ⊠K D0.

(4.10)

Thinking of ⊠K2 𝑀 K ⊠K D0 as a type-𝐴𝐴 bimodule, which is denoted by K2 [I⋑]. Tensoring
K2 [I⋑] to a distinguished component has the effect as transforming the type-𝐷 module to the type-𝐴

module.

XΛ1+Λ2 (𝑌1#𝑌2, 𝐿1#𝐿2)Lℓ

1+ℓ

2 −1 ⊠K D0 =

=

(cid:16)

(cid:16)

XΛ1 (𝑌1, 𝐿1)Lℓ

1 ⊗F XΛ2 (𝑌2, 𝐿2)Lℓ

2

XΛ1 (𝑌1, 𝐿1)Lℓ

1 ⊠K XΛ2 (𝑌2, 𝐿2)Lℓ

2 −1

(cid:17) ⊠ K2 [I⋑]
(cid:17)

When both 𝐿1 and 𝐿2 have just one component, 4.10 is isomorphic to X𝜆1+𝜆2 (𝑌1#𝑌2, 𝐾1#𝐾2).

We can restate Theorem 4.2 as:

X𝜆1+𝜆2 (𝑌1#𝑌2, 𝐾1#𝐾2) ≃ X𝜆1 (𝑌1, 𝐾1)K ⊠ KX𝜆2 (𝑌2, 𝐾2).

Gluing in the solid torus for the remaining link components, there is a similar statement of

the pairing theorem for the link surgery formula. Let CΛ1 (𝐿1) and CΛ2 (𝐿2) be the link surgery

hypercubes of Manolescu and Ozsváth. Write

CΛ𝑖 (𝐿𝑖) (cid:27) Cone

(cid:18)

C0(𝐿𝑖)

𝐹 −𝐾𝑖 +𝐹 𝐾𝑖

(cid:19)

C1(𝐿𝑖)

Here C𝜈 (𝐿𝑖) consists of the complexes of all points of the cube Eℓ𝑖 such that the coordinate for 𝐾𝑖
is 𝜈 ∈ {0, 1}. Also, we are writing 𝐹 𝐾𝑖 (resp. 𝐹−𝐾𝑖 ) for the sum of the hypercube maps for all

sublinks (cid:174)𝑁 ⊂ 𝐿𝑖 which contain 𝐾𝑖 (resp. −𝐾𝑖).

Theorem 4.2 is equivalent to

Theorem 4.3. The surgery hypercube C𝑌1#𝑌2,Λ1+Λ2 (𝐿1#𝐿2) is homotopy equivalent to the (ℓ1+ℓ2−1)-

dimensional hypercube

(cid:18)

Cone

C0(𝑌1, 𝐿1) ⊗ C0(𝑌2, 𝐿2)

𝐹 𝐾

1 ⊗𝐹 𝐾

2 +𝐹 −𝐾

1 ⊗𝐹 −𝐾
2

C1(𝑌1, 𝐿1) ⊗ C1(𝑌2, 𝐿2)

(cid:19)

48

For each 𝜈 ∈ {0, 1}, the above complexes C𝜈 (𝐿1) ⊗ C𝜈 (𝐿2) are equipped with the tensor product
differential 𝐷 𝜈
1

𝑗 is the total differential of the hypercube C𝜈 (𝐿 𝑗 ) (i.e. the

2, where 𝐷 𝜈

⊗ id + id ⊗𝐷 𝜈

sum of the internal differentials as well as the hypercube maps).

4.3 Computation of d-invariants

In this section, we apply the connected sum formula to compute 𝑑𝑡𝑜 𝑝 (𝐻𝑔1,𝑔2). We decompose

the link in Figure 1.1 into three summands in Figure 4.3, where we have two copies of the Borromean

knot and a Hopf link in the middle.

Figure 4.3 Connected sum decomposition of link surgery

4.3.1 Type-D module of Borromean knot

The knot surgery complex of Borromean knot with Z2 [𝑈] coefficient is computed in [OS08],

we rephrase it here with the Z2 [𝒰, 𝒱] coefficient.

Denote the genus 𝑔 Borromean knot by 𝐵𝑔.

CF K (𝐵𝑔) (cid:27) Λ∗𝐻1(Σ𝑔; Z2) ⊗ Z2 [𝒰, 𝒱],

such that for a generator x ∈ Λ𝑘 𝐻1(Σ𝑔; Z2), 𝐴(x) = gr(x) = 𝑘 − 𝑔.

Note that 𝐵𝑔 ⊂ #2𝑔 (𝑆2 × 𝑆1). Under the identification 𝐻1(Σ; Z2) (cid:27) 𝐻1(#2𝑔 (𝑆2 × 𝑆1)) and
𝐻𝐹∞(#2𝑔 (𝑆2 × 𝑆1) (cid:27) Λ∗𝐻1(Σ𝑔; Z2) ⊗ Z2 [𝒰, 𝒰−1, 𝒱, 𝒱−1], the action of 𝛾 ∈ 𝐻1(Σ; Z2) is given

by the formula

𝛾 · x = 𝜄𝛾x ⊗ 𝒱 + 𝑃𝐷 (𝛾) ∧ x ⊗ 𝒰.

Let {𝛼∗

𝑖 , 𝛽∗
𝑖 }

𝑔
𝑖=1 be a symplectic basis of homology classes, there is an induced map

𝐼 : Λ𝑘 𝐻1(Σ, Z2) → Λ𝑘 𝐻1(Σ, Z2),

49

which commutes with wedge product and satisfies 𝐼 (𝛼𝑖) = 𝛽𝑖 and 𝐼 (𝛽𝑖) = 𝛼𝑖. Together with the

Hodge star operator

∗ : Λ𝑘 𝐻1(Σ; Z2) → Λ2𝑔−𝑘 𝐻1(Σ; Z2),

the flip map ℎ is by the following formula. For x ∈ Λ𝑘 𝐻1(Σ; Z2),

ℎ : x → (∗𝐼x) ⊗ 𝑣2(𝑘−𝑔).

Note that, the knot surgery complex splits to the summands generated by the pairs (𝑥, ℎ(𝑥)).

The top generator of the 𝐻1-action can be represented by 𝑥0 = 𝛼1 ∧ 𝛽1 ∧ · · · ∧ 𝛼𝑔 ∧ 𝛽𝑔 or 𝑥1 = 1.

Hence, to compute 𝑑𝑡𝑜 𝑝 (𝐻𝑔1,𝑔2), we just need the type-𝐷 module for the pair (x0, x1), and let us
denote it by 𝐵K
𝑔 .

𝑔 is generated by Span(x0, x1) ⊗ I. We denote the generator in the I𝑖 idempotent by x𝑖
𝐵K

𝑗 . The

differential is given by

𝛿1(x0

0) = x1

0 ⊗ 𝜎 + x1

𝛿1(x0

1) = x1

1 ⊗ 𝜎 + x1

1 ⊗ 𝒱2𝑔𝜏,
0 ⊗ 𝒱−2𝑔𝜏.

4.3.2 Type-𝐷 𝐴 bimodule of Hopf link

We recall the type-𝐷 𝐴 bimodule for the negative Hopf link K1ZK2

(𝜆1,0) from Chapter.17 of

[Zem21a] in this section.

K1ZK2

(𝜆1,0) can be depicted by the following diagram:

Z0,0

𝐴−direction

Z1,0

𝐷−direction

Z0,1

Z1,1

The module is generated by two elements a and d, such that the Alexander gradings 𝐴 = ( 𝐴1, 𝐴2)

and Maslov gradings gr = (grw

, grz) are

𝐴(a) = (0, 1) + (−

𝐴(d) = (1, 0) + (−

, −

, −

1
2
1
2

),

),

1
2
1
2

gr(a) = (0, 0),

gr(d) = (0, 0).

50

For each summand, we have

(i) Z0,0 is generated by ⟨a⟩[𝒰1] ⊕ ⟨d⟩ [𝒱] over F[𝒰2, 𝒱2],

(ii) Z1,0 is generated by ⟨d⟩[𝒱, 𝒱−1] over F[𝒰2, 𝒱2],

(iii) Z0,1 is generated by ⟨a⟩[𝒰1] ⊕ ⟨d⟩ [𝒱] over F[𝒰2, 𝒱2, 𝒱−1

2

],

(iv) Z1,1 is generated by ⟨d⟩[𝒱, 𝒱−1] over F[𝒰2, 𝒱2, 𝒱−1

2

].

The type-𝐷 structure maps are depicted as:

Z0,0

Z1,0

𝑚1
2

Z0,0

𝐿

1𝑝1
2

+−𝐿

1𝑝1
2

Z1,0

𝑚1
2

𝛿1
1 =

𝐿

2 𝑓 1
1

+−𝐿

2 𝑓 1
1

𝐿

2𝑔1
1

+−𝐿

2𝑔1
1

𝛿1
2 =

Z0,1

Z1,1

𝑚1
2

Z0,1

𝐿

1𝑞1
2

+−𝐿

1𝑞1
2

Z1,1

𝑚1
2

Z0,0

Z1,0

𝛿1
3 =

−𝐻𝜔1
3

Z0,1

Z1,1

The type-𝐷 maps are computed in Chapter.17 of [Zem21a].

Proposition 4.2 ([Zem21a]). Give the negative Hopf link framing (𝜆1, 0). The structure maps of
the minimal model K1ZK2

(𝜆1,0) are as follows:

(i) For the summand Z0,0 and Z1,0, the 𝑚1

2 is given by

1 a ⊗ 𝒰2𝒱2

a) 𝑚1

2(𝒰1, 𝒰𝑛

b) 𝑚1

2(𝒱1, 𝒰𝑛

c) 𝑚1

2(𝒱1, 𝒱𝑚

d) 𝑚1

2(𝒰1, 𝒱𝑚

1 a ⊗ 1.

𝒰𝑛−1

1 a) =

1 a) = 𝒰𝑛+1


d ⊗ 𝒱2

1 d) = 𝒱𝑚+1




1 d) =

𝒱𝑚−1
1

1

a ⊗ 𝒰2

d ⊗ 1.

d ⊗ 𝒰2𝒱2

if 𝑚 > 0

if 𝑛 > 0

if 𝑛 = 0.

if 𝑚 = 0.

51

For the summand Z0,1 and Z1,1, the 𝑚1

2 is given by

2(𝒰𝑖
𝑚1

1𝒱
1

𝑗

, 𝒱𝑛

1 d) = 𝒱
1

𝑗+𝑛−𝑖

d ⊗ 𝒰𝑖

2𝒱𝑖

2

.

(ii) The maps 𝐿1𝑝1

2 and 𝐿1𝑞1

2 are given by the same formulas as each other, as are −𝐿1𝑝1

2 and −𝐿1𝑞1
2.

They are determined by the following formulas:

d ⊗ 𝒰𝑖

2𝒱𝑖+1

2

and 𝐿1𝑝1

2(𝜎1, 𝒱

a) 𝐿1𝑝1

b) −𝐿1𝑝1

1

1a) = 𝒱−𝑖−1
2(𝜎1, 𝒰𝑖
1a) = 𝒱−𝑖−1+𝜆1
2(𝜏1, 𝒰𝑖
2(𝜏1, −) = 0 and −𝐿1𝑝1

1

c) 𝐿1𝑝1

2(𝜎1, −) = 0.

𝑗
𝑗
1 d ⊗ 1.
1 d) = 𝒱
𝑗+𝜆1

d ⊗ 𝒰

d ⊗ 1 and −𝐿1𝑝1

2(𝜏1, 𝒱

𝑗
1 d) = 𝒱
1

𝑗+1
2 𝒱
2

𝑗

.

(iii) The maps for ±𝐿2 are as follows:

1a ⊗ 𝜎2 and 𝐿2𝑓 1

1 (𝒱

𝑗
1 d) = 𝒱

1 d ⊗ 𝜎2.

𝑗

a) 𝐿2𝑓 1

b) −𝐿2𝑓 1

1 (𝒰𝑖
1 (𝒰𝑖

1a) = 𝒰𝑖
1a) = 𝒰𝑖+1

1 a ⊗ 𝜏2 and

−𝐿2𝑓 1

1 (𝒱

𝑗
1 d) =

a ⊗ 𝒱−1

2

𝜏2

if 𝑗 = 0

𝑗−1

d ⊗ 𝜏2

𝒱
1

if 𝑗 > 0.





c) 𝐿2𝑔1

d) −𝐿2𝑔1

1 d ⊗ 𝜎2.

1 (𝒱𝑖
1 (𝒱𝑖

1 d) = 𝒱𝑖
1 d) = 𝒱𝑖−1

1 d ⊗ 𝜏2.

(iv) The map −𝐻𝜔1

3 is determined by the relations

, 𝒰𝑖

𝜆1+𝑚−𝑖−2
1a) = min(𝑖 + 1, 𝑚)𝒱
1

d ⊗ 𝒰𝑚−1

1 𝒱𝑚−1

1

𝜏2

−𝐻𝜔1

−𝐻𝜔1

−𝐻𝜔1

1

1

3(𝜏1, 𝒱𝑚
3(𝜏1, 𝒰𝑛
3(𝜏1, 𝒰𝑛
3(𝜏1, 𝒱𝑚

1

1

−𝐻𝜔1

, 𝒰𝑖

1a) = 0
𝑗

, 𝒱

, 𝒱

1 d) = min(𝑛, 𝑗)𝒱
1 d) = 0,

𝑗

1

𝑗−𝑛+𝜆1−1

d ⊗ 𝒰

𝑗−1
1 𝒱
2

𝑗−2

𝜏2

and that −𝐻𝜔1

3 vanishes if an algebra input is a multiple of 𝜎1. The map −𝐻𝜔1

3 also vanishes

on pairs of algebra elements with other configurations of idempotents.

52

Below is the summand K1ZK2

(0,0) · I0, where an arrow decorated with 𝑎|𝑏 from x to y means that

2(𝑎, x) = y ⊗ 𝑏 and we set 𝑈2 = 𝒰2𝒱2.
𝛿1
𝒱1|𝑈2
𝒰1|1

𝒱1|𝑈2
𝒰1|1

𝒰2
1 a

𝒰2a

· · ·

a

𝒱1|𝒱2
𝒰1|𝒰2

d

𝒱1|1
𝒰1|𝑈2

𝒱1d

𝒱1|1
𝒰1|𝑈2

𝒱2
1 d

· · ·

𝜏1|1

𝜎1|𝑈2

2 𝒱2

𝜏1|1

𝜎1|𝑈2𝒱2

𝜏1|1

𝜎1|𝒱2

𝜏1|𝒰2

𝜏1|𝒰2𝑈2

𝜎1|1

𝜎1|1

𝜏1|𝒰2𝑈2
2

𝜎1|1

· · · 𝒱−3
1 d

𝒱1|1
𝒰1|𝑈2

𝒱−2
1 d

𝒱1|1
𝒰1|𝑈2

𝒱−1
1 d

𝒱1|1
𝒰1|𝑈2

d

𝒱1|1
𝒰1|𝑈2

𝒱1d

𝒱1|1
𝒰1|𝑈2

𝒱2
1 d

· · ·

Note that, K1ZK2

(0,0) · I1 is the localization of K1ZK2

(0,0) · I0 at 𝒱2.

4.3.3 Link Floer complex

We first compute the type-𝐷 module of connected sum of genus 𝑔1 Borromean knot and Hopf

link, which is the type-𝐷 module of the dual knot of 0-surgery on the Borromean knot.

We compute 𝐵K
𝑖 𝒰𝑚

Let x𝑚
𝑖

:= x0

𝑔 ⊠ KZK
0,0
1 a, y𝑚
:= x0

· I0 first.
𝑖 𝒱𝑚

𝑖

1 d, and z𝑚

𝑖

box tensor.

The differentials are given by

:= x1

𝑖 𝒱𝑚

1 d. Then x, y, and z are the generators of the

𝛿(x𝑚

𝛿(x𝑚

𝛿(y𝑚

𝛿(y𝑚

1

0

0 ) = z−𝑚−1
1 ) = z−𝑚−1
0 ) = z𝑚
1 ) = z𝑚

𝑚+2𝑔
0 + z
1

+ z−𝑚−1+2𝑔
1

+ z−𝑚−1−2𝑔

0

⊗ 𝒰𝑚

⊗ 𝒰𝑚

2

2 𝒱𝑚+1
2 𝒱𝑚+1
2
⊗ 𝒰𝑚+1

2

2 𝒱𝑚
2 𝒱𝑚

2

𝑚−2𝑔
1 + z
1

⊗ 𝒰𝑚+1

It can be depicted as following, where the coefficient are labeled on the arrow. The arrow

without coefficient has coefficient 1.
· · ·

· · ·

x2𝑔−1
0

y0
1

𝑔
x
0

𝑔−1
y
1

· · ·

x0
0

y2𝑔−1
1

· · ·

𝒰2𝑔−1𝒱2𝑔 𝒰2

𝒰

𝑔
2 𝒱

𝑔+1
2 𝒰

𝑔−1
𝑔
2 𝒱
2

𝒱2 𝒰2𝑔

2 𝒱2𝑔−1

2

· · ·

z−2𝑔
0

z0
1

· · ·

z−𝑔−1
0

𝑔−1
z
1

· · ·

z−1
0

z2𝑔−1
1

· · ·

53

𝑠2

C0,1
0,1
C1,1
0,0

C0,0
1,1
C1,0
1,0

𝑠1

Figure 4.4 C0,0(𝐵𝑔1

, 𝐵𝑔2) with a summand

𝑔 ⊠ KZK
𝐵K
0,0
generators by ˜x𝑚

· I1 is the localization of 𝐵K
𝑖 , ˜y𝑚

𝑖 , and ˜z𝑚
𝑖 .
Then the type-𝐷 structure map is given by

𝑔 ⊠ KZK
0,0

· I1 by 𝒱−1. Let us denote the corresponding

𝛿1(x𝑚

𝛿1(y𝑚

⊗ 𝜏

𝑖

𝑖 ⊗ 𝜎 + ˜y𝑚−1
˜y𝑚

𝑖 ) = ˜x𝑚
𝑖 ⊗ 𝜎 + ˜x𝑚+1



𝑖 ⊗ 𝜎 + ˜z𝑚−1
𝑖 ) = ˜z𝑚

𝑖 ⊗ 𝜎 + ˜x0
˜y0

𝑖 ) =

𝑖

𝑖

⊗ 𝜏

𝛿1(z𝑚

⊗ 𝜏 for 𝑚 > 0,

𝑖 ⊗ 𝜏 for 𝑚 = 0

We then apply the connected sum formula again and get the link surgery complex

C := C0,0(𝐵𝑔1

, 𝐵𝑔2) = ((𝐵K
𝑔1

⊠ KZK

0,0) ⊗F 𝐵K
𝑔2

) ⊠ K2 𝑀 K ⊠K D0.

Let use C𝜀1,𝜀2

𝑠1,𝑠2 to denote the summand of C0,0(𝐵𝑔1

, − 1

2) and at (𝜀1, 𝜀2) vertex of the hypercube. We use Figure 4.4 to represent C0,0(𝐵𝑔1
𝑠1,𝑠2 . We also include a subcomplex in Figure 4.4.

(− 1
2
where each dot at (𝑠1, 𝑠2) represent ⊕𝜀1,𝜀2 C𝜀1,𝜀2
We now describe the generators of C𝜀1,𝜀2.

, 𝐵𝑔2), which has Alexander grading (𝑠1, 𝑠2) +
, 𝐵𝑔2),

(i) C0,0 is generated by x𝑚

𝑖, 𝑗 := x𝑚

𝑖 ⊗ x0

𝑗 and y𝑚

𝑖, 𝑗 := y𝑚

𝑖 ⊗ x0
𝑗 .

54

(ii) C1,0 is generated by z𝑚

𝑖, 𝑗 := z𝑚

𝑖 ⊗ x1
𝑗 .

(iii) C0,1 is generated by ˜x𝑚

𝑖, 𝑗 := ˜x𝑚

𝑖 ⊗ x0

𝑗 and ˜y𝑚

𝑖, 𝑗 := ˜y𝑚

𝑖 ⊗ x0
𝑗 .

(iv) C1,1 is generated by ˜z𝑚

𝑖, 𝑗 := ˜z𝑚

𝑖 ⊗ x1
𝑗 .

The gradings at these generators are:

(i) 𝐴(x𝑚

𝑖, 𝑗 ) = 𝐴( ˜x𝑚

𝑖, 𝑗 ) = ( 𝐴(x𝑖), 𝐴(x 𝑗 ) + 1) + (− 1
2

(ii) 𝐴(y𝑚

𝑖, 𝑗 ) = 𝐴( ˜y𝑚

𝑖, 𝑗 ) = ( 𝐴(x𝑖 + 1), 𝐴(x 𝑗 )) + (− 1
2

, − 1

2) and gr(x𝑚

𝑖, 𝑗 ) = gr( ˜x𝑚

𝑖, 𝑗 ) = gr(x𝑖) + gr(x 𝑗 ),

, − 1

2) and gr(y𝑚

𝑖, 𝑗 ) = gr( ˜y𝑚

𝑖, 𝑗 ) = gr(x𝑖) + gr(x 𝑗 ),

(iii) 𝐴(z𝑚

𝑖, 𝑗 ) = 𝐴(˜z𝑚

𝑖, 𝑗 ) = ( 𝐴(x𝑖 + 1), 𝐴(x 𝑗 )) + (− 1
2

, − 1

2) and gr(z𝑚

𝑖, 𝑗 ) = gr(˜z𝑚

𝑖, 𝑗 ) = gr(x𝑖) + gr(x 𝑗 ).

Recall that, for an element e, the coefficient ring acts on the grading by the following:

𝐴(𝒰

𝑘1
1 𝒱

𝑙1
1 𝒰

𝑘2
2 𝒱

𝑙2
2 e) = 𝐴(e) + (𝑙1 − 𝑘1, 𝑙2 − 𝑘2),

gr(𝒰

𝑘1
1 𝒱

𝑙1
1 𝒰

𝑘2
2 𝒱

𝑙2
2 e) = gr(e) − 2𝑘1 − 2𝑘2.

We use 𝑓 ±𝐿𝑖 to denote the corresponding maps in the link surgery formula. We also define the

flip map 𝜏 on the set {0, 1}, such that 𝜏(0) = 1 and 𝜏(1) = 0.

(i) 𝑓 𝐿1 (x𝑚

2

𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘

2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙

2

2

2

𝑓 𝐿1 (x𝑚

𝑓 𝐿1 (y𝑚

𝑓 𝐿1 (y𝑚

) = z−𝑚−1
𝑖, 𝑗
) = z−𝑚−1
𝑖, 𝑗
) = z𝑚

2

⊗ 𝒰𝑘+𝑚

2 𝒱𝑙+𝑚+1
2 𝒱𝑙+𝑚+1
⊗ 𝒰𝑘+𝑚
2 𝒱𝑙
2 𝒱𝑙

2

2

2

(𝑖), 𝑗 ⊗ 𝒰𝑘
(𝑖), 𝑗 ⊗ 𝒰𝑘

) = z𝑚

(ii) 𝑓 −𝐿1 (x𝑚

2

𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘

2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙

2

2

2

) = z

) = ˜z

) = z

) = ˜z

2

⊗ 𝒰𝑘

𝐴(x𝑖)−𝐴(x𝜏 (𝑖) )−𝑚−1
𝜏(𝑖), 𝑗
𝐴(x𝑖)−𝐴(x𝜏 (𝑖) )−𝑚−1
𝜏(𝑖), 𝑗
𝐴(x𝑖)−𝐴(x𝜏 (𝑖) )+𝑚
𝜏(𝑖), 𝑗
𝐴(x𝑖)−𝐴(x𝜏 (𝑖) )+𝑚
𝜏(𝑖), 𝑗

2 𝒱𝑙
2 𝒱𝑙
2
𝒱𝑙+𝑚
⊗ 𝒰𝑘+𝑚+1
2
2
𝒱𝑙+𝑚
⊗ 𝒰𝑘+𝑚+1
2
2

⊗ 𝒰𝑘

𝑓 −𝐿1 ( ˜x𝑚

𝑓 −𝐿1 (y𝑚

𝑓 −𝐿1 ( ˜y𝑚

(iii) 𝑓 𝐿2 (x𝑚

2

𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘

2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙

2

2

𝑓 𝐿2 (y𝑚

𝑓 𝐿2 (z𝑚

) = ˜x𝑚

) = ˜y𝑚

) = ˜z𝑚

2

𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘

2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙

2

2

55

(iv) 𝑓 −𝐿2 (x𝑚

2

𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘
𝑖, 𝑗 ⊗ 𝒰𝑘

2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙
2 𝒱𝑙

2

2

2

𝑓 −𝐿2 (y𝑚

𝑓 −𝐿2 (y0

𝑓 −𝐿2 (z𝑚

) = ˜x𝑚+1

) = ˜y𝑚−1

) = ˜x0

) = ˜z𝑚−1

𝑖,𝜏( 𝑗) ⊗ 𝒰𝑘
𝑖,𝜏( 𝑗) ⊗ 𝒰𝑘
𝑖,𝜏( 𝑗) ⊗ 𝒰𝑘
𝑖,𝜏( 𝑗) ⊗ 𝒰𝑘

𝑙+𝐴(𝑥 𝑗 )−𝐴(𝑥 𝜏 ( 𝑗 ) )
2 𝒱
2
𝑙+𝐴(𝑥 𝑗 )−𝐴(𝑥 𝜏 ( 𝑗 ) )
2 𝒱
2
𝑙+𝐴(𝑥 𝑗 )−𝐴(𝑥 𝜏 ( 𝑗 ) )−1
2 𝒱
2
𝑙+𝐴(𝑥 𝑗 )−𝐴(𝑥 𝜏 ( 𝑗 ) )
2 𝒱
2

, when 𝑚 > 0

Based on the description above, one can check we have the following properties of the map in

the link surgery complex.

Proposition 4.3.

(i) 𝑓 𝐿1 induces isomorphism when 𝑠1 ≥ 1 + 𝑔1,

(ii) 𝑓 𝐿2 induces isomorphism when 𝑠2 ≥ 1 + 𝑔2,

(iii) 𝑓 −𝐿1 induces isomorphism when 𝑠1 ≤ −𝑔1,

(iv) 𝑓 −𝐿2 induces isomorphism when 𝑠2 ≤ −𝑔2.

4.3.4 Computation of d-invariants

We first quotient out the acyclic subcomplex in C to reduce it to a finitely generated module.

To do it, we use the homological algebra argument in Chapter.4 of [MO10]. We will assign a

specific filtration to the complex, and then prove that the associated graded complex is acyclic,

which implies that the complex itself is acyclic.

Following the convention in [MO10], a filtration F on a R-module A is a collection of R-

submodules {F 𝑖 (A) | 𝑖 ∈ Z} of A such that F 𝑖 (A) ⊆ F 𝑗 (A) for all 𝑖 ≤ 𝑗. A filtration is called

bounded above if F 𝑖 (A) = A for 𝑖 ≫ 0, and bounded below if F 𝑖 (A) = 0 for 𝑖 ≪ 0. A filtration

is called bounded if it is both bounded above and bounded below.

If A is equipped with a differential 𝜕 that turns it into a chain complex, we say that the chain

complex (A, 𝜕) is filtered by F if 𝜕 preserves each submodule F 𝑖 (A). The associated graded

complex grF A is defined as

grF (A) =

(cid:202)

𝑖∈Z

(cid:0)F 𝑖 (A)/F 𝑖−1(A)(cid:1),

(4.11)

equipped with the differential induced from F .

56

If F is a bounded filtration on a chain complex (A, 𝜕), a standard result from homological

algebra says that if grF (A) is acyclic, then A is acyclic as well.

A standard way to construct bounded filtrations is as follows. If A is freely generated over R

by a collection of generators 𝐺, a bounded map F : 𝐺 → Z defines a bounded filtration on A by

letting F 𝑖 (A) be the submodule generated by the elements 𝑔 ∈ 𝐺 with F (𝑔) ≤ 𝑖.

Suppose now that we have a direct product of R-modules

A =

A𝑠,

(cid:214)

𝑠∈𝑆

indexed over a countable set 𝑆. Suppose further that each A𝑠 is a free module over R with a set of

generators 𝐺 𝑠. Assume that A is equipped with a differential 𝜕.

In this situation, an assignment

(cid:216)

F :

𝐺 𝑠 → Z

𝑠∈𝑆
specifies bounded filtrations on each A𝑠. Together these produce a filtration on A given by

F 𝑖 (A) =

F 𝑖 (A𝑠).

(cid:214)

𝑠∈𝑆

This filtration on A is generally neither bounded above nor bounded below. It is bounded above

provided that there exists 𝑖 ≫ 0 such that F 𝑖 (A𝑠) = A𝑠 for all 𝑠; that is, if F (𝑔) ≤ 𝑖 for all

𝑔 ∈ 𝐺 𝑠, 𝑠 ∈ 𝑆. We have the following:

Lemma 4.3.1. Consider a module A = (cid:206)𝑠∈𝑆 A𝑠, where each A𝑠 is a freely generated over R by
a set of generators 𝐺 𝑠. Suppose F : (cid:208)𝑠∈𝑆 𝐺 𝑠 → Z defines a filtration on A that is bounded above.
Further, suppose A is equipped with a differential 𝜕, and that the associated graded complex

grF (A) is acyclic. Then A itself is acyclic.

Let C(𝑠1,𝑠2)≤(𝑔1,𝑔2) denote the subcomplex (cid:206)𝑠1,𝑠2≤(𝑔1,𝑔2) 𝐶𝜀1,𝜀2

𝑠1,𝑠2 and C1 denote the quotient com-

plex C/C(𝑠1,𝑠2)≤(𝑔1,𝑔2).

57

𝑠2

𝑠1

Figure 4.5 C(𝑠1,𝑠2)≤(2,2), for (𝑔1, 𝑔2) = (2, 2) case

Lemma 4.3.2. 𝐻∗(C1) = 0.

Proof. Given a generator x in C𝜀1,𝜀2

𝑠1,𝑠2 , define the filtration by

F0,0(x) = −𝑠1 − 𝑠2.

This filtration is bounded above on C1. Hence, by Lemma 4.3.1, we just need to show the associated
graded complex is acyclic. The associated graded complex are generated by C𝜀1,𝜀2
𝑠1,𝑠2 , such that either
𝑠1 > 𝑔1 or 𝑠2 > 𝑔2. The remaining differential are 𝑓 𝐿1 and 𝑓 𝐿2. Then by Proposition 4.3, at least
one of 𝑓 𝐿1 or 𝑓 𝐿2 induces isomorphism. Hence, the associated graded complex is acyclic.
□

Hence, we have 𝐻∗(C) (cid:27) 𝐻∗(C(𝑠1,𝑠2)≤(𝑔1,𝑔2)).
Let C𝑠1≤−𝑔1 denote the subcomplex of C(𝑠1,𝑠2)≤(𝑔1,𝑔2) consist of C𝜀1,𝜀2

𝑠1,𝑠2 , such that 𝑠1 ≤ −𝑔1.

Lemma 4.3.3. 𝐻∗(C𝑠1≤−𝑔1) = 0.

Proof. Using the filtration F0,1 = 𝑠1 + 𝑠2 + 𝜀1, which is bounded above, the associated graded

complex are

58

𝑠2

𝑠1

Figure 4.6 C2, for (𝑔1, 𝑔2) = (2, 2) case

C0,1
𝑠1,𝑠2

𝑓 −𝐿

1

C0,0
𝑠1,𝑠2

𝑓 −𝐿

1

,

C1,1
𝑠1,𝑠2−1

C1,0
𝑠1,𝑠2−1

C1,1
𝑠1,𝑔2

C1,0
𝑠1,𝑔2 .

when 𝑠2 + 𝜀1 ≤ 𝑔2 and

𝑓 −𝐿2 induces isomorphism when 𝑠1 ≤ −𝑔1, the associated graded
Since by Proposition 4.3,
complex with 𝑠2 + 𝜀1 ≤ 𝑔2 is acyclic. For 𝑠2 + 𝜀1 = 𝑔2, 𝑓 𝐿2 induces an isomorphism, hence it is
□

also acyclic.

Let C2 := C1/C𝑠1≤−𝑔1, then we have 𝐻∗(C2) (cid:27) 𝐻∗(C).
Similarly, we can quotient out the subcomplex C𝜀1,𝜀2

𝑠1,𝑠2 , such that 𝑠1 + 𝜀2 > 1 − 𝑔1 and 𝑠2 ≤ 1 − 𝑔2.

Let us denote the quotient complex by C3.

C0,0
1−𝑔1,𝑠2

Note that, 𝑓 −𝐿1 also induces an isomorphism on

𝑓 −𝐿

1

, when 𝑠2 ≤ −𝑔2. Let us quotient

C1,0
1−𝑔1,𝑠2−1

59

𝑠2

10

𝑠1

Figure 4.7 C4, for (𝑔1, 𝑔2) = (2, 2) case

it and denote the quotient complex by C4. Based on the discussion above, we have

Proposition 4.4. 𝐻∗(C4) (cid:27) 𝐻∗(C).

By localizing 𝒰2, it is not hard to see that the towers can be represented by C1−𝑔1,−𝑔2 +
(𝑠1,𝑠2)∈𝑆 ( 𝑓 −𝐿1 + 𝑓 −𝐿2)(C0,0
𝑠1,𝑠2), where 𝑆 is a subset of lattice in [1 − 𝑔1, · · · , 𝑔1] × [1 − 𝑔2, · · · , 𝑔2].

(cid:201)

Hence, to compute the d-invariants, we need to find the top grading of these different representations

of the towers.

Given a framed link (𝐿, Λ), let 𝑊Λ(𝐿) be the corresponding cobordism 4-manifold and 𝔷s the
Spin𝑐 structure on 𝑊Λ(𝐿) corresponds to the Alexander grading s. The absolute grading of the

elements in the link surgery formula are given by the following formula:

Theorem .4 ([Zem23]). Suppose 𝐿 ⊂ 𝑆3 is a link with integer framing Λ and 𝔰 ∈ Spin𝑐 (𝑆3

Λ(𝐿))
Λ(𝐿), 𝔰) ≃ XΛ(𝐿, 𝔰) is absolutely graded if on each

is torsion. Then the isomorphism CF−(𝑆3

X𝜀
Λ(𝐿, s) ⊂ XΛ(𝐿) we use the Maslov grading

˜gr := grw +

𝑐1(𝔷s)2 − 2𝜒(𝑊Λ(𝐿)) − 3𝜎(𝑊Λ(𝐿))
4

+ |𝐿| − |𝜀|

where grw is the internal Maslov grading from CF L (𝐿).

60

In our case, for the Alexander grading s = (𝑠1, 𝑠2), 𝑐1(𝔷s)2 = −8𝑠1𝑠2. Since the representations

are in either C0,1 or C1,0. Given an element e𝑠1,𝑠2 ∈ 𝐶𝜀1,𝜀2
𝑠1,𝑠2 ,

˜gr = grw − 2𝑠1𝑠2 − 2.

Without loss of generality, we assume that 𝑔1 ≥ 𝑔2. Given an element e = Σ𝑠1,𝑠2e𝑠1,𝑠2, such that

e represents a top generator of the towers. Suppose 𝑒𝑠1,𝑠2 ∈ C𝜀1,𝜀2
of the top generator of C𝜀1,𝜀2

𝑠1,𝑠2 . It is not hard to see that

𝑠1,𝑠2 , let ˜gr(e𝑠1,𝑠2) denotes the grading

˜gr(e) = min𝑠1,𝑠2 ˜gr(e𝑠1,𝑠2).

We claim that for any e that represents the top generator of the towers,

max( ˜gr(e)) = 𝑔1 − 𝑔2 − 2.

Proof. Let us first check e ∈ C1,0

1−𝑔1,−𝑔2

. C1,0

1−𝑔1,−𝑔2

is generated by 4 elements, z−2𝑔1
0,0

⊗ 𝒰2𝑔2
2

, z−2𝑔1
0,1 ,

z0
1,0

(cid:201)

, and z0

⊗ 𝒰2𝑔2
1,1.
2
Hence max( ˜gr(e)) = ˜gr(z−2𝑔1
0,1
Given a representation e other than C1,0
(𝑠1,𝑠2) C𝜀1,𝜀2

𝑠1,𝑠2 , such that 𝑠1𝑠2 > 0 for all (𝑠1, 𝑠2) ∈ 𝑆. Hence, we have

) = 𝑔1 − 𝑔2 − 2(𝑔1 − 1)𝑔2 − 2 ≤ 𝑔1 − 𝑔2 − 2.

1−𝑔1,−𝑔2

. Note that, e cannot be contained in the subcomplex

max( ˜gr(e)) = max𝑆 (min(𝑠1,𝑠2)∈𝑆 ˜gr(e𝑠1,𝑠2)) ≤ max(grw(e𝑠1,𝑠2)) − 2,

where 𝑠1𝑠2 ≥ 0.

Let e{𝑖, 𝑗 } denotes the elements which tensors with x𝑖 and x 𝑗 . Since the representations

contain ( 𝑓 −𝐿1 + 𝑓 −𝐿2)(C0,0

𝑠1,𝑠2), both x𝑖, 𝑗 and x𝜏(𝑖),𝜏( 𝑗) are contained in e, for a given (𝑖, 𝑗). Hence,

max(grw(e𝑠1,𝑠2)) = 𝑔1 − 𝑔2.

Hence, to prove the claim, we only need to find an element e with ˜gr(e) = 𝑔1 − 𝑔2 − 2.

in C1,0

Let e be the representations on the slope 𝑠1 + 𝑠2 = 1, such that, when 𝑠1 is even, we take elements
𝑠1,𝑠2, and when when 𝑠1 is odd, we take elements in C0,1
𝑠1,𝑠2.
When 𝑠1 is even, the top generators of C1,0

𝑠1,𝑠2 are

z−𝑔1−1+2𝑘
0,1

𝑔2+1−2𝑘
⊗ 𝒱
2

,

61

where 𝑘 ∈ [⌊ −𝑔2

𝑔2−1
2 ⌋].
When 𝑠1 is odd, the top generators of C0,1

2 ⌋, ⌊

𝑠1,𝑠2 are

˜y

𝑔1+2𝑘
1,0

⊗ 𝒱−𝑔2−2𝑘
2

,

where 𝑘 ∈ [⌊ 1−𝑔2

2 ⌋, ⌊

𝑔2
2 ⌋].

We have the minimal grading of these generators as 𝑔1 − 𝑔2. Hence, ˜gr(e) = 𝑔1 − 𝑔2 − 2.

□

62

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[OS08]

[OS10]

[Ras04]

[Rud76]

Peter Ozsváth and Zoltán Szabó. Holomorphic triangles and invariants for smooth
four-manifolds. Advances in Mathematics, 202(2):326–400, 2006.

Peter Ozsváth and Zoltán Szabó. Knot floer homology and integer surgeries. Algebraic
& Geometric Topology, 8(1):101–153, 2008.

Peter S Ozsváth and Zoltán Szabó. Knot floer homology and rational surgeries.
Algebraic & Geometric Topology, 11(1):1–68, 2010.

Jacob Rasmussen. Lens space surgeries and a conjecture of goda and teragaito.
Geometry & Topology, 8(3):1013–1031, 2004.

Lee Rudolph. How independent are the knot-cobordism classes of links of plane curve
singularities. Notices Amer. Math. Soc, 23:410, 1976.

[Zem21a]

Ian Zemke. Bordered manifolds with torus boundary and the link surgery formula.
arXiv preprint arXiv:2109.11520, 2021.

[Zem21b]

Ian Zemke. The equivalence of lattice and heegaard floer homology. arXiv preprint
arXiv:2111.14962, 2021.

[Zem23]

Ian Zemke.
arXiv:2308.15658, 2023.

A general heegaard floer surgery formula.

arXiv preprint

64

[Zha23a]

Chen Zhang. D invariants obstruction to sliceness of a class of algebraically slice
knots. arXiv preprint arXiv:2311.06944, 2023.

[Zha23b] Chen Zhang. D invariants of splicing of higher genus borromean knots. In preparation,

2023.

65

APPENDIX A

D INVARIANTS COMPUTATION VIA LATTICE COHOMOLOGY

When 𝐾 is an algebraic knot, Σ2(𝐾2,𝑞) is a graph manifold. In this case, we can compute the

d-invariants using lattice cohomology. In this appendix, we give the computation of the d-invariants

of Σ2(𝑇2,3;2,𝑞).

Plumbing diagram of Σ2(𝑇2,3;2,𝑞)

In this section, we give a plumbing diagram description for Σ2(𝑇2,3;2,𝑞), for 𝑞 ≥ 7, using the

algorithm from section 3.2.2, which is illustrated in Figure A.1. Let 𝑞 = 2𝑙 + 1, we have the

plumbing diagram for Σ2(𝑇2,𝑞) in Figure A.1 (a) with the framing −1 for the singular fiber. 𝑇2,3 has

the plumbing description as in Figure A.1 (b), where the arrow denotes 𝑇2,3 with Seifert framing

−6. Splicing them together, we obtain the plumbing diagram in Figure A.1 (c). Applying Kirby

calculation to it, we get the simple plumbing diagram in Figure A.1 (d).

Lattice cohomology

In [OS03b], Ozsváth and Szabó gave an algorithm to compute Heegaard Floer homology of a

plumbed three-manifold obtained from a plumbing tree which is negative definite and with at most

two bad vertice (which is a vertex such that the weight exceeds minus the valence). This algorithm

is later formalized by Némethi in [Ném05] and [Ném08], where he defined 𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑐𝑜ℎ𝑜𝑚𝑜𝑙𝑜𝑔𝑦

for the plumbing diagram HF(𝑌 (𝐺)) and proved it is isomorphic to the Heegaard Floer homology

for 𝑎𝑙𝑚𝑜𝑠𝑡 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 graphs. This isomorphism is generalized to all the pluming diagram by Zemke

in [Zem21b]. Since the plumbing diagram for Σ2(𝑇2,3;2,𝑞) has two bad vertices, we will use the

d-invariants formula in [OS03b] to compute.

Given a plumbing diagram 𝐺, let us use 𝑣 to denote its vertices. The plumbing diagram gives

rise to a four-manifold with boundary 𝑋 (𝐺). Let 𝑌 (𝐺) be the boundary of 𝑋 (𝐺).

𝐻2(𝑋; Z) is spanned by the spheres which correspond to each vertex 𝑣. Taking Poincaré dual

to each sphere, we can also view 𝑣 as a basis for 𝐻2(𝑋, 𝑌 ; Z). We also have the following exact

66

sequence:

0 → 𝐻2(𝑋, 𝑌 ; Z) → 𝐻2(𝑋; Z) → 𝐻1(𝑌 ; Z) → 0

(A.1)

Note that we identify 𝐻2(𝑋; Z) with a subset of 𝐻2(𝑋, 𝑌 ; Q). The set of characteristic elements

is defined by

Char = Char(𝐺) := {𝑘 ∈ 𝐻2(𝑋; Z)|𝑘 (𝑣) + (𝑣, 𝑣) ∈ 2Z}

for every 𝑣. The first Chern class gives an identification of the set of Spin𝑐 structures of 𝑋 with
Char(𝐺). Using the exact sequence A.1, the set of Spin𝑐 structures over 𝑌 is identified with the

set of 𝐻2(𝑋, 𝑌 ; Z)-orbits in Char(𝐺). Let us denote the characteristic classes corresponding to the
Spin𝑐 𝔰 over 𝑌 by Char𝔰 (𝐺). We have the following formula to compute d-invariants:

Proposition A.1 ([OS03b]). Let 𝐺 be a negative-definite graph with at most two bad points, and
fix a Spin𝑐 structure 𝔰 over 𝑌 . Then,

𝑑 (𝑌 (𝐺), 𝔰) = max{𝑘∈Char𝔰 (𝐺)}

𝑘 2 + |𝐺 |
4

.

Computation

To compute the d-invariants in the above expression, we need to represent the Spin𝑐 structure

first. Let us denote the plumbing diagram of two-fold branched cover of 𝑇2,3;2,𝑝 by 𝑀𝑝. From

section 3.2, we have

𝐻1(𝑌 (𝑀𝑝)) = Z/𝑝Z,

which has an one-to-one correspondence with Spin𝑐 (𝑀𝑝).

67

The intersect form 𝐴𝑝 of 𝑀𝑝 is

−3

1

1

−2

1

1 −2

2𝑙−6
(cid:122)(cid:125)(cid:124)(cid:123)
. . .




















































−3



1

−2

1

1 −2

1

where we have (2𝑙 − 6)’s −2 vertices in the middle linked as Hopf link.
2𝑙−6
(cid:32)(cid:32)(cid:32)
(cid:32)(cid:32)(cid:32)
(cid:123)
(cid:125)(cid:124)
(cid:122)
0, · · · , 0, 0, 0, −1)

Denote the vector (−1, 0, 0,

𝑇

by w𝑝. Then for any characteristic vector 𝑘, it

can be represented by

𝐴𝑝 𝑘 = w𝑝 + 2Z.

Since 𝑝 is a prime number, by the discussion from section A, we can use the following set to

represent Spin𝑐 (𝑌 (𝑀𝑝):

𝑘𝑖 = 𝐴𝑝

−1(w𝑝 + 𝑖I),

here 𝑖 ∈ {0, 1, · · · , 𝑝} and I = (1, 0, 0, · · · )𝑇
Using [𝑘] to denote the 𝐻2(𝑋, 𝑌 ; Z)-orbit of 𝑘, hence they represent the same Spin𝑐 structure

.

on 𝑌 . For simplicity, let us use 𝜒𝑘 to denote the function 𝜒𝑘 : 𝐿 → Z:

𝜒𝑘 (𝑥) := −(𝑘 (𝑥) + (𝑥, 𝑥))/2,

and 𝑚𝑘 = min𝑥∈𝐿 𝜒𝑘 (𝑥). For any 𝑘 ′ ∈ [𝑘], we can find a 𝑥 ∈ 𝐿, s.t. 𝑘 ′ = 𝑘 + 2𝑥. (𝑘 ′)2 =

(𝑘 + 2𝑥, 𝑘 + 2𝑥) = (𝑘, 𝑘) − 8𝜒𝑘 (𝑥). Hence,

𝑘 2 − 8min𝜒𝑘 = max
𝑘 ′ ∈[𝑘]

(𝑘 ′)2.

68

Now we get a new expression for the d-invariants:

𝑑 (𝑌 (𝑀), [𝑘]) = −2𝑚𝑘 + 𝑘 2+|𝐺 |

4

.

Then the relative d-invariants ¯𝑑 (𝑀, [𝑘𝑖]) = −2(𝑚𝑘𝑖 − 𝑚𝑘0) + 𝑘𝑖

2

2−𝑘0
4

.

We give an explicit computation of the relative d-invariants here.

𝑘𝑖

2 = (w𝑝 + 𝑖I) 𝐴𝑝

−1(w𝑝 + 𝑖I)

𝑖 − 𝑘0
𝑘 2

2 = 2𝑖𝐼𝑇 𝐴𝑝

−1w𝑝 + 𝑖2𝐼𝑇 𝐴𝑝

−1

For the first term, we need to solve the equation 𝐴𝑝v𝑝 = w𝑝, then 𝑖𝐼𝑇 𝐴𝑝

−1w𝑝 is the first element

of v𝑝. After solving the equation, we get v𝑝 = (1, 1, 2,

2𝑙−6
(cid:32)(cid:32)(cid:32)
(cid:32)(cid:32)(cid:32)
(cid:122)
(cid:123)
(cid:125)(cid:124)
2, · · · , 2, 2, 1, 1)

𝑇

and 𝑖𝐼𝑇 𝐴𝑝

−1w𝑝 = 1.

The second term 𝐼𝑇 𝐴𝑝

−1𝐼 = 𝐴𝑝

−1

11 and it is easy to compute, which is the (1,1) minor of 𝐴𝑝.

One can check that 𝐼𝑇 𝐴𝑝

−1𝐼 = −

𝑝−4
𝑝 . Then we have:

¯𝑑 (𝑀, [𝑘𝑖]) = −2(𝑚𝑘𝑖 − 𝑚𝑘0) −

𝑝−4
4𝑝 𝑖2 + 𝑖
2.

Let 𝑥 = (𝑎1, 𝑎2, · · · , 𝑎𝑟), 2𝑚𝑘𝑖 = max𝑥 (𝑘𝑖 (𝑥) + (𝑥, 𝑥)),

(𝑘𝑖 (𝑥) + (𝑥, 𝑥)) = (𝑖 − 1)𝑎1 − 𝑎𝑛 − 3𝑎1

2 − 3𝑎𝑛

2 + 2𝑎1𝑎3 + · · · + 2𝑎𝑛−2𝑎𝑛

= −𝑎1

2 + (𝑙 − 1)𝑎1 − 2(𝑎2 −

− (𝑎𝑛−2 − 𝑎𝑛−1)2 − 2(𝑎𝑛−1 −

2 − · · · − 2𝑎𝑛−1
2 − 2𝑎2
1
1
2
2
1
𝑎𝑛−2)2 −
2

𝑎3)2 −

(𝑎3 − 2𝑎1)2 − (𝑎2 − 𝑎3)2 − · · ·
1
2

(𝑎𝑛−2 − 2𝑎𝑛)2 − 𝑎𝑛

2 − 𝑎𝑛

(A.2)

Hence, 2𝑚𝑘𝑖 = −max𝑎1 ((𝑖 − 1)𝑎1 − 𝑎1). In particular, 2𝑚𝑘0 achieves at the vertex ˜𝑥 with 𝑎1 = 0

and

Linearly independence

2𝑚𝑘𝑖 − 2𝑚𝑘0 ≤ −2𝜒𝑘0 ( ˜𝑥) − 2𝜒𝑘𝑖 ( ˜𝑥) = −𝑖𝑎1 = 0.

(A.3)

Using the computation above, we can prove the linearly independence as in Chapter 3.

Since 𝑑 (𝐿 ( 𝑝, 1), 𝑗) = − 1

4 + (2 𝑗−𝑝)2
4𝑝

, we have

¯𝑑 (𝐿( 𝑝, 1), 𝑗) =

𝑗 2
𝑝 − 𝑗.

69

Suppose we have 𝑖, 𝑗, s.t.

¯𝑑 (𝑌 (𝑀), [𝑘𝑖]) + ¯𝑑 (𝐿 ( 𝑝, 1), 𝑗) = 0, then

𝑝−4
4𝑝 𝑖2 − 𝑖
𝑝−4
𝑖2 −
4

𝑗 2− 𝑗 𝑝

𝑝 ≡ 0(𝑚𝑜𝑑Z),
2 +
𝑝𝑖
2 + 𝑗 2 ≡ 0(𝑚𝑜𝑑𝑝).

Hence, 𝑖 must be an even number, suppose 𝑖 = 2𝑛, 𝑛 ∈ (0, 1, · · · , 𝑝−1

2 ), then we have

𝑗 2 − 4𝑛2 ≡ 0(𝑚𝑜𝑑𝑝),

( 𝑗 − 2𝑛) ( 𝑗 + 2𝑛) ≡ 0(𝑚𝑜𝑑𝑝),

which means, we have 𝑗 = 2𝑛 or 𝑝 − 2𝑛. Together, we have 𝑝 possible solutions. However,

when we put it back to the equation, we have

𝑗 2− 𝑗 𝑝

2 +

𝑝−4
4𝑝 𝑖2 − 𝑖
2(𝑚𝑘𝑖 − 𝑚𝑘0) = 𝑛(3 − 𝑛).

𝑝 = 𝑛(𝑛 − 3),

When 0 < 𝑛 < 3, 𝑛(3 − 𝑛) > 0, but by equation A.2, 2(𝑚𝑘𝑖 − 𝑚𝑘0) ≤ 0. Hence, we have less than

p solutions, which obstructs the existence of the metabolizer.

70

Figure A.1 Plumbing diagram

71