RIBBONS, SATELLITES, AND EXOTIC PHENOMENA IN HEEGAARD
FLOER HOMOLOGY
by
GARY GUTH
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2023
DISSERTATION APPROVAL PAGE
Student: Gary Guth
Title: Ribbons, Satellites, and Exotic Phenomena in Heegaard Floer Homology
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Robert Lipshitz Chair
Boris Botvinnik Core Member
Ben Elias Core Member
Laura Fredrickson Core Member
Daniela Vallega-Neu Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded June 2023
ii
© 2023 Gary Guth
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Gary Guth
Doctor of Philosophy
Department of Mathematics
June 2023
Title: Ribbons, Satellites, and Exotic Phenomena in Heegaard Floer Homology
We study properties of surfaces embedded in 4-manifolds by way of Heegaard
Floer homology. We begin by showing link Floer homology obstructs concordance
through ribbon homology cobordisms; this extends the work of Zemke and
Daemi-Lidman-Vela–Vick-Wong. In another direction, we consider the effect of
satellite operations on concordances. We show that the map induced by a satellite
concordance is determined by the pattern and the map induced by the original
concordance map. As an application, we produce the first examples of stably exotic
behavior in the four-ball, i.e. we produce exotic disks whose exotic behavior persists
under many 1-handle stabilizations. As a second application, in joint work with
Hayden-Kang-Park, we show that the positive Whitehead doubling pattern is
injective on the class of ĤFK -distinguishable disks in B4: we show that for any
disks D,D′ in B4 which are distinguished by their induced maps on ĤFK , their
positive Whitehead doubles are also distinguished. In particular, Wh+(D) and
Wh+(D′) are exotic.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Gary Guth
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
University of California, Berkeley, CA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2023, University of Oregon
Bachelors of Arts, Mathematics, 2017, University of California, Berkeley
AREAS OF SPECIAL INTEREST:
Low Dimensional Topology
PROFESSIONAL EXPERIENCE:
Graduate Student Fellow, University of Oregon, 2018-2023
PUBLICATIONS:
Guth, G. (2021). Ribbon Homology Cobordisms and Link Floer Homology.
arXiv:2111.09925.
Guth, G. (2022). For Exotic Surfaces with Boundary, One Stabilization is not
Enough. arXiv:2207.11847.
Guth, G., Hayden, K., Kang, S., Park, J. (2023). Doubled Disks and Satellite
Surfaces. work in progress.
v
ACKNOWLEDGEMENTS
First and foremost, I am indebted to my advisor, Robert Lipshitz. Without
his patient teaching I would understand far less mathematics. His many
suggestions, careful reading of my work, and thoughtful feedback have been
invaluable. Most of all, I am thankful for his kindness and support through the
stress and turbulence of graduate school.
Thank you to all of you at the University of Oregon who have taught me
so much, and in particular, thank you to the faculty who taught my graduate
courses. I am especially thankful for my fellow low dimensional topologists – Holt,
Jesse, Siavash, Champ and Neda. I found my mathematical footing largely thanks
to our COVID-era Zoom seminars, and have continued to be motivated by our
conversations. Beyond our shared mathematical interests, I am grateful for the
community we fostered and for your friendship. Additionally, I am thankful for
my cohort, who survived the (grueling) first years of graduate school with me,
especially Diego, Nico, and Samantha.
I am grateful for those in the larger mathematical community who have
graciously shared their knowledge and wisdom. Thank you to Irving Dai, Kyle
Hayden, Maggie Miller, and Ian Zemke for answering many mathematical questions
over the years and for sharing about their experiences on the job market and as
postdocs. Thank you to Jennifer Hom, Peter Kronheimer, Ciprian Manolescu,
and Peter Ozsváth for their guidance and support in pivotal moments of my
career. Thank you as well to my collaborators Kyle Hayden, Sungkyung Kang,
and JungHwan Park.
vi
Finally, I am thankful for my friends and family. To my oldest friends,
Willy, Ben, and Chris, I simply cannot imagine (nor remember) life without your
friendship; thank you for the joy you bring to my life. Bret and Annika, thank you
for your endless support (and just as importantly, for all the games which continue
to keep me sane). Thank to my grandmother, Sylvia, for welcoming me to Oregon,
and for (mercilessly) teaching me never to lose track of the number of trump cards
which have been played. Thank you to Mom, Dad, and Ryan for your boundless
love and support. Thank you, Georgianna. I love you more, most, times infinity.
vii
For my grandmother, Sylvia, who made the Northwest feel close to home.
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Four-manifolds and their Surfaces . . . . . . . . . . . . . . . . . . 1
Topology in Dimension 3.5 . . . . . . . . . . . . . . . . . . . . . . 5
Satellite Operations and Exotic Behavior . . . . . . . . . . . . . . 9
Stability of Exotic Behavior . . . . . . . . . . . . . . . . . . . . . . 14
Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . . . 16
Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 18
II. BACKGROUND MATERIAL . . . . . . . . . . . . . . . . . . . . . . . 22
Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . . . 22
Link and Knot Floer Homology . . . . . . . . . . . . . . . . . . . . 25
Cobordisms and Functoriality . . . . . . . . . . . . . . . . . . . . . 27
Bordered Floer Homology . . . . . . . . . . . . . . . . . . . . . . . 37
III. RIBBON HOMOLOGY CONCORDANCES . . . . . . . . . . . . . . . 42
First Homology Action as Link Cobordism Maps . . . . . . . . . . 42
Maps induced by ribbon homology concordances . . . . . . . . . . 48
Torsion and Link Floer Homology . . . . . . . . . . . . . . . . . . 58
ix
Chapter Page
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
IV. SATELLITE CONCORDANCES
AND BORDERED FLOER HOMOLOGY . . . . . . . . . . . . . . . . 66
1- and 3-handle maps: . . . . . . . . . . . . . . . . . . . . . . . . . 67
2-handle map: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
No Cancelation Lemma . . . . . . . . . . . . . . . . . . . . . . . . 74
V. INJECTIVE SATELLITE OPERATORS . . . . . . . . . . . . . . . . . 78
VI. STABLY EXOTIC SURFACES . . . . . . . . . . . . . . . . . . . . . . 82
Cabled concordances . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Two notions of stabilization distance . . . . . . . . . . . . . . . . . 84
Concordances induced by cables . . . . . . . . . . . . . . . . . . . 88
An upper bound for the stabilization distance . . . . . . . . . . . . 89
A lower bound for the stabilization distance . . . . . . . . . . . . . 91
APPENDIX: DIRECT COMPUTATIONS . . . . . . . . . . . . . . . . . . . . 95
Computing the morphism complex . . . . . . . . . . . . . . . . . . 96
From concordance maps to complement maps . . . . . . . . . . . . 99
From complement maps to cabled concordance maps . . . . . . . . 104
x
Chapter Page
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xi
LIST OF FIGURES
Figure Page
1 A three-dimensional 1-handle and a canceling 2-handle. The cocore {pt}×
∂D2 ⊂ D1 × D2 of the 1-handle intersects the core ∂D2 × {pt} ⊂
D2×D1 of the 2-handle in a single point, so the two may be canceled. 3
2 Two surfaces S1 and S2 intersect in a pair of points. This intersection is
eliminated by finding a Whitney disk which can be used push S2 away
from S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 An immersed disk with boundary K. By pushing part of this disk into the
interior of the four-ball, we can remove the ribbon singularities to obtain
an embedded disk. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 The positive Whitehead double pattern in the solid torus. . . . . . . . 11
5 The negative Whitehead double of a slice disk for the Stevedore knot. . 12
6 A cobordism between Σ0 and Σ1, where the intermediate surface Σ̃ is visible
as Σ0 with a collection of 1-handles attached. . . . . . . . . . . . . 15
7 A Heegaard diagram for S1 × S2#S1 × S2. The α curves are drawn in
red and the β curves are drawn in blue. . . . . . . . . . . . . . . . . 23
8 The graph Γ realizing the action of a closed curve γ in Y . The cyclic ordering
is indicated by the dashed arrow. . . . . . . . . . . . . . . . . . . . 30
9 The decorated link cobordisms (S1 × D3,M) (left) and (D2 × S2,M ′)
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10 A surface Σ with Σz-sub region a bigon. . . . . . . . . . . . . . . . . . 36
11 A schematic of the decorated surface in S2 × S1 × {0} described in the
text preceding Lemma 3.1.1. . . . . . . . . . . . . . . . . . . . . . . 43
12 A schematic of the decorated surface Fγ. In general, K×{0} and γ might
be linked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
13 When the link cobordism map on the right is followed by the action of γ,
it becomes equivalent to the map on the left. . . . . . . . . . . . . . 46
xii
Figure Page
14 A procedure for trading handles of a concordance for handles of the ambient
manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
15 A concordance from the unknot to K#K with no bands. . . . . . . . . 64
16 By pinching along the dotted line, we see a dynamic bigon map is homotopic
to the composition of a monogon map with a triangle map. . . . . . 70
17 A (g, n)-stabilization along (B4, S0). The case (g, n) = (2, 2) is shown. . 84
18 Swimming one band through another. . . . . . . . . . . . . . . . . . . . 85
19 An isotopy taking a 1-handle stabilization of D to a 1-handle stabilization
of D′. A swim move occurs in frame 6. (Continued in Figure 20) . . 86
20 The remainder of the isotopy between the stabilizations of D and D′. Swim
moves occur in frames 2 and 4. . . . . . . . . . . . . . . . . . . . . 87
21 An isotopy of D ∪ (v ∪ v∗). . . . . . . . . . . . . . . . . . . . . . . . . . 89
22 Part 1 of an isotopy between p-fold stabilizations of Dp and D
′
p . . . . 91
23 Part 2 of an isotopy between p-fold stabilizations of Dp and D
′
p . . . . 92
24 The complex CFK−(J), i ∈ {1, 2}. . . . . . . . . . . . . . . . . . . . . . 95
25 On the left is the summand of CFK−(J) containing FD(1) and FD′(1). On
the right is a model for the corresponding summand of ĈFD(S3 − J). 96
26 The type-D structure C associated to a unit box. . . . . . . . . . . . . 98
27 A doubly pointed bordered Heegaard diagram for the longitudinal unknot
in the solid torus, (T∞, λ). . . . . . . . . . . . . . . . . . . . . . . . 100
28 The full complex ĈFD(S3 − J). . . . . . . . . . . . . . . . . . . . . . . 102
29 A doubly pointed bordered Heegaard diagram for the (p, 1)-cable in the
solid torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
xiii
LIST OF TABLES
Table Page
1 Bi-gradings of generators of CFK−(J). . . . . . . . . . . . . . . . . . . . 95
xiv
CHAPTER I
INTRODUCTION
Four-manifolds and their Surfaces
In this section, we provide a brief overview of four-manifold topology, aiming
to motivate the questions explored in this dissertation. Roughly, a manifold is a
space which, locally, is indistinguishable from our own Euclidean space – objects
like 3-space, the surface of the earth, planes, and donuts. Topology concerns itself
with those features of spaces which are preserved by gentle deformations. To a
topologist, a circle is equivalent to an ellipse, as the circle can simply be stretched
to match the dimensions of the ellipse, or the the ellipse compressed until it forms
a perfect circle. However, the surface of a ball is somehow fundamentally different
from the surface of a donut; no amount of stretching or twisting will create a hole
in the sphere nor close off the hole in the donut.
The notion of gentle deformation is formalized as follows: let X and Y
be two n-dimensional manifolds, and let f be a function from X to Y which
has an inverse. We say that f is a homeomorphism if both f and its inverse
are continuous. If such a map between X and Y exists, we say that X and Y
are homeomorphic. If f and its inverse are smooth as well, we say that f is a
diffeomorphism, and that X and Y are diffeomorphic. A priori, homeomorphism
is a weaker relation than diffeomorphism, yet, in dimensions 1, 2, and 3 these
two relations are equivalent; any two manifolds which are homeomorphic are also
diffeomorphic. In higher dimensions, however, homeomorphic manifolds need not be
1
diffeomorphic, and the interplay between these two notions of equivalence continues
to be a central theme in the study of manifolds.
In particular, much of the richness of four-dimensional topology arises from
the stark difference between these two notions of equivalence; four-manifolds X and
Y which are homeomorphic but not diffeomorphic are called exotic. Dimension
four is the smallest dimension in which exotic phenomena can appear. While
exotic manifolds exist in infinitely many dimensions, exotic behavior in dimension
four is especially peculiar. For instance, if n ≠ 4, Euclidean space Rn admits
a unique smooth structure, i.e. any manifold homeomorphic to Rn is, in fact,
diffeomorphic to Rn; on the hand, there are uncountably many smooth four-
manifolds homeomorphic to R4 but not diffeomorphic to it. In any dimension other
than four, the n-dimensional hyper-sphere Sn admits a finite number of smooth
structures, and the exact number of such structures is computable. However, the
number of smooth structures on S4 is completely mysterious.
What is it about dimension four which makes the smooth topology so
complicated? In short, the answer is the way in which surfaces (two dimensional
manifolds) can be embedded in four-manifolds. The question of classifying smooth,
simply connected n-manifolds was revolutionized by Smale, with his proof of the h-
cobordism theorem for n > 4. Two n-manifolds X0, X1 are cobordant if there exists
an (n + 1)-manifold W such that ∂W = X0 ⨿ X1. If the inclusion of Xi ↪→ W
is a homotopy equivalence, we say that W is an h-cobordism. Smale proved that if
two simply connected n-manifolds X and Y , n > 4, can be connected by a simply
connected h-cobordism W , they are, in fact, diffeomorphic.
Any smooth, compact manifold can be constructed from simple pieces called
handles. An n-dimensional k-handle is simply an n-dimensional disk Dk ×Dn−k; n-
2
dimensional k-handles are attached to boundaries of n-manifolds along Sk−1×Dn−k.
An (n + 1)-dimensional cobordism W between X0 and X1 can be decomposed into
handles, so that
W ∼= X0 × [0, 1] ∪H1 ∪H2 ∪ . . . ∪Hn−1,
where Hk is a union of (n + 1)-dimensional k-handles. At times, handles can cancel
in pairs: roughly a k-handle creates a k-dimensional cavity which can potentially
be filled in by a (k + 1)-handle. See Figure 1. Smale argues that if W is an h-
cobordism, any handle decomposition for W can be simplified until all handles
cancel in pairs, leaving a trivial decomposition of W as:
W ∼= X0 × [0, 1].
It then follows that X ∼0 = X1, since ∂W ∼= ∂(X × [0, 1]).
FIGURE 1 A three-dimensional 1-handle and a canceling 2-handle. The cocore
{pt} × ∂D2 ⊂ D1 ×D2 of the 1-handle intersects the core ∂D2 × {pt} ⊂ D2 ×D1 of
the 2-handle in a single point, so the two may be canceled.
A pair of handles h = Dk × Dn+1−k and h = Dk+1 × Dn−k1 2 cancel
exactly when the cocore of h1, S1 = {pt} × ∂Dn+1−k ⊂ h1 and the core of h2,
S2 = ∂D
k+1×{pt} ⊂ h2 intersect in a single point. The assumption that W is an h-
3
cobordism guarantees that, after perhaps sliding handles over one another, S1 and
S2 will intersect once algebraically. Smale carefully argues that when the dimension
of Xi is at least 5, h1 and h2 can be smoothly deformed until S1 and S2 intersect in
a single point geometrically.
Smale’s proof relies on the “Whitney trick”. Since the algebraic intersection
of S1 and S2 is one, all additional intersection points appear in pairs. A pair of
oppositely oriented intersection points p and q determines a loop γ in W , by
concatenating a path λ1 in S1 from p to q and a path λ2 in S2 from q to p. Since W
is assumed to be simply connected, this loop can be contracted to a point, and this
contraction determines a disk D, with boundary γ. Since X0 and X1 are assumed
to have dimension greater than 4, we can deform this disk until it contains no self-
intersections and is disjoint from S1 and S2. The intersection points p and q can
then be eliminated by pushing S1 along this disk D. For a schematic, see Figure 2.
FIGURE 2 Two surfaces S1 and S2 intersect in a pair of points. This intersection is
eliminated by finding a Whitney disk which can be used push S2 away from S1.
This argument uses the assumption on the dimension in an essential way: in
dimension four, the Whitney disk may intersect itself or the 2- and 3-handles which
we hope to cancel as there are insufficiently many dimensions to eliminate these
intersections.
4
However, the groundbreaking work of Freedman showed that Smale’s
program can be carried out in dimension four by dropping the requirement that
Whitney disks be smoothly embedded, and settle for topologically embedded disks
[Fre82]. As a consequence, Freedman proved that h-cobordant four-manifolds
are homeomorphic. Moreover, the homeomorphism type of a four-manifold X is
determined by its intersection form, which is a bilinear form that records the ways
in which embedded surfaces intersect in X. Just a year later, with the advent of
gauge theory, Donaldson proved that the question of classifying smooth, simply
connected four-manifolds was much more complicated [Don83]; many topological
four-manifolds fail to admit smooth structures and those that do admit smooth
structures very often admit infinitely many.
In an essential way, the landscape of four dimensional topology is shaped by
the ways in which surfaces can be embedded in four-manifolds. This thesis aims to
continue to tease out the relations between four-manifolds and their surfaces.
Topology in Dimension 3.5
There is much to be gained by considering the interplay of three and four-
dimensional topology. Every three manifold can be realized as the boundary of
some four-manifold, and much can be deduced about the four-manifold by studying
its boundary. For instance, a theorem of Freedman and Quinn [FQ90] states that
any three manifold Y which has the same integral homology as S3 bounds a unique
contractible four-manifold up to homeomorphism (such a three-manifold is called
an integer homology sphere. The story is quite different smoothly, however, and
many integer homology spheres do not bound smooth contractible manifolds, or
even manifolds with the homology of B4, integer homology balls. Two integer
5
homology spheres Y1 and Y2 are homology cobordant if there is a smooth four-
manifold W with ∂W = −Y1 ⨿ Y2 and H∗(W,Yi;Z) = 0 for i ∈ {1, 2}. Integer
homology 3-spheres which bound an integer homology ball are exactly those
three-manifolds which are cobordant to S3. The set of integer homology spheres
modulo homology cobordism forms a group under connected sum, called the three-
dimensional integer homology cobordism group, usually denoted Θ3Z.
The 3-dimensional homology cobordism group highlights yet again the
peculiar nature of low dimensional topology. The analogous n-dimensional
homology cobordism groups are trivial for n ̸= 3 by work of Kervaire [Ker69].
However, Θ3Z was proven to be non-trivial by Rokhlin, who exhibited a surjective
homomorphism µ : Θ3Z → Z/2 [Roh52]. Work of Fintushel-Stern and Furuta showed
that Θ3Z is infinitely generated [FS85, FS90, Fur90] and recent work of Dai-Hom-
Stoffregen-Truong showed it contains a Z∞ summand [DHST18].
In a similar vein, natural questions about four-space arise when studying knot
theory. We define a knot K to be an embedding of the 1-sphere into a 3-manifold
Y ,
K : S1 ↪→ Y 3.
We say that K is the unknot if K bounds a disk in Y ; moreover, the unknot is
uniquely characterized by this property. However, if we think of Y as the boundary
of a 4-manifold W , we can consider surfaces with boundary K which are allowed to
occupy space in the four-manifold W . By allowing our surfaces to live in four-space,
it is no longer true that the unknot is characterized by bounding a disk. In fact,
many knots in S3 bound disks which are embedded in the four-ball, B4. Such knots
are called slice.
6
Some slice disks can be visualized in three-dimensions. For example, consider
the immersed disk in Figure 3. This disk has two ribbon singularities, where the left
band of the surface passes through two separate sheets of this surface. By pushing
small neighborhoods of these arcs where the surface intersects itself into the interior
of the four-ball we can eliminate these singularities, leaving us with an embedded
disk in the four-ball with boundary K.
FIGURE 3 An immersed disk with boundary K. By pushing part of this disk into
the interior of the four-ball, we can remove the ribbon singularities to obtain an
embedded disk.
A generalization of this phenomena is called concordance. Knots K1 and K2
are called concordant if there is an embedded annulus C ⊂ [1, 2] × S3 with the
property that C ∩ {i} × S3 = Ki, for i ∈ {1, 2}. Notice that a knot is slice if
and only if it is concordant to the unknot. The set of knots in S3 modulo smooth
concordance forms a group, C, with respect to connected sums. Much work has
been invested in understanding the structure of this group.
A concordance C is called ribbon if there are no local maxima with respect to
the projection [1, 2] × S3 → [1, 2]. A long standing conjecture of Gordon, which
was recently proved by Agol, posits that the relation of ribbon concordance defines
a partial order on the set of knots in S3 [Ago22, Gor81].
7
A homology concordance between knots K1 and K2 in three-manifolds Y1 and
Y2 respectively is a pair (W,C) such that
1. W is a homology cobordism from Y1 to Y2;
2. C is an annulus embedded in W so that C ∩ Yi = ki for i ∈ {1, 2}.
The set of knots in S3 modulo homology concordance forms a group, CZ,
under connected sums, and the set of knots in arbitrary homology spheres modulo
homology concordance also forms a group, denoted ĈZ. There are natural maps
C → CZ → ĈZ.
The first map is surjective, while the second is injective. The kernel of the first
map is quite mysterious; an element of the kernel would be a knot in S3 which is
not slice in B4, but is slice in some other homology 4-ball. No examples of such
knots are known. The cokernel of the second map is also interesting to consider; an
element of the cokernel of this map is a knot in a homology 3-sphere which is not
homology concordant to any knot in S3. The cokernel is highly nontrivial by work
of Levine and Hom-Levine-Lidman and even contains a Z∞ summand by work of
Zhou [Lev16, HLL22, Zho20].
A cobordism W from Y1 to Y2 is called ribbon if it admits a handle
decomposition consisting of 1- and 2-handles (i.e. contains no three handles).
Equivalently, W can be equipped with a Morse function with no critical points
of index three. A homology concordance (W,C) is called simultaneously ribbon if
there is a Morse function f : W → R satisfying:
1. W is ribbon with respect to f (f has no critical points of index three.)
8
2. The restriction of f to C is Morse
3. f |C has no critical points of index two.
Recent work of Huber and Friedl-Misev-Zenter shows that ribbon homology
cobordism defines a partial order on the set of irreducible homology spheres
[Hub22, FMZ22]. This and Agol’s work lead to the natural question:
Question 1.2.1. Does simultaneous ribbon homology cobordism define a partial
order on the set of pairs (Y,K) of knots K in homology 3-spheres?
We provide evidence for this in Chapter 3.
Satellite Operations and Exotic Behavior
In Section 1.2, we introduced the smooth concordance group, which was the
set of knots in S3 studied up to smooth concordance equipped with the operation
of connected sum. This equivalence relation can be weakened by considering locally
flat concordances. An embedded surface Σ ⊂ X4 is locally flat if every point x in
Σ contains a neighborhood U such that the pair (U,U ∩ Σ) is homeomorphic to
(R4,R2 × {0}). We will write CTOP for the group of knots in S3 modulo locally flat
concordance under connected sum. The natural map
C → CTOP ,
has a nontrivial kernel, i.e. there are knots in S3 which are not smoothly slice, but
are topologically slice. In fact, by work of Freedman-Quinn [FQ90], any knot K
with trivial Alexander polynomial is topologically slice. Conversely, there are many
examples of knots with trivial Alexander polynomial which are not smoothly slice,
as can be shown using gauge or Floer theory.
9
One particularly interesting class of examples arises by studying satellite
knots. Let K be a knot in S3 and let P be a knot in the solid torus. Define the
satellite of K with pattern P in S3 to be the result of removing a neighborhood
of the knot K in S3 and replacing it with the solid torus containing the knot P ,
gluing according to the Seifert framing of K. More succinctly,
(S3, P (K)) = ((S3 − ν(K)) ∪ (S1 ×D2), P ).
The knot K is called the companion knot and P is called the pattern knot.
The Alexander polynomial of a satellite knot is determined by the Alexander
polynomials of the companion and the pattern:
∆P (K) = ∆P (t) ·∆ wK(t ).
Here, w is the winding number of P , which is defined as follows: P represents a
class in the first homology of S1 × D2, and therefore [P ] is some multiple of a
generator for H1(S1 ×D2) ∼= Z; define w to be this integer.
Consider the pattern shown in Figure 4. This pattern is called the positive
Whitehead double pattern, P = Wh+. Though Wh+ is nontrivial in S1 × D2, it
becomes unknotted after embedding (S1 × D2,Wh+) ↪→ S3. Therefore, since the
Whitehead doubling pattern has winding number zero, we have that for any knot
K,
∆Wh+(K)(t) = ∆U(t) ·∆K(t0) = ∆K(1).
The Alexander polynomial for any knot K evaluated at t = 1 is equal to ±1.
Therefore, by Freedman-Quinn, Wh+(K) is topologically slice. In many cases,
10
Wh+(K) can be shown not to be smoothly slice. Conjecturally, a knot K is
smoothly slice if and only if Wh+(K) is smoothly slice.
FIGURE 4 The positive Whitehead double pattern in the solid torus.
This example has a nice reformulation. Given a concordance C : K1 → K2,
and a satellite pattern P ⊂ S1 × D2, define the satellite concordance P (C) :
P (K1) → P (K2) as the result of deleting a neighborhood of the concordance from
S3 × [1, 2] and replacing it with (S1 ×D2 × [1, 2]), P × [1, 2]):
(S3 × [1, 2], P (C)) = ((S3 × I − ν(C)) ∪ (S1 ×D2 × [1, 2]), P × [1, 2]).
Therefore, a satellite pattern P determines a well-defined operator on C (or CTOP ),
namely [K] 7→ [P (K)]. As an operator on CTOP , Wh+ is trivial, since after applying
the pattern, all knots become topologically slice. However, its behavior on C is
much more mysterious; the conjecture above can be reformulated by asking whether
the operator Wh+ is injective.
There is a natural four-dimensional analogue of this phenomenon. Suppose D
is a slice disk for a knot K. By puncturing D, we obtain concordance C : U → K.
Given a satellite pattern P , the procedure outlined above produces a concordance
P (C) : P (U) → P (K).
11
If the pattern P satisfies the property that P ↪→ S1 × D2 ↪→ S3 is the unknot, we
say that P is an unknotted pattern. In other words, P is unknotted when P (U) =
U . Therefore, the concordance C : U → K gives rise to a concordance
P (C) : U → P (K).
By capping off (S3, U) with (B4, D2), we obtain a slice disk for P (K), which we
will call the satellite of D with pattern P , denoted P (D). For an example, see
Figure 6.
FIGURE 5 The negative Whitehead double of a slice disk for the Stevedore knot.
Extending the work of Freedman-Quinn, Conway-Powell show that if D is a
slice disk for K with the additional property that
π 41(B − ν(D)) ∼= Z
then D is unique up to topological isotopy rel boundary [CP21]. Two smooth
embeddings f, g : Σ → X are isotopic if there is a one parameter family of
diffeomorphisms Ht : X → X such that H0 is the identity and H1 ◦ f = g. A
topological isotopy is analogous, replacing the family of diffeomorphisms with a
family of homeomorphisms. We often will identify these embeddings with their
12
images and say that Σ0 = f(Σ) and Σ1 = g(Σ) are isotopic (in the appropriate
category). Surfaces Σ0 and Σ1 in X are called exotic if they are topologically
isotopic but not smoothly. In the case in which our surfaces have boundary, we
will consider isotopies which restrict to the identity on the boundaries.
Just as closed 4-manifolds may exhibit exotic behavior, surfaces embedded in
a given four-manifold may be equivalent but not smoothly. In particular, given a
pair of slice disks D1 and D2 for K, the positive Whitehead doubles of D1 and D2
are necessarily topologically isotopic by Conway-Powell, but can often be shown to
be nonisotopic smoothly. In analogy with the case of knots in S3, we conjecture:
Conjecture 1.3.1. Let D1 and D2 be slice disks for K. D1 and D2 are smoothly
isotopic red boundary if and only if Wh+(D1) and Wh
+(D2) are smoothly isotopic
red boundary.
As before, this question can be reformulated in terms of satellite operations.
Following [JZ21], let Surf0(K) be the set of isotopy classes of connected, properly
embedded genus-0 surfaces in the four-ball with boundary. An unknotted pattern P
induces a map
P : Surf0(K) → Surf0(P (K)),
in the obvious way, by taking a slice disk for K to the satellite disk for P (K)
determined by the pattern. In this language, we can reframe Conjecture ?? as
asking whether the operator Wh+ is injective. In Chapter 5, in joint work with
Hayden-Kang-Park, we provide evidence for this conjecture (as well as evidence
that much larger families of unknotted satellite operators are injective) by studying
the behavior of invariants coming knot Floer homology.
13
Stability of Exotic Behavior
Though exotic behavior abounds in dimension four, it tends to be unstable,
which is to say that exotic behavior tends to vanish after enlarging the objects in
some way. The most famous example is due to Wall.
Theorem 1. [Wal64] Let X0 and X1 be simply connected, oriented exotic 4-
manifolds. Then for sufficiently large k, X #k(S20 × S2) and X1#k(S2 × S2) are
diffeomorphic.
In order to motivate an analogous operation for surfaces, we will sketch a
proof of this argument.
Proof. If X0 and X1 are exotic, they are cobordant through some 5-manifold W ;
we can assume that W has no 1- or 4-handles (else we can do surgery to eliminate
them), meaning W is built entirely from five-dimensional 2- and 3-handles. The 2-
handles are attached along nullhomotopic circles, which has the effect of splitting
off S2×S2 summands on the boundary. Hence, ∂ k 2+(X0× I ∪ 2-handles) = X0# S ×
S2; a dual argument shows that ∂−(X1 × I ∪ 3-handles) = X0#kS2 × S2. The result
then follows by observing that ∂+(X0 × I ∪ 2-handles) ∼= ∂−(X1 × I ∪ 3-handles).
Work of Gompf [Gom84] extends Wall’s result to oriented 4-manifolds with
arbitrary fundamental groups. A famous open conjecture states that a single
stabilization is always enough to eliminate such exotic behavior. The operation
of taking a connected sum with S2 × S2 is called a stabilization.
The work of Hosokawa-Kawauchi and Baykur-Sunukjian [HK79, BS16]
shows that exotic surfaces Σ0 and Σ1 contained in a four-manifold X also become
smoothly isotopic after increasing the genus of the two surfaces by attaching
14
“tubes” (or, more precisely, attaching the boundaries of three-dimensional 1-
handles). This operation is also called stabilization (or sometimes, internal
stabilization).
Theorem 2. Let Σ0 and Σ1 be smoothly embedded surfaces in a four-manifold X.
Let Σ̃0 and Σ̃1 be obtained from Σ0 and Σ1 by k internal stabilizations. Then, for
sufficiently large k, Σ̃0 and Σ̃1 are smoothly isotopic.
The proof is nearly identical to the four-manifold case. Since Σ0 and Σ1 are
homologous, they are cobordant through a three-manifold Y in X. Y can be built
from three dimensional 1- and 2-handles, and Σ̃0 and Σ̃1 can be identified both as
the union of Σ0 together with the boundary of the 1-handles and as the union of Σ1
with the boundary of the 2-handles. For this reason, increasing the genus of the two
surfaces is a natural analogue of stabilization in the surface case.
FIGURE 6 A cobordism between Σ0 and Σ1, where the intermediate surface Σ̃ is
visible as Σ0 with a collection of 1-handles attached.
15
These examples demonstrate the general principle that exotic behavior is
unstable. One naturally asks how unstable: how many times is it necessary to
stabilize exotic pairs before they become equivalent? A long standing conjecture
states that, in both of these cases, a single stabilization should always be enough
to eliminate the exotic behavior. Much work has been done in providing evidence
for this conjecture; for most known examples, it has been confirmed that a single
stabilization is enough (for example, see [Auc03, AKMR15, BS16, Bay18]) and,
under certain homological conditions, there are cases where the exotic behavior is
known to dissolve under a single stabilization [AKM+17]. However, in Chapter 6,
we provide counterexamples in the case of surfaces with boundary.
Heegaard Floer Homology
The results of this dissertation rely heavily on a collection of invariants
coming from Heegaard Floer homology. Heegaard Floer homology, introduced by
Ozsváth and Szabó [OS04b, OS06, OS04a], is a collection of invariants of three-
and four-manifolds as well as knots and surfaces embedded within them. We
will provide a much more thorough review of the salient aspects of the theory in
Chapter 2, though, for the casual reader, we have included a structural summary of
these invariants here.
Any three-manifold Y can be represented by a combinatorial object called a
Heegaard diagram, H, which consists of a closed surfaces Σ of genus g decorated
with two collections of embedded curves {α1, . . . , αg} and {β1, . . . , βg} with the
property that when Σ is thickened and three-dimensional 2-handles are attached
along the α and β curves the resulting object is homeomorphic to Y − (B3 ⨿B3).
16
To this Heegaard diagram H (together with a choice of basepoint on Σ,
Ozsváth-Szabó define a chain complex, CF−(H), which is an F2[U ]-module. There
are several variants:
CF∞(H) = CF−(H)⊗F[U ] F[U,U−1],
CF+(H) = CF∞(H)/CF−(H),
ĈF (H) = CF−(H)/U · CF−(H).
The differential counts certain holomorphic disks in the g-fold symmetric product of
Σ. The homology of this complex is denoted HF ◦(Y ) (◦ ∈ {−,∞,+,̂}), and, up to
canonical isomorphism does not depend on the choices made in the construction of
the complex.
Moreover, these invariants fit into the structure of a topological quantum
field theory (TQFT). A TQFT associates to an n-manifold an algebraic object, and
to a cobordism between n-manifolds, a morphism between objects. (Formally, a
TQFT is a symmetric monoidal functor from a cobordism category to an abelian
category.) In the context of Heegaard Floer homology, a cobordism W from Y1 to
Y2 (together with a graph embedded in W connecting the basepoints), there is an
induced map
F ◦W : HF (Y1) → HF (Y2).
These maps are well behaved: the product cobordism Y × [0, 1] : Y → Y induces
the identity map on HF ◦(Y ), and given a decomposition of W as W1 ∪ W2 where
W1 : Y0 → Y1 and W2 : Y1 → Y2, the cobordism maps satisfy a composition law
[OS06, Zem15].
17
Given a knot K in a three manifold Y , Ozsváth-Szabó and, independently,
Rasmussen define a complex CFL−(Y,K) [OS04a, Ras03]. We will follow the
conventions of [Zem19d] and view CFL−(Y ) as an F[U, V ]-module. Just as in the
closed 3-manifold case, the chain homotopy type of this complex is an invariant of
the pair (Y,K). Moreover, if W is a cobordism from Y1 to Y2 and Σ is a surface
embedded in W such that Σ ∩W = K1 ∪K2, where Ki ⊂ Yi for i ∈ {1, 2} there is
an associated map
F −W,Σ : HFL (Y1, K1) → HFL(Y2, K2).
This map depends on some additional data; see chapter 2 for more details. The
map FW,Σ is an invariant of the pair (W,Σ) [Zem19d].
Summary of Results
In this section, we provide a brief overview of the layout of this dissertation.
In Chapter 2, we review some background on Heegaard Floer homology. In Chapter
3, we analyze the cobordism maps associated to ribbon homology concordances.
The main theorem is an obstruction to the existence of such cobordisms, building
off of work of [Zem19b, DLVVW19].
Theorem 3. Let (W,F) : (Y0, K0) → (Y1, K1) be a ribbon Z-homology concordance
where F = (C,A) is a surface decorated by A, a pair of parallel arcs. Then the
induced map
FW,F ,s : HFL−(Y0, K0, s|Y0) → HFL−(Y1, K1, s|Y1)
is a split injection.
We will also consider an algebraic reduction of HFL−(Y,K, s), denoted
HFL−(Y,K, s). HFL−(Y,K, s) is a finitely generated F[V ]-module, and therefore
18
can be decomposed into a free summand and a torsion summand, which is denoted
HFL−red(Y,K, s).
Definition 1.6.1. Let K be a nullhomologous knot in a 3-manifold Y . Define the
torsion order of K in Y to be the quantity
OrdV (Y,K, s) = min{d ∈ N : V d · HFL−red(Y,K, s) = 0}.
Juhász-Miller-Zemke [JMZ20] use the the torsion order of knots in S3 to
give bounds on many topological invariants of knots, including the fusion number,
the bridge index, and the cobordism distance. We prove an analogue of [JMZ20,
Theorem 1.2] in the ribbon homology cobordism setting.
Theorem 4. Suppose (W,Σ) : (Y0, K0) → (Y1, K1) is a Z-homology link cobordism
such that W is ribbon with respect to a Morse function h : W → R compatible with
Σ. Suppose Σ has m critical points of index 0 and M critical points of index 2 with
respect to h|Σ. Then
OrdV (Y0, K0, s|Y0) ≤ max{M,OrdV (Y1, K1)}+ 2g(Σ).
When W is a product, we also have
OrdV (Y1, K1, s|Y1) ≤ max{m,OrdV (Y0, K0)}+ 2g(Σ).
We use Theorem 4 to prove some results about ribbon cobordisms between
knots in homology cobordant 3-manifolds, and consider some generalizations of the
fusion number in the context of ribbon homology cobordisms.
19
In Chapter 4, we make use of bordered Floer homology to show that maps
induced by satellite concordances are determined by the “companion” concordance
and the type-A structure associated to the pattern knot in the solid torus.
Theorem 5. Let C : K → K ′ be a smooth concordance. Then, there exists a map
F : ĈFD(S3 −K) → ĈFD(S3 −K ′) induced by C, such that for any pattern knot
P in the solid torus, the following diagram commutes up to homotopy:
CFA−(HP )⊠ ĈFD(S3 −K) ≃ CFK−(KP )
id⊠F FCP
CFA−(H )⊠ ĈFD(S3 −K ′) ≃ − ′P CFK (KP ),
where HP doubly pointed, bordered Heegaard diagram for P ⊂ S1 ×D2, KP and K ′P
are satellites of K and K ′, and CP is the concordance induced by P . The horizontal
arrows are given by the pairing theorem [LOT18].
After proving Theorem 13, we describe a sufficient condition on the
type-A structure of the satellite pattern to determine whether the pattern is
“HFK -injective”; in short, we give a criterion for determining whether a pair
of concordances which are distinguishable by knot Floer homology will remain
distinguishable after applying the satellite.
In Chapter 5, we turn to applications. In particular, we show:
Theorem 6. Let K be a knot in S3 with slice disks D1, D2. If D1 and D2 are
ĤFK-distinguishable, then so are Wh+(D1) and Wh
+(D2). Moreover, Wh
+(D1)
and Wh+(D2) are an exotic pair.
In fact, the first statement of Theorem 6 holds for many satellite patterns
(positive cables, the Mazur patterns, and generalized doubling patterns). Finally,
20
we prove that there exist exotic surfaces in the four-ball with arbitrarily large
stabilization distance.
Theorem 7. For any p, there exists a knot Jp which bounds a pair of exotic disks
Dp and D
′
p which remains exotic after p− 1 internal stabilizations.
Theorem 7 also (somewhat trivially) gives examples of higher genus surfaces
with arbitrarily large stabilization distance.
21
CHAPTER II
BACKGROUND MATERIAL
Heegaard Floer Homology
A Heegaard splitting for a three-manifold Y is a decomposition of Y as
Y = H1 ∪ϕ H2,
where H1 and H2 are handlebodies of genus g (i.e. regular neighborhoods of the
g-fold wedge of circles) and ϕ is an orientation reversing homeomorphism from ∂H1
to ∂H2. Every three-manifold possesses such a decomposition by an application of
Morse theory: simply choose a self-indexing Morse function f : Y → [0, 3] and
define H1 = f
−1([0, 3 ]), H2 = f
−1([3 , 3]).
2 2
The data of a Heegaard splitting can be recorded combinatorially by a
Heegaard diagram. A Heegaard diagram consists of a tuple H = (Σ,α,β), which
consists of
1. A closed surface Σ of genus g;
2. Two collections of pairwise disjoint, closed curves α = {α1, . . . , αg} and β =
{β1, . . . , βg} in Σ such that both {αi} and {βi} represent linearly independent
classes in H1(Σ,Z).
The three-manifold Y can be reconstructed from α and β as follows: thicken Σ to
Σ × [0, 1]; attach thickened disks along the α-curves in Σ × {0} and thickened disks
along the β-curves in Σ × {1}; the homological assumptions on the α- and β-curves
22
FIGURE 7 A Heegaard diagram for S1×S2#S1×S2. The α curves are drawn in red
and the β curves are drawn in blue.
guarantee that the resulting three-manifold has boundary S2 ⨿ S2, which can be
filled with three-balls.
Heegaard Floer homology is an invariant of closed 3-manifolds defined by
Ozsváth and Szabó [OS04b]. Given a pointed Heegaard diagram H = (Σ,α,β, z)
for Y , one considers the symmetric product
Symg(Σ) = Σ×g/Sg,
the quotient of the g-fold product of Σ by the symmetric group. Symg(Σ) is a
smooth manifold, and in fact, inherits a complex structure from Σ. The attaching
curves α and β determine half-dimensional submanifolds
Tα = α1 × . . .× αg Tβ = β1 × . . .× βg.
CF−(Y, z) is freely generated as an F[U ]-module by intersection points in Tα ∩
Tβ, and the differential counts holomorphic disks connecting intersection points,
weighted by U -powers determined the algebraic intersection of the disk with the
subvariety
Vz = {z} × Symg−1(Σ).
23
Let D2 be the unit disk in the complex plane. Let e1 be the arc in ∂D
2 with
positive real part and let e2 be the arc in ∂D
2 with negative real part. Given
intersection points x,y ∈ Tα ∩ Tβ, a Whitney disk from x to y is a map
ϕ : D2 → Symg(Σ)
such that
1. ϕ(−i) = x and ϕ(i) = y,
2. ϕ(e1) ⊂ Tα and ϕ(e2) ⊂ Tβ.
The set of homotopy classes of Whitney disks from x to y is denoted π2(x,y).
Given ϕ ∈ π2(x,y) and a path of complex structures Js on Symg(Σ), we define
MJs(ϕ) to be the moduli space of Js-holomorphic Whitney disks representing ϕ.
The complex disk D2 has an R-action given by translation, and we define
M̂Js(ϕ) = MJs(ϕ)/R.
The Maslov index of ϕ, µ(ϕ) is the expected dimension of this moduli space.
Ozsváth and Szabó prove that for a generic path of complex structures, if (Σ,α,β)
is a Heegaard diagram with attaching curves in general position, then for each
ϕ ∈ π2(x,y) with µ(ϕ) = 1, M̂Js(ϕ) is a compact, zero dimensional manifold.
Finally, we can define a differential on CF−(Y, z) as
∑ ∑
∂(x) = #M̂(ϕ)Unz(ϕ)y.
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1
24
One proves that ∂2 = 0 by studying the of the moduli spaces M̂(ϕ) with ϕ ∈
π2(x,w) and µ(ϕ) = 2.
Ozsváth and Szabó prove that the chain homotopy type of CF−(Y, z) does
not depend on the choices made in the construction (the complex structure
and choice of Heegaard splitting) and is therefore a well-defined three-manifold
invariant.
The Heegaard Floer homology groups split over SpinC-structures of the three-
manifold. In [OS04b], Ozsváth and Szabó define a map
s : T ∩ T → SpinCz α β (Y )
by interpreting SpinC-structures on Y as homology classes of non-vanishing vector
fields on Y (in the sense of [Tur97]); once we choose a Morse function inducing our
Heegaard splitting, an intersection point x determines flowlines from the index one
to index two critical points and the basepoint determines a flowline connecting the
index zero and three critical points. Outside a neighborhood of these flowlines, the
gradient vector field is non-vanishing, and this homology class is defined to be the
SpinC-structure associated to the intersection point x. Intersection points which
are connected by a Whitney disk are necessarily in the same SpinC-structure, so we
have that ⊕
CF ◦(Y ) = CF ◦(Y, s).
s∈SpinC(Y )
Link and Knot Floer Homology
There is a refinement of this invariant for knots and links in three-manifolds.
Several versions of link Floer homology exist in the literature. Knot Floer
25
homology is an invariant of knots in 3-manifolds defined by Ozsváth and Szabó
[OS04a] and independently by Rasmussen [Ras03]. The extension to links is due to
Ozsváth and Szabó in [OS08a]. We review the definitions in order to establish the
conventions we will be following.
We can encode the data of a knot in a three-manifold by placing additional
basepoints on our Heegaard diagram. Once again, choose a self-indexing Morse
function f : Y → [0, 3] which induces a Heegaard splitting of Y with one index zero
critical point and index three critical point. A pair of basepoints w and z on the
Heegaard surfaces determines a knot in Y in the following way: there is a unique
flow-line connecting the index zero and three critical points which passes through
w (and similarly for z). The union of these two paths is a knot in Y . Similarly,
we can encode a link in Y by choosing more w and z basepoints on our Heegaard
surface.
Definition 2.2.1. A multi-based link L = (L,w, z) in a 3-manifold Y is an oriented
link L with two collections of basepoints w and z such that each component of L
has at least one w- and one z- basepoint and the basepoints alternate between w
and z as one travels along the link.
The link Floer complexes are constructed by choosing a multi-pointed
Heegaard diagram (Σ,α,β,w, z) for (Y,L), where α = (α1, ..., αg+n−1) and
β = (β1, ..., βg+n−1) are the attaching curves, g is the genus of Σ, and n = |w| = |z|.
Denote by Tα and Tβ the half dimensional tori α1× ...×αg+n−1 and β1× ...×βg+n−1
in Symg+n−1(Σ). The link Floer complex splits over SpinC-structures for Y and is
generated by intersection points in Tα∩Tβ. Following [Zem18], we define CFL−(Y, s)
to be the free F2[U, V ]-module generated by intersection points x in Tα ∩ Tβ with
sw(x) = s. As in the three-manifold case, the differential is defined by counting
26
holomorphic disks of Maslov index 1, with the U and V variables recording the
algebraic intersections of the disks with the basepoints: let
∑ ∑
∂(x) = #M̂(ϕ)Unw(ϕ)V nz(ϕ)y,
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1
and extend F2[U, V ]-linearly. Note, CFL−(Y, s) could also have been defined as
generated by intersection points x with sz(x) = s. By [Zem18, Lemma 3.3],
sw(x) − sz(x) = PD[L], where [L] is the fundamental class of the link. Hence,
when the homology class of the link is trivial in H1(Y ;Z), the maps sw and sz
agree, so either choice yields the same complex. However, for links which are not
nullhomologous, the two complexes may differ. For a more general set up, see
[Zem18, Section 3].
Cobordisms and Functoriality
Heegaard Floer homology (as well as knot and link Floer homology) have
the structure of a topological quantum field theory. In short, four-dimensional
cobordisms between three-manifolds induced functorial maps between chain
complexes. Similarly, decorated link cobordisms induce maps between link Floer
complexes. Since we will primarily be interested in surfaces in this thesis, we will
emphasize those maps induced by link cobordisms.
Definition 2.3.1. A decorated link cobordism from (Y0,L0) = (Y0, (L0,w0, z0))
to (Y1,L1) = (Y1, (L1,w1, z1)) is a pair (W,F) = (W, (Σ,A)) with the following
properties:
1. W is an oriented cobordism from Y0 to Y1
27
2. Σ is an oriented surface in W with ∂Σ = −L0 ∪ L1
3. A is a properly embedded 1-manifold in Σ, dividing it into subsurfaces Σw
and Σz such that w0,w1 ⊂ Σw and z0, z1 ⊂ Σz.
In [Zem18], it is shown that a decorated link cobordism (W,F) from (Y0,L0)
to (Y ,L ) and a SpinC1 1 -structure s on W , give rise to a map
F −W,F ,s : CFL (Y0,L0, s| ) → CFL−Y0 (Y1,L1, s|Y1),
and these maps are functorial [Zem18, Theorem B] in the following sense:
1. Let (W,F) be the trivial link cobordism, i.e. W = Y ×[0, 1], Σ = L×[0, 1] and
A is a collection of arcs p× [0, 1] where p ⊂ L−(w∪z) and consists of exactly
one point in each component of L− (w ∪ z). Then FW,F ,s = idCFL−(Y,L,s| ).Y
2. If (W,F) can be decomposed into the union of two decorated link cobordisms
(W1,F1) ∪ (W2,F2) and s1 and s2 are SpinC-structures on W1 and W2
respectively which agree on their common boundary, then
∑
FW2,F2,s2 ◦ FW1,F1,s1 ≃ FW,F ,s.
s∈SpinC(W ),
s|W =si i
The decorated link cobordism maps are defined as compositions of maps
associated to handle attachments to the embedded surfaces and to the ambient
4-manifold.
In general, it is quite difficult to compute the decorated link cobordism maps.
In some simple cases, however, the link cobordism maps can be computed in terms
of the graph cobordism maps defined in [Zem15].
28
Definition 2.3.2. If (Y0,w0) and (Y1,w1) are 3-manifolds with a collection of
basepoints w0 and w1, a ribbon graph cobordism between them is a pair (W,Γ)
such that W is a cobordism from Y0 to Y1 and Γ is a graph embedded in W with
the properties that Γ ∩ Yi = wi, each basepoint wi has valence 1 in Γ, and at each
vertex, the edges of Γ are given a cyclic ordering.
A ribbon graph cobordism (W,Γ) : (Y0,w0) → (Y1,w1) gives rise to two maps:
FAW,Γ,s, F
B − −
W,Γ,s : CF (Y0,w0, s|Y0) → CF (Y1,w1, s|Y1).
These two maps satisfy
FAW,Γ,s ≃ FB ,W,Γ,s
where Γ is the graph obtained by reversing the cyclic ordering at each of the
vertices. The map FAW,Γ,s depends on the interaction between the graph and the α-
curves while FB depends on the interaction of the graph and the β -curves. When
W,Γ,s
Γ is simply a path, these maps agree with the original cobordism maps defined by
Ozsváth and Szabó [Zem15, Theorem B].
The graph cobordism maps encode the action of Λ∗H1(Y )/Tors on the
Heegaard Floer complexes. Recall that, given a closed loop γ ⊂ Y , the action of
[γ] ∈ H1(Y )/Tors on HF−(Y,w) is induced by a map
A : CF−γ (Y,w) → CF−(Y,w)
defined bv ∑ ∑
Aγ(x) = a(γ, ϕ)#M̂(ϕ)Unw(ϕ)y.
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1
29
FIGURE 8 The graph Γ realizing the action of a closed curve γ in Y . The cyclic
ordering is indicated by the dashed arrow.
Roughly speaking, the quantity a(γ, ϕ) is the intersection number of γ and the
portion of the boundary of a domain for ϕ which lies on an α-curve. This map
satisfies A2γ ≃ 0 and can be realized by the graph cobordism (Y × [0, 1],Γ), where
the graph Γ is shown in Figure 8.
If (W,F) : (Y0,L0) → (Y1,L1) is a decorated link cobordism and Γ ⊂ Σ is
a ribbon graph, we say that Γ is the ribbon 1-skeleton of Σw if Γ ⊂ W , Γ ∩ Yi =
wi, Σw is a regular neighborhood of Γ in Σ, and the cyclic orders of Γ agree with
the orientation of Σ. A ribbon 1-skeleton of Σz is defined in exactly the same way.
There are natural chain isomorphisms
CFL−(Y,L, s)⊗ ∼ −F2[U,V ] F2[U, V ]/(V − 1) = CF (Y,w, t)
and
CFL−(Y,L, s)⊗F F [U, V ]/(U − 1) ∼ −2[U,V ] 2 = CF (Y, z, t− PD[L]).
30
Under these identifications, a link cobordism map FW,F ,s induces two maps on
CF−(Y ), denoted
FW,F ,s|U=1 and FW,F ,s|V=1.
These maps agree with the maps induced by the graph cobordism maps associated
to the ribbon 1-skeletons of Σ.
Theorem 8. [Zem18, Theorem C] If (W,F) is a decorated link cobordism, and
Γw ⊂ Σw and Γz ⊂ Σz are ribbon 1-skeleta, then
F AW,F ,s|U=1 ≃ FW,Γz,s−PD[Σ] :
and
FW,F ,s|V=1 ≃ FBW,Γ ,s,w
under the identifications above.
For a full discussion on Zemke’s graph TQFT framework, see [Zem15] or, for
an overview, see [Zem19c, Section 9.2].
In the following situation, the decorated link cobordism maps are determined
by the corresponding graph cobordism map. Let F be a closed surface in W :
Y0 → Y1, which is decorated by A, as in Definition 2.3.1. Choose disjoint disks
D0 and D1 in F which each intersect A in a single arc, and perturb F so that it
intersects Yi in Di. Remove each Di, leaving a decorated cobordism F0 between
doubly pointed unknots U1 and U2. Let pi denote the center of the disk Di. Identify
CFL−(Yi,Ui, s) with CF−(Yi, pi, s) ⊗F[W ] F[U, V ], where W acts on F[U, V ] as
UV . Under this identification, a graph cobordism map FW,Γ,s induces a map
CFL−(Y − F2[U,V ]0,U0, s|Y0) → CFL (Y1,U1, s|Y1), which we write as FW,Γ,s| . In this
31
case, the link cobordism map induced by (W,F0) : (Y0,U0) → (Y1,U1) is relatively
simple.
Proposition 2.3.3. [Zem19c, Proposition 9.7] Let F = (Σ,A) be a closed
decorated link cobordism, and let (W,F0) be the link cobordism obtained from F
by the procedure outlined above. Define ∆A to be
⟨c1(s),Σ⟩ − [Σ] · [Σ] χ(Σw)− χ(Σz)
+ .
2 2
Then, V ∆A · FB W,Γ ,s|
F[U,V ] ∆A ≥ 0
w
FW,F0,s ≃ U−∆A · FA F[U,V ]W,Γz,s−PD[Σ]| ∆A ≤ 0,
where Γw and Γz are ribbon 1-skeleta.
It will also be useful to understand how the link cobordism maps change
under surgery operations. If (W,F) is a link cobordism and γ is a closed curve
in A, we can simultaneously do surgery on γ in W and Σ to obtain a new link
cobordism (W (γ),F(γ)), i.e. remove a regular neighborhood of γ ⊂ (W,Σ), which
can can identified with (S1×D3, S1×D1) and replace it with (D2×S2, D2×S0). The
surface obtained by surgery on γ naturally inherits a decoration A(γ), so denote
the new decorated surface F(γ) = (Σ(γ),A(γ)) (see Figure 9). If the curve γ
represents a non-divisible element of H1(W ;Z) then
FW,F ,s ≃ FW (γ),F(γ),s(γ)
by [Zem19a, Proposition 5.4]. The assumption that [γ] is non-divisible guarantees
that there is a is the unique SpinC-structure s(γ) on W (γ) which extends a given
32
FIGURE 9 The decorated link cobordisms (S1 × D3,M) (left) and (D2 × S2,M ′)
(right).
SpinC-structure on W −N(γ). An analogous result holds for surgeries on collections
of curves γ1, ..., γn which will be of use in the proof of Theorem ??.
Proposition 2.3.4. Let (W,F) be a link cobordism. Let γ1, ..., γn be closed curves
in A and let (W (γ1, ..., γn),F(γ1, ..., γn)) be the surgered link cobordism. If the
restriction map H1(W − ⨿N(γi)) → H1(⨿∂N(γi)) is surjective, then there is a
unique SpinC-structure s(γ1, ..., γn) extending s|W−⨿N(γ ) for each s ∈ SpinC(W ) andi
FW,F ,s ≃ FW (γ1,...,γn),F(γ1,...,γn),s(γ1,...,γn).
Finally, recall in [OS06, Proof of Theorem 3.1], Ozsváth and Szabó define an
extended cobordism map:
FW,s : Λ
∗H1(W ;Z)/Tors⊗ CF−(Y0, s| −Y0) → CF (Y1, s|Y1).
A Heegaard triple (Σ,α,β,γ) gives rise to a cobordism Xα,β,γ. Since the natural
map H1(∂Xα,β,γ) → H1(Xα,β,γ) is surjective, a given element h ∈ H1(Xα,β,γ) is in
the image of some (h1, h2, h3) ∈ H1(∂Xα,β,γ) ∼= H1(Yα,β) ⊕ H1(Yβ,γ) ⊕ H1(Yα,γ).
33
Then, by utilizing the H1/Tors-action on ∂Xα,β,γ, define a map
Λ∗H1(Xα,β,γ;Z)/Tors⊗ CF−(Y −α,β, sα,β)⊗ CF (Yβ,γ, sβ,γ) → CF−(Yα,γ, sα,γ),
by
Fα,β,γ(h⊗ x⊗ y) = Fα,β,γ((h1 · x)⊗ y) + Fα,β,γ(x⊗ (h2 · y))− h3 · Fα,β,γ(x⊗ y).
This action induces a map on homology. By decomposing W = W1 ∪ W2 ∪ W3
into the 1-, 2-, and 3-handle attachment cobordisms, the extended cobordism map
is defined to be
FW,s(h⊗ x) = FW3,s ◦ FW2,s(h⊗ FW1,s(x)),
This map satisfies a version of the usual SpinC-composition law:
Proposition 2.3.5. [OS06, Proposition 4.20] If W = W1 ∪ W2, and ξ1 ∈
Λ∗H1(W
∗
1;Z)/Tors and ξ2 ∈ Λ H1(W2;Z)/Tors, then
∑
FW2,s2(ξ2 ⊗ FW1,s1(ξ1 ⊗ ·)) = FW,s((ξ3 ⊗ ·),
s∈SpinC(W ),
s|W =si i
where ξ ∗3 ∈ Λ H1(W ;Z)/Tors is the image of ξ1 ⊗ ξ2 under the natural map.
There is an H1(Y ;Z)/Tors-action on multi-pointed Heegaard diagrams as well
[Zem15, Equation 5.2]
Aγ : CFL−(Y,L, s) → CFL−(Y,L, s),
34
defined ∑ ∑
A (x) = a(γ, ϕ)#M̂(ϕ)Unw(ϕ)V nz(ϕ)γ y.
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1
Using this action, we can define extended link cobordism maps
FW,F ,s : (Λ
∗H1(W )/Tors⊗ F )⊗ CFL−2 (Y −0,L0, s|Y0) → CFL (Y1,L1, s|Y1).
in exactly the same way. Note, that since the link Floer TQFT is defined with
coefficients in F2, we need to tensor the exterior algebra generated by H1(Y )/Tors
with F2.
We will make use of an algebraic variant of CFK−, which is usually denoted
CFK−. Define CFK−(Y,K, s) be the F2[U ]-module obtained from CFK−(Y,K, s) by
setting V = 0 with differential
∑ ∑
∂(x) = #M̂(ϕ)Unw(ϕ)y.
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1,
nz(ϕ)=0
Let HFK−(Y,K, s) be the homology of this complex.
Secondary invariants of a surface
We now recall the definition of the secondary invariant τ of Juhász and Zemke
[JZ21]. Let Σ be a surface in B4 with boundary K = (K,w, z), a doubly based
knot. Decorate Σ by a single arc such that the z-subregion Σz ⊂ S is a bigon. Let
F = (Σ,A) be the resulting decorated surface. Then, there is an induced map
F −B4,F : F[U, V ] → CFK (K,w, z).
35
FIGURE 10 A surface Σ with Σz-sub region a bigon.
Definition 2.3.6. [JZ21, Definition 4.4] Let (K,w, z) be a doubly based knot in S3
and let Σ and Σ′ be two disks in B4 with boundary K, decorated as above. Then,
define
τ(Σ,Σ′) = min{n : Un · [FB4,F(1)] = Un · [F −B4,F ′(1)] ∈ HFK (K)}.
Remark 2.3.7. This invariant, τ (and its relatives ν, Vk,Υ), are called “secondary
invariants” because they are defined as surface analogues of existing knot
invariants.
A key result of [JZ21] states this invariant provides a lower bound for the
stabilization distance.
Theorem 9. [JZ21, Theorem 1.1] Let K be in a knot in S3, and let Σ,Σ′ be disks
in B4 with boundary K. Then,
τ(Σ,Σ′) ≤ µ(Σ,Σ′),
where µ(Σ,Σ′) is the stabilization distance between S and S ′.
36
Bordered Floer Homology
Before defining the invariants of bordered three-manifolds, we provide a brief
review of the relevant algebraic structures.
Fix a ground ring k of characteristic two. An A∞ algebra A over k is a
graded k -module A equipped with k -linear maps
µ : A⊗ii → A[2− i],
for i ≥ 1 satisfying
∑ n∑−j+1
µi(a1 ⊗ . . .⊗ aℓ−1 ⊗ µj(aℓ ⊗ . . .⊗ aℓ+j−1)⊗ aℓ+j ⊗ . . .⊗ an) = 0,
i+j=n+1 ℓ=1
for n ≥ 1. An A∞ algebra is strictly unital if there is an element 1 ∈ A such that
µ2(a, 1) = µ2(1, a) = a and µi(a1, . . . , ai) = 0 if i ̸= 2 and aj = 1 for some j.
A right A∞ module M over A is a graded k -module M equipped with
operations
mi : M ⊗ A⊗(i−1) → M [2− i],
satisfying ∑
mi(mj(x⊗ a1 ⊗ . . .⊗ aj−1)⊗ . . .⊗ an−1)
i+j=n+1
∑ ∑n−j
+ µi(a, a1 ⊗ . . .⊗ aℓ−1 ⊗ µj(aℓ ⊗ . . .⊗ aℓ+j−1)⊗ . . .⊗ an−1) = 0.
i+j=n+1 ℓ=1
We say M is strictly unital if for any x ∈ M , m2(x, 1) = x and mi(x ⊗ a1 ⊗ . . . ⊗
ai−1) = 0 if i > 2 and some aj = 1.
37
Let A be a dg algebra and let N be a graded k -module with a map
δ1 : N → (A⊗N)[1],
such that
(µ2 ⊗ I ) ◦ (I ⊗ δ1) ◦ δ1N A + (µ1 ⊗ IN) ◦ δ1 = 0.
The pair (N, δ1) is called a tyep D structure over A. A type D structure
homomorphism is a k -module map f 1 : N1 → A⊗N2 satisfying
(µ2 ⊗ IN2) ◦ (IA ⊗ f 1) ◦ δ2N + (µ2 ⊗ IN2) ◦ (IA ⊗ δ1 ) ◦ f 1N + (µ1 ⊗ IN2) ◦ f 1 = 0.1 2
The map δ1 can be iterated to define maps
δk : N → (A⊗k ⊗N)[k],
where δ0 = IN and δi = (I 1 i−1A⊗(i−1) ⊗ δ ) ◦ δ . Similarly, a type D homomorphism f 1
can be used to define maps
∑
fk : N1 → (A⊗k ⊗N2)[k − 1], fk(x) = (I j 1 iA⊗(i−1) ⊗ δN ) ◦ (I ⊗i ⊗ f ) ◦ δ .2 A N2
i+j=k−1
Given an A∞ module M over A and a type D structure (N, δ1), we define the
k -module M⊠N = M ⊗k N , equipped with differential
∑∞
∂⊠(x⊗ y) = (m kk+1 ⊗ IN)(x⊗ δ (y)).
k=0
38
We can represent the differential on the box tensor product graphically as
δ
∂⊠ = .
m
Given a map of type D structure f 1 : N1 → N2, we define a map IM ⊠ f 1 : M ⊠
N1 → M⊠N2 by
∑∞
(I ⊠ f 1M )(x⊗ y) = (mk+1 ⊗ IN2)(x⊗ fk(y)).
k=1
Graphically, this map is represented
δN1
f 1
IM ⊠ f 1 = .
δN2
m
Bordered Floer homology is a package of invariants of 3-manifolds with
parametrized boundary. Bordered Floer homology associates to a surface F a
differential graded algebra A(F ) and to a 3-manifold Y with boundary together
with an identification φ : ∂Y → F a left differential graded module over
39
A(−F ), the type D module of Y , denoted ĈFD(Y ) and a right A∞-module over
A(F ), the type A module of Y , denoted ĈFA(Y ). Much like classical Heegaard
Floer homology, the bordered Floer invariants are defined by representing a 3-
manifold with parametrized boundary by a kind of Heegaard diagram and counting
holomorphic disks which, in the bordered case, may asymptotically approach the
boundary.
Bordered Floer homology has a pairing theorem [LOT18, Theorem 1.3], which
recovers the hat-version of the Heegaard Floer homology of the manifold obtained
by gluing bordered manifolds along their common boundary. Given 3-manifolds Y1
and Y2 with ∂Y ∼ ∼1 = F = ∂Y2, there is a homotopy equivalence
ĈF (Y1 ∪ Y2) ≃ ĈFA(Y1)⊠A(F ) ĈFD(Y2).
Bordered Floer theory also recovers knot Floer homology [LOT18, Theorem
11.21]. Given a doubly pointed bordered Heegaard diagram (H1, w, z) for
(Y1, ∂F,K) and a bordered Heegaard diagram (H2, z) with ∂Y ∼1 = F ∼= −∂Y2,
then
HFK−(Y1 ∪ Y2, K) ∼= H∗(CFA−(H1, w, z)⊠A(F ) ĈFD(H2, z)).
Bordered Floer theory, therefore, gives an effective way to study satellites. Let KP
be the satellite of K with pattern P . ĈFD(S3 − K) is determined by CFK−(K)
[LOT18, Chapter 11], HFK−(KP ) can be computed by finding a doubly pointed
bordered Heegaard diagram HP for the pattern P in the solid torus and computing
the box tensor product CFA−(HP )⊠ ĈFD(S3 −K).
The last important result from bordered Floer theory which we will utilize, is
the morphism spaces pairing theorem, which gives a means of recovering classical
40
Heegaard Floer homology in terms of the Hom functor rather than the tensor
product functor [LOT11]. If Y1 and Y2 are 3-manifolds with ∂Y ∼1 = F ∼= ∂Y2, then
ĈF (−Y1 ∪ Y2) ≃ MorA(−F )(ĈFD(Y1), ĈFD(Y2)),
where the latter object is the chain complex of A(−F )-linear maps from ĈFD(Y1)
to ĈFD(Y2) equipped with differential
d(φ) = ∂ ◦ φ+ φ ◦ ∂ .
ĈFD(Y2) ĈFD(Y1)
Having concluded the requisite background material, we proceed towards the
proofs of Theorems 3 and 4.
41
CHAPTER III
RIBBON HOMOLOGY CONCORDANCES
First Homology Action as Link Cobordism Maps
It is helpful in geometric arguments that the H1/Tors-action can be realized
as a graph cobordism map on CF−. In the same way, it is beneficial to realize the
H1(Y )/Tors-action on CFL− as a link cobordism map. By Theorem 8, Σw and Σz
should be ribbon 1-skeleta of the graph in Figure 8. Our strategy will be to try to
compute the decorated link cobordism map obtained by tubing on a torus whose
longitude is a curve which represents the class in H1(Y )/Tors.
Construct a closed decorated link cobordism F inside of S2 × S1 × [−1, 1] as
follows. Let Σ be the boundary of a tubular neighborhood of {pt} × S1 × {0} ⊂
S2 × S1 × {0}. Let the dividing circles A be two parallel closed curves on the
boundary of Σ obtained by isotoping {pt} × S1 × {0} radially, so that A divides
Σ into Σw and Σz which are both annuli. As in the text preceding Proposition
2.3.3, choose disks D1 and D2 which intersect the same dividing arc {pt} × S1, and
take (W,F0) to be the link cobordism obtained by isotoping F and removing the
disks D1 and D2. See Figure 11. Note that the graph shown in Figure 8 is a ribbon
1-skeleton for both Σw and Σz.
Lemma 3.1.1. For (S2 × S1 × [−1, 1],F0) the link cobordism described above
F + −S2×S1×[−1,1],F ,s (θ ) = θ and F
−
0 0 S2×S1×[−1,1],F ,s (θ ) = 0,0 0
where θ+ (θ−) is the generator of CFL−(S2 × S1,U , t0) of higher (lower) grading, s0
is the torsion SpinC-structure on S2 × S1 × [−1, 1] and t0 = s0|S2×S1.
42
FIGURE 11 A schematic of the decorated surface in S2 × S1 × {0} described in the
text preceding Lemma 3.1.1.
Proof. By Proposition 2.3.3, FS2×S1×[−1,1],F ,s is determined by its reduction to a0 0
graph cobordism map: concretely, if FBS2×S1×[−1,1],Γ ,s is the corresponding graphw
map, then
F ∆A B F2[U,V ]S2×S1×[−1,1],F ,s ≃ V · F0 0 S2×S1×[−1,1],Γ ,s |w 0
under the identification of CFL−(Y,U, s0) with CF−(Y,w, s0) ⊗F2[W ] F2[U, V ] as
before. The quantity ∆A, given by
⟨c1(s0), [Σ]⟩ − [Σ] · [Σ] χ(Σw)− χ(Σz)
+ ,
2 2
vanishes, since [Σ] is nullhomologous in S1 × S2, and Σw and Σz are both cylinders.
By construction, FBS2×S1×[−1,1],Γ ,s is the graph cobordism shown in Figure 8.w 0
Hence, FBS2×S1×[−1,1],Γ ,s is just the map Aγ. It is straightforward to verify thatw 0
the action of [{pt} × S1] ∈ H (S2 × S11 ) takes θ+ to θ− and θ− to zero.
We now turn to the case of a nullhomologous knot K embedded in an
arbitrary 3-manifold Y. Let (Y × [−1, 1],FY ) be Morse-trivial, in the sense defined
in Section ??. The idea is to modify FY by by tubing on a torus with a dividing
43
FIGURE 12 A schematic of the decorated surface Fγ. In general, K × {0} and γ
might be linked.
arc which represents a class in H1(Y ;Z)/Tors. Let γ ⊂ Y ×{0}− (K×{0}) be such
a curve. Choose a path λ in Y × {0} connecting γ to a point p in K × {0} which
lies on one of the dividing arcs.
Let T be the boundary of a tubular neighborhood of γ in Y × {0}. Decorate
T with two parallel circles isotopic to γ. Tube T and Σ together along λ. Denote
the resulting surface Σγ. Decorate the tube with two parallel arcs, connected to one
of the circles parallel to γ on one end, and to one of the dividing arcs on FY on the
other end. A schematic of this decoration, which is denoted Aγ, is shown in Figure
12.
Lemma 3.1.2. For the decorated surfaces (Y × [−1, 1],FY ) and (Y × [−1, 1],Fγ)
described above, we have that
FY×[−1,1],Fγ ,s(·) ≃ FY×[−1,1],F ,s([γ]⊗ ·).Y
Proof. Decompose (Y × [−1, 1],Fγ) as (X,FX) ◦ (Nλ(γ),F) where X = ((Y ×
[−1, 1]) − Nλ(γ)), F = Fγ ∩ Nλ(γ), and FX = Fγ ∩ X. F is a punctured torus
whose decoration is shown in Figure 13. Given a SpinC-structure s on Y × [−1, 1],
its restriction to X is torsion on ∂Nλ(γ), which can be extended by s0, the unique
44
SpinC-structure on Nλ(γ). By considering another Mayer-Vietoris sequence, it is
not hard to see that this extension is unique. The composition law then implies
that
FY×[−1,1],Fγ ,s ≃ FX,FX ,s| ◦ FX Nλ(γ),F ,s0 .
The map FN (γ),F ,s0 associated to the cobordism (Y,K) → (Y ⨿S2×S1, K⨿U)λ
can be computed as the composition of a 0-handle/birth map, followed by a 1-
handle map, followed by the map FS2×S1×[−1,1],F ,s computed above: given an0 0
element x ∈ CFL−(Y,L, s), the 0-handle/birth map simply introduces a pair
of intersection points c+, c− on a genus 0 Heegaard diagram for S3, and takes
x 7→ x ⊗ c+. Attaching a 1-handle, with both feet attached to the new 0-handle
corresponds to the map x ⊗ c+ 7→ x ⊗ θ+ [Zem18, Section 5]. By Lemma 3.1.1,
F + −S2×S1×[−1,1],F ,s (x⊗ θ ) = x⊗ θ . All together then,0 0
F −N (γ),F ,s0(x) = x⊗ θ .λ
On the other hand, consider the cobordism FN (γ),D,s0 where D is a disk decoratedλ
with a single arc, followed by the action of [γ]: this can be computed as the
composition of the 0-handle/birth and 1-handle maps followed by the action of
[γ]. Just as before, the 0-handle/birth and 1-handle maps take an intersection point
x to x⊗ θ+ and the action of [γ] takes this element to x× θ−. Hence,
FN (γ),F ,s0(x) = [γ] · Fλ Nλ(γ),D,s0(x).
See Figure 13 for a comparison of these two cobordism maps.
45
FIGURE 13 When the link cobordism map on the right is followed by the action of
γ, it becomes equivalent to the map on the left.
By definition of the extended link cobordism maps, [γ] · FN (γ),D,s0(x) =λ
FN (γ),D,s0([γ] ⊗ x). Combining these observations with the composition law forλ
the extended cobordism maps shows
FY×[−1,1],Fγ ,s(x) ≃ FX,F ,s| ◦ FN (γ),F ,s0(x)X X λ
≃ FX,F ,s| ◦ FN (γ),D,s0([γ]⊗ x)X X λ
≃ FX,F ,s (1⊗ FX X Nλ(γ),D,s0([γ]⊗ x))
≃ FY×[−1,1],F ,s([γ]⊗ x)Y
as desired.
Note that the choice of path λ did not matter, since the diffeomorphism type
of the neighborhood Nλ(γ) did not depend on λ.
Let (W,F) : (Y0, K0) → (Y1, K1) be a link cobordism which is concordance
Morse-trivial. Decompose W as a composition of handle attachments W3 ◦W2 ◦W1.
Let γ be a curve in W which represents an element of H1(W )/Tors. Homotope the
curve γ so it is contained in the boundary of W1, which is denoted Ỹ = ∂+W1. In
46
this dimension and codimension, such a homotopy can be taken to be an isotopy.
Let F̃γ be the decorated surface in Ỹ × [−1, 1] described above which realizes
the action of [γ]. Let (W,Fγ) be the link cobordism (W3,F3) ◦ (W2,F2) ◦ (Ỹ ×
[−1, 1], F̃γ) ◦ (W1,F1), where Fi is F ∩Wi.
Lemma 3.1.3. Let (W,Fγ) be the decorated surface described above. Then,
FW,F ,s([γ]⊗ ·) ≃ FW,Fγ ,s(·).
Proof. We will make use of the decomposition of (W,Fγ) as
(W3,F3) ◦ (W2,F2) ◦ (Ỹ × [−1, 1], F̃γ) ◦ (W1,F1).
FW,Fγ can be computed as the composition:
FW,Fγ ,s(x) ≃ FW3,F3,s| ◦ FW W2,F2,s| ◦ F3 W2 Ỹ×[−1,1],F ,s| ◦ FW1,F1,s| (x).γ Ỹ ×[−1,1] W1
By Lemma 3.1.2, this map is chain homotopic to
FW3,F3,s| ◦ FW2,F2,s| ◦ FỸ×[−1,1],F ,s| ([γ]⊗ FW W W1,F1,s| (x)).3 2 Ỹ Ỹ ×[−1,1] W1
Since (Ỹ × [−1, 1],FỸ ) is the identity,
FỸ×[−1,1],F ,s| ([γ]⊗ FW1,F1,s| (x)) = [γ] · FW W1,F× − 1 1,s| (x) + 0,Ỹ Ỹ [ 1,1] W1
and hence this map can be rewritten as:
FW3,F3,s| ◦ FW W2,F2,s| ([γ] · FW W1,F1,s| (x)).3 2 W1
47
But, since γ was chosen as to lie in Ỹ , this is by definition equal to the map
FW,F ,s([γ]⊗ x).
Maps induced by ribbon homology concordances
As is typical in proving results of this kind, our strategy will be to compare
the double of a ribbon Z-homology concordance to a Morse-trivial link cobordism.
Recall that the double of a link cobordism (W,Σ) : (Y0, K0) → (Y1, K1) is the
link cobordism (D(W ), D(Σ)) = (W,Σ) ∪(Y1,K1) (W,Σ), where (W,Σ) is the link
cobordism obtained by turning (W,Σ) around and reversing the orientation.
It will be helpful to have a version of the “sphere tubing” property of the
link cobordism maps [Zem19b, Lemma 3.1], [MZ19, Lemma 4.2]; if (W,F) is a link
cobordism in a homology cobordism and S is a null-homologous 2-sphere embedded
in the complement of the link cobordism, S can be tubed to the embedded surface
without changing the induced map.
Proposition 3.2.1. Let F = (Σ,A) be a decorated link cobordism in a homology
cobordism W , and let S ⊂ W be a smoothly embedded, nullhomologous sphere
disjoint from F . Let F ′ be a decorated cobordism obtained by connecting Σ and S
by a tube whose feet are disjoint from A. Then,
FW,F ,s ≃ FW,F ′,s.
Proof. Factor FW,F ,s and FW,F ′,s through a regular neighborhood N(S) of S. Since
S is nullhomologous, N(S) can be identified with D2 × S2. F ′ intersects N(S) in
a disk D′ and ∂N(S) in an unknot. We can perturb F so it meets N(S) in a disk
D and ∂N(S) in an unknot as well. Let D and D′ be the disks D and D′ decorated
48
with a single dividing arc. Since S is nullhomologous, the restriction of a given
s ∈ SpinC(W ) to N(S) will be torsion. By [Zem19b, Lemma 3.1], FN(S),D,s| doesN(S)
not depend on the choice of embedded disk, and so FN(S),D,s| ≃ FN(S)) N(S),D′,s| .N(S)
Moreover, since W is a homology cobordism, the map H2(W ) → H2(∂−W )
is an isomorphism, and, in particular, an injection. It follows from the following
diagram that the map H2(W ) → H2(W −N(S)) is injective as well.
H2
∼
(W ) = H2(∂−W )
H2(W −N(S))
Therefore, a given SpinC-structure on W − N(S) will extend over N(S),
and moreover will extend uniquely. By the composition law for the link cobordism
maps,
FW,F ,s ≃ FW−N(S),F−N(S),s| − ◦ FW N(S) N(S),D,s|N(S))
≃ FW−N(S),F−N(S),s|W− ◦ FN(S) N(S),D′,s|N(S))
≃ FW,F ′,s,
as desired.
We will prove Theorem 3 in two steps: we will prove the theorem for link
cobordisms (W,Σ) which are concordance Morse-trivial and then argue that we can
always reduce to this case.
The first step will follow from Proposition 2.3.4. Let (W,F) : (Y0, K0) →
(Y1, K1) be a ribbon Z-homology concordance which is concordance Morse-trivial
and decorated by a pair of parallel arcs. Decompose W = W1 ∪ W2 into the 1-
49
and 2-handle cobordisms. A key observation of [DLVVW19] is that the product
cobordism Y0 × [−1, 1] and the double of W can be obtained by two different
surgeries on the same intermediate manifold X = D(W1). X can be described
explicitly as (Y0 × [−1, 1])#n(S1 × S3). It is not hard to see that surgery on S1 × S3
along S1 × {pt} yields S4, and so that Y0 × [−1, 1] can be obtained from surgery
on X is straightforward. That D(W ) can also be obtained by surgery on X follows
from the following lemma.
Lemma 3.2.2. [DLVVW19, Proposition 5.1] Let W : Y0 → Y1 be a cobordism
corresponding to attaching 2-handles along curves γ1, . . . , γn ⊂ Y0. Then, the double
of W can be obtained from Y0 × [−1, 1] by doing surgery on γ1, . . . , γn ⊂ Y0 × {0}.
To apply Proposition 2.3.4, there is a homological condition on the collection
of surgery curves α1, . . . , αn, which must be satisfied, namely the restriction map
H1(X −⨿N(αi)) → H1(⨿∂N(αi)) must be surjective.
Lemma 3.2.3. If α1, . . . , αn ⊂ X are either the attaching curves of the 2-handles
of W or the core curves S1 × {pt} of the S1 × S3 summands, which we denote
γ1, ..., γn and η1, ..., ηn respectively, the restriction map H
1(X − ⨿N(αi)) →
H1(⨿∂N(αi)) is surjective.
Proof. Let α1, . . . , αn be either set of curves. Inclusions induce the following
commutative diagram:
H1(X) H1(⨿∂N(αi))
H1(W −⨿N(αi))
The map H1(X − ⨿N(αi)) → H1(⨿∂N(αi)) is surjective if the map
H1(X) → H1(⨿∂N(αi)) is. The curve ηi runs over the ith 1-handle geometrically
50
once, and since W is a Z-homology cobordism, after some handle slides, we can
arrange that the curve γi runs over the ith 1-handle algebraically once. In either
case, this implies that the composition
H1(#n(S1 × S3)) → H1(X) → H1(⨿∂N(αi)),
is an isomorphism. Therefore, the map H1(X) → H1(⨿∂N(αi)) is surjective as
desired.
We can now establish the theorem for the case where (W,Σ) is concordance
Morse-trivial.
Proposition 3.2.4. Suppose (W,F) : (Y0, K0) → (Y1, K1) is a ribbon Z-
homology concordance which is concordance Morse-trivial where F = (C,A) is
the concordance decorated by a pair of parallel arcs. Then, every s ∈ SpinC(W )
has a unique extension D(s) ∈ SpinC(D(W )) and the map induced by the double of
(W,F)
F : HFL−D(W ),D(F),D(s) (Y0, K0, s|Y0) → HFL−(Y0, K0, s|Y0)
is the identity.
Proof. Decompose W as W1 ∪ W2 where Wi is the cobordism corresponding to the
attachment of the i-handles, i = 1, 2. Let Ỹ = ∂+W1 which can be identified with
Y0#
n(S1 × S2), where n is the number of 1-handles (and since W is a Z-homology
cobordism, n is also the number of 2-handles). Let X = W1 ∪ W1 be the double
of W1, which is diffeomorphic to (Y0 × [−1, 1])#n(S1 × S3). Define a decorated
surface Fα = (Σα,Aα) in X as follows: Let α1, . . . , αn be a collection of curves in
51
X. Isotope each αi so that it is embedded in Ỹ . In Section 3.1, we constructed a
decorated surface Fα ⊂ Ỹ × [−1, 1] which realized the action of αi. Define Fα toi
be the decorated surface obtained stacking these surfaces on top of one another, i.e.
Fα = Fαn ∪· · ·∪Fα1 . Let (X,FX) = (W1,F ∩W1)∪(Ỹ × [−1, 1],Fα)∪(W1,F∩W1).
Let ηi be the curve S
1 × {pt} in the ith S1 × S3 summand of X. By taking
αi to be ηi, we obtain a decorated surface Fη in X, with the curves η1, . . . , ηn ⊂ Aη.
Apply Proposition 2.3.4 to see that
FX,Fη ,s ≃ FX(η1,...,ηn),Fη(η1,...,ηn),s(η1,...,ηn).
Doing surgery on X along the curves η1, . . . , ηn yields Y0 × [−1, 1]#nS4 which is,
of course, diffeomorphic to the product Y0 × [−1, 1]. Recall that Fη was defined
by tubing on tori decorated by parallel copies of ηi. Doing surgery on the curves
ηi in these tori yields spheres embedded in the S
4 summands (and are therefore
nullhomologous.) Therefore, (Y0 × [−1, 1],Fη(η1, . . . , ηn)) can be built from the
Morse-trivial link cobordism (Y0 × [−1, 1],FY0×[−1,1]) by tubing on a collection of
spheres. By Proposition 3.2.1, the link cobordism map does not detect tubing on
nullhomologous spheres, and hence
FX(η1,...,ηn),Fη(η1,...,ηn),s(η1,...,ηn) ≃ FY0×[−1,1],FY ×[−1,1],s(η1,...,ηn).0
By Lemma 3.2.2, D(W ) can be obtained from X by doing surgery on the
attaching curves γ1, . . . , γn of the 2-handles of W . Now take the αi to be the
attaching curves γ1, ..., γn for the 2-handles for W , and consider the decorated
link cobordism Fγ in X which realizes the action of γ1, . . . , γn. Just as above,
52
Proposition 2.3.4 shows:
FX,Fγ ,s ≃ FX(γ1,...,γn),Fγ(γ1,...,γn),s(γ1,...,γn) ≃ FD(W ),Fγ(γ1,...,γn),s(γ1,...,γn).
Again, the surface Fγ(γ1, . . . , γn) is obtained by tubing on the spheres that arise by
doing surgery on the tori in Fγ. Here is a geometric argument that these spheres
are nullhomologous in D(W ). The torus Ti corresponding to γi was defined by
isotoping γi into ∂+W1 and taking the boundary of a regular neighborhood of γi
in ∂+W1. Hence, [Ti] = 0 ∈ H2(X). By attaching a thickened disk along a meridian
of γi, we obtain cobordism from Ti to the sphere Si obtained by surgery on γ.
Therefore, [Si] = [Ti] = 0 in H2(X). Since Si is disjoint from γi it also represents
a class in H2(X − N(γi)). X is obtained from X − N(γi) by attaching a 3- and
4-handle, so the relative homology group H1(X,X − N(γi)) = 0, implying the map
induced by inclusion H2(X −N(γi)) → H2(X) is injective. In particular, this means
[Si] is trivial in H2(X −N(γi)), and therefore trivial in H2(X(γi)) as well. Applying
this argument to each γi shows that all spheres Si are nullhomologous in D(W ).
Therefore, we can again apply Proposition 3.2.1 to see that
FD(W ),Fγ(γ1,...,γn),s(γ1,...,γn) ≃ FD(W ),D(F),s(γ1,...,γn),
where D(F) is the decorated cobordism obtained by doubling F . Since (W,F) was
concordance Morse-trivial, (D(W ), D(Σ)) is as well.
Altogether, we have shown that the maps induced by (Y0 × [−1, 1],FY0×[−1,1])
and (D(W ), D(F)) are chain homotopic to the maps induced by the link
53
cobordisms (X,Fη) and (X,Fγ) respectively. Hence, we simply need to show that
FX,Fη ,s ≃ FX,Fγ ,s.
By Lemma 3.1.3, we have that
FX,Fη ,s(·) ≃ FX,F ,s([η1] ∧ · · · ∧ [ηn]⊗ ·)X
and
FX,Fγ ,s(·) ≃ FX,FX ,s([γ1] ∧ · · · ∧ [γn]⊗ ·),
where (X,FX) is concordance Morse-trivial. We will show that
FX,F ,s([η1] ∧ · · · ∧ [ηn]⊗ ·) ≃ FX X,F ,s([γ1] ∧ · · · ∧ [γn]⊗ ·),X
This is effectively proven in [DLVVW19, Theorem 4.10], but we will recall
their argument for the convenience of the reader. By [DLVVW19, Proposition 5.1],
[η1] ∧ · · · ∧ [ηn] = [γ1] ∧ · · · ∧ [γn] ∈ (Λ∗H1(X)/Tors/⟨H1(Y0)/Tors⟩)⊗ F2,
where ⟨H1(Y0)/Tors⟩ is the ideal generated by elements of H1(Y0)/Tors. Therefore,
[η1] ∧ · · · ∧ [ηn] and [γ1] ∧ · · · ∧ [γn] differ by an element of Λn(H1(X)/Tors) ∩
⟨H1(Y0)/Tors⟩ ⊗ F2. By the linearity of the link cobordism maps, it suffices to show
that FX,F ,s(x⊗ ξ) = 0 for any ξ ∈ Λn(H1(X)/Tors) ∩ ⟨H1(Y0)/Tors⟩ ⊗ F2.X
(∧Λn(H1(X) )/Tors) ∩ ⟨H1(Y0)/Tors⟩ ⊗ F2 is generated by elements of the form
ω ∧ i∈S[ηi] where ω is a wedge of elements in H1(Y0)/Tors and S is a proper
subset of {1, ..., n}. So, we can take ξ to be of this form. Since S is a proper subset
54
of {1, ..., n} there is some ηj which does not appear as a factor of ξ. We can realize
the map FX,F ,s(ξ ⊗ ·) as a link cobordism map FX,F ,s. By our construction of FX ξ ξ,
we can assume that Fξ is disjoint from the jth S1 × S3 summand, of which ηj is the
core. Therefore, (X,Fξ) can be decomposed as (X − Tj,Fξ − D) ∪ (Tj,D), where
T = (S1 × S3j )−D4 is the jth S1 × S3 summand, and D is a disk decorated with a
single dividing arc. Hence, the map FX,F ,s factors through Fξ Tj ,D,s| .Tj
But, the map FT ,D,s| is trivial. This map can be computed as the followingj Tj
composition: first, a 0-handle with a birth disk (which is identified with D) is
born, then a 1-handle is attached with both feet on the 0-handle, with a trivially
embedded annulus followed by 3-handle, again with a trivial annulus. The 1- and
3-handle maps take x to x⊗ θ+ and x⊗ θ+ to zero respectively, where θ+ is the top
graded generator of CFL−(S1 × S2,U , t0).
Therefore, it has been established that
FY0×[−1,1],F × − ≃ FY [ 1,1],s(η1,...,ηn) D(W ),D(F),s(γ0 1,...,γn).
The map on the left is induced by a Morse-trivial link cobordism, and therefore
induces the identity map on HFL−(Y0, K0, t0), where t0 is the restriction of
s(η1, . . . , ηn) to Y0. Spin
C-structures on Y0 extend uniquely over W and D(W )
since they are Z-homology cobordisms. Therefore s(γ1, ..., γn) is the unique SpinC
structure extending s over D(W ). Therefore, for any s ∈ SpinC(W ), the map
FD(W ),D(F),D(s) induces the identity.
Having completed the proof in the case that (W,Σ) is concordance Morse-
trivial we show that we can always reduce to this case. Given any concordance
C ⊂ W , there is a Morse function f on W with the property that f |C has no
55
critical points (for instance, simply identify C ∼= S1 × I and extend the projection
S1 × I → I to a Morse function on W .) However, it is not immediately clear
that the extension can be chosen in such a way as to give W the structure of a
ribbon cobordism. The following proposition shows that birth-saddle pairs of a
concordance can be traded for four-dimensional 1- and 2-handle pairs.
Proposition 3.2.5. Let K0 and K1 be ribbon concordant in a homology cobordism
W . Then, there is handle decomposition of the pair (W,Σ) with the property that
W is ribbon and the handle decomposition for Σ is trivial.
Proof. Choose a Morse function on (W,Σ) giving rise to a banded link diagram.
This decomposition of Σ consists of a collection of birth circles U1, ..., Un and bands
B1, ..., Bn, and after some band slides, we can assume the band Bi has one foot on
Ui and the other on K0.
Consider a birth circle Ui. If a band Bj runs through Ui (for j ≠ i), it can
be swum through the band Bi, so we can assume that the only band which runs
through Ui is Bi. Introduce a canceling 1- and 2-handle pair near Ui. Slide all
strands of Bi which run through Ui over the newly introduced 2-handle. Each
strand of Bi which ran through Ui now also runs through the 1-handle we have
added. Now, contract the band Bi until it no longer runs through Ui. In doing so,
we drag along an arc of K0 through Ui, possibly many times. Slide Ui under the
1-handle, unlinking it from K0. The band Bi can be swum past the 2-handle, at
which point, Ui and Bi can be cancelled. See Figure 14. Repeat for all Ui and Bi.
This process has yielded some new ribbon Z-homology concordance (W,Σ) :
(Y0, K
′ ′
0) → (Y1, K1) which is concordance Morse-trivial. It remains to show that
this process has not altered the original knots. By erasing the new 1- and 2-handle
pairs, we obtain the knot K ′0 ⊂ Y0, which is clearly isotopic to K0. On the other
56
hand, if we do surgery on the whole diagram, we obtain a knot K ′1 in a surgery
diagram for Y . To see K ′1 1 is isotopic to K1, we can slide an arc of K
′
1 near where
we canceled the birth circle and band over the black 0-framed surgery curve. Any
arcs which pass through the black 0-framed surgery curve can now be slid over
the red 0-framed surgery curve. Iterating this process, we obtain a knot which
is isotopic to the result of band surgery on the bands B1, ..., Bn in the original
diagram, which is exactly K1.
FIGURE 14 A procedure for trading handles of a concordance for handles of the
ambient manifold.
Proof of Theorem 1. This now follows immediately from the previous propositions.
For any ribbon Z-homology concordance (W,F), the doubled link cobordism
induces the same map as a concordance Morse-trivial link cobordism (D(W ),F ′),
and this map induces the identity map on HFL−(Y0, K0, s|Y0). As s extends
uniquely over D(W ), the composition law implies that FW,F ,s has a left inverse,
namely FW,F ,s. Hence, FW,F ,s is a split injection.
57
Torsion and Link Floer Homology
Let CFL−(Y,K, s) be the F2[V ]-module obtained from CFL−(Y,K, s) by
setting U = 0 with differential
∑ ∑
∂(x) = #M̂(ϕ)V nz(ϕ)y.
y∈Tα∩Tβ ϕ∈π2(x,y),
µ(ϕ)=1,
nw(ϕ)=0
Let HFL−(Y,K, s) be the homology of this complex.
A key property of the link Floer TQFT which is utilized in [JMZ20] is the
following.
Lemma 3.3.1. [JMZ20, Lemma 3.1] Let (W,F) be a decorated link cobordism. Let
FV be a link cobordism obtained by adding a tube to the Σz region. Then,
FW,FV ,s ≃ V · FW,F ,s.
Proof. Choose a neighborhood the tube diffeomorphic to the 4-ball. By the
composition law, it suffices to show that
FB4,F ∩B4 ≃ V · FB4,F ,V
where F is a pair of disks which bound a 2-component unlink in ∂B4 decorated by
a dividing arc. FV ∩ B4 is obtained from FB4,F by adding a tube connecting the
two disk with feet in the z-region. This map can be computed as the composition
of two z-band maps. This computation is carried out in [Zem18, Section 8.2], and
the resulting map is multiplication by V .
58
With this tool at our disposal, we can prove an analogue of [JMZ20,
Proposition 4.1].
Proposition 3.3.2. Let (W,Σ) : (Y0, K0) → (Y1, K1) be a Z-homology link
cobordism. Let h : W → R be a Morse function compatible with Σ with respect
to which W is ribbon. Suppose that h|Σ has m critical points of index 0, b critical
points of index 1, and M critical points of index 2. Let F be a decoration of Σ such
that Σw is a regular neighborhood of an arc from K0 to K1. Then,
V M · F b−mD(W ),D(F),D(s) ≃ V · idHFL−(Y0,K0,s| ) .Y0
Proof. The Morse function h induces a movie presentation for (W,Σ).
1. m birth disks appear disjoint from K0, with boundaries U1, . . . , Um.
2. n 4-dimensional 1-handles are attached whose feet are disjoint from Σ.
3. m fusion bands B1, ..., Bn are attached which connect K0 and U1, . . . , Um.
After some band slides, the band Bi has one foot in Ui and the other in K0.
4. b−m additional bands Bm+1, ..., Bb are attached.
5. n four-dimensional 2-handles are attached along curves γ1, . . . , γn.
6. M death disks appear capping off unknotted components U1, . . . , UM in the
link obtained by doing band surgery.
By playing this movie forward, and then again in reverse, we obtain a movie for
(D(W ), D(F)).
Consider the link cobordism which is obtained by deleting steps (5)-(8), i.e.
remove the 2-handles, the deaths, the dual births, and the dual 2-handles. The
59
resulting four-manifold X is the double of the cobordism W1 = (Y1 × [0, 1]) ∪
1-handles. The resulting surface is D(F ∩ W1). Let Gγ be the surface obtained
from D(F ∩ W1) by tubing on tori T1, ..., Tn which are the boundaries of regular
neighborhoods of the γi curves in ∂+W1 as in the proof of Proposition 3.2.4.
Surgery on the curves γ1, . . . , γn yields a link cobordism (D(W ),G). The decorated
cobordism G can be obtained from D(F) by attaching M tubes from the deaths
disks to the dual birth disks and tubing on n nullhomologous spheres which are the
result of surgery on the tori Ti. We can arrange for the feet of the tubes to be sit in
the subsurface Σz. Attaching the nullhomologous spheres has no effect on the link
cobordism map, and attaching the tubes has the result of multiplying by V M by
Lemma 3.3.1. Therefore, by Proposition 2.3.4,
F MX,Gγ ,t ≃ V · FD(W ),D(F),D(s),
where t is determined by the fact that t(γ1, . . . , γn) = D(s).
Now consider the cobordism obtained by deleting steps (4)-(9) and again
tubing on the tori corresponding to the γi curves. The ambient four-manifold is still
X, but removing steps (4) and (9) has the effect of removing the bands Bm+1, ..., Bb
and their duals from Gγ. Call this surface Hγ. Since the bands Bm+1, ..., Bb and
their duals form a collection of b − m tubes, another application of Lemma 3.3.1
shows
F b−mX,Gγ ,t ≃ V · FX,Hγ ,t.
60
But, surgery on (X,Hγ) along γ1, . . . , γn yields the link cobordism (D(W ),H)
which is the double of a ribbon homology concordance. So by Proposition 3.2.4
FX,Hγ ,t ≃ idCFL−(Y0,K0,s| ) .Y0
Altogether then, we have that
V M · F b−mD(W ),D(F),D(s) ≃ V · idHFL−(Y0,K0,s| ),Y0
as desired.
Proof of Theorem 4. This now follows by an argument identical to that of [JMZ20,
Theorem 1.2].
Unlike the inequality of [JMZ20, Theorem 1.2], this does not give the
symmetric result
OrdV (Y1, K1, s|Y1) ≤ max{m,OrdV (Y0, K0, s|Y0)}+ 2g(Σ),
since W is not ribbon as we have defined it, unless W = Y0 × [0, 1].
Applications
We do have some immediate applications. Theorem 4 gives a clear
relationship between the torsion orders of ribbon homology cobordant knots.
Corollary 3.4.1. Let (W,Σ, s) : (Y0, K0) → (Y1, K1) be a ribbon Z-homology
cobordism, then
OrdV (Y0, K0, s|Y0)−OrdV (Y1, K1, s|Y1) ≤ 2g(Σ).
61
Recall that the fusion number Fus(K) of a ribbon knot K in S3 is the
minimal number of bands in a handle decomposition of ribbon concordance C from
the unknot U to K in S3 × [0, 1]. By [JMZ20], the torsion order of K in S3 provides
a lower bound for the fusion number of K. There are a few possible generalizations
we will consider.
Let K be a knot in a 3-manifold Y . Suppose that that K is ribbon in Y , in
the sense that there is a concordance (Y × [0, 1], C) : (Y × {0}, U) → (Y × {1}, K)
where U is the boundary of a disk in Y and C is an annulus which is ribbon with
respect to the projection Y × [0, 1] → [0, 1].
Definition 3.4.2. We define the fusion number of K in Y , which we denote
FusY (K), to be the minimal number of bands in a ribbon concordance from U
to K in Y × [0, 1].
Corollary 3.4.3. If K is ribbon in Y , then
OrdV (Y,K, s) ≤ FusY (K).
Proof. Let (Y × [0, 1], C) : (Y, U) → (Y,K) be a ribbon concordance with b =
FusY (K) bands (and therefore b local minima as well). Theorem 2 then implies
OrdV (Y,K, s) ≤ max{b,OrdV (Y, U, s)} = b,
since HFL−(Y, U, s) is torsion free.
In another direction, one could also consider ribbon concordances in Z-
homology cobordisms. Given a Z-homology concordance (W,Σ) : (Y0, U) → (Y1, K),
one can always find a Morse function h on W compatible with Σ so that h|Σ is
62
ribbon, so we will continue to require that the ambient manifold is ribbon as well.
However, by imposing the condition that the ambient manifold is ribbon, we have
introduced an asymmetry which makes generalizing the fusion number to ribbon
Z-homology concordances somewhat subtle; in S3 × [0, 1], concordances from the
unknot with no local maxima can be turned around and viewed as concordances to
the unknot with no local minima. However, this is clearly not the case for a ribbon
homology concordance (W,Σ) : (Y0, U) → (Y1, K), as W is not ribbon.
Therefore, since ribbon homology link cobordisms to and from the unknot
differ, we can consider both cases: on the one hand, we have ribbon Z-homology
concordances (W,Σ) : (Y ′, U) → (Y,K) (where W is a ribbon Z-homology
cobordism and Σ is an annulus with no local maxima), and on the other, link
cobordisms (W,Σ) : (Y,K) → (Y ′, U) where W is a ribbon Z-homology cobordism
and Σ is an annulus with no local minima.
For the latter notion, Theorem ?? immediately implies the following.
Corollary 3.4.4. Let K be a knot in a 3-manifold Y . If (W,Σ) : (Y,K) → (Y ′, U)
is a link cobordism such that W is a ribbon Z-homology cobordism and Σ is an
annulus with no local minima and b index 1 critical points, then
OrdV (Y,K, s) ≤ b,
for any s ∈ SpinC(Y ).
Let Fus∧(Y,K) be minimal number of bands over all link cobordisms of the
form (W,Σ) : (Y,K) → (Y ′, U) where W is a ribbon Z-homology cobordism and
Σ is an annulus with no local minima. The previous result, of course, implies that
OrdV (Y,K, s) ≤ Fus∧(Y,K).
63
Let us now turn to ribbon Z-homology concordances (W,Σ) : (Y, U) →
(Y ′, K). A little care is needed in defining a fusion number in this context, as we
must also take into account the handle decomposition of the ambient manifold, as
the following example illustrates.
FIGURE 15 A concordance from the unknot to K#K with no bands.
Example 3.4.5. Let K be a trefoil, and consider K#K. There is a concordance
from the unknot to K#K in S3×I with a single birth and band. This concordance,
of course, has the minimal number of bands, else K#K would be isotopic to
the unknot. However, this birth-band pair can be eliminated in the homology
cobordism obtained from S3 × [0, 1] obtained by attaching a canceling 1- and 2-
handle pair. Consider the following movie which is shown in Figure 15:
1. t = 0: An unknot U sits in S3.
2. t = 1: A 1-handle is attached away from U . This has the result of doing 0-
surgery on an unknot unlinked with U .
3. t = 3/2: U is isotoped in ∂(S3 × [0, 1] ∪ 1-handle).
4. t = 2: A 2-handle is attached to ∂(S3 × [0, 1] ∪ 1-handle).
64
5. t = 5/2: An isotopy in ∂(S3× [0, 1]∪1-handle∪2-handle) pulls the knot K#K
away from the 1- and 2-handle. This can be done by sliding an arc in K and
in K over the 2-handle.
No bands are needed in this example, but Ord (S3V , K#K) = OrdV (S
3, K) > 0.
Hence, OrdV (S
3, K) cannot possibly be a lower bound on the number of bands
required in such link cobordisms.
This example illustrates that handles of the surface can be traded for handles
in the ambient manifold. In light of these observations, we define Fus∨(Y,K) to
be minimal number of bands plus 2-handles over all ribbon homology concordances
(W,Σ) : (Y ′, U) → (Y,K). However, OrdV (Y,K, s) cannot be a lower bound for
Fus∨(Y,K) by work of Hom-Kang-Park.
Example 3.4.6. Let K be a ribbon knot in S3 with fusion number 1. By [HKP20,
Theorem 1], the torsion order of the (p, 1)-cable of K, which we denote Kp,1, is p.
However, there is a Kirby diagram for the complement of a ribbon disk of Kp,1 with
a single 2-handle and two 1-handles. By replacing one of the dotted unknots with a
unknot U we obtain ribbon Z-homology cobordism from the unknot to Kp,1 with no
bands and one 2-handle. Therefore, Fus∨(S3, Kp,1) = 1, but Ord 3V (S ,Kp,1) = p.
65
CHAPTER IV
SATELLITE CONCORDANCES
AND BORDERED FLOER HOMOLOGY
The material in this section draws from forthcoming work joint with Hayden-
Kang-Park.
In the section we prove Theorem 13. Let C : K → K ′ be a concordance.
Given a pattern knot P ⊂ S1 ×D2, we obtain a concordance between the satellites
of K and K ′ as follows. Remove a neighborhood of C in S3 × I . The Seifert
framing of K determines an identification φ : ∂ν(K) → S1 × ∂D2. Define the
satellite concordance CP to be (S
3 × I − C,∅) ∪φ×id (S1 ×D2 × I, P × I).
Since it requires no additional effort, we prove Theorem 13 for homology
concordances, i.e. link cobordisms (W,C) : (Y,K) → (Y ′, K ′) where Y and Y ′
are integer homology spheres, C is an annulus, and W is a homology cobordism.
Let (W,C) : (Y,K) → (Y ′, K ′) be a homology concordance. Let F = (C,A)
be the annulus C decorated with a pair of parallel arcs running from K to K ′. This
decorated concordance induces a map CFK−(Y,K) → CFK−(Y ′, K ′), denoted
FW,C , which is defined as a composition of elementary cobordism maps [Zem18].
Choose a Morse function f : W → R on W so that f |C has no critical points and
choose a gradient-like vector field which is tangent to C. This induces a handle
decomposition for (W,C) which only involves four-dimensional handles.
Since the restriction of this Morse function to the surface has no critical
points, this also produces a handle decomposition for the complement of C; the
attaching curves for the handles are already embedded in the complement of K, so
the cobordism W − C is built simply by attaching the same handles to Y −K.
66
Proposition 4.0.1. Let (W,C) : (Y,K) → (Y ′, K ′) be a homology concordance.
Then, given a handle decomposition for (W,C) as above, there exists a map FW−C
with the property that for any pattern knot P in the solid torus, the following
diagram commutes up to homotopy:
CFA−(HP )⊠ ĈFD(Y −K) ≃ CFK−(Y,KP )
IH ⊠FP W−C FW,CP
CFA−(HP )⊠ ĈFD(Y ′ −K ′) ≃ CFK−(Y ′, K ′P ).
Here, HP is a doubly pointed Heegaard diagram for the knot P in the solid
torus, KP and K
′ ′
P are the satellites of K and K with pattern P , (W,CP ) is the
concordance induced by the pattern, and the horizontal homotopy equivalences are
given by the pairing theorem for knot Floer homology [LOT18].
The map FW−C : ĈFD(Y − K) → ĈFD(Y ′ − K ′) will be defined in
the standard way, as a composition of maps corresponding to handle attachments
[OS06].
1- and 3-handle maps:
The maps associated to 1- and 3-handle attachments are simplest to define.
For simplicity, let Y be an integer homology sphere. We will write Y (S0) for the
result of S0-surgery on Y (which is, of course, diffeomorphic to Y#S1 × S2).
Lemma 4.1.1. Let F : CFK−(Y,K −P ) → CFK (Y (S0), K ′P , t0) be the 1-
handle cobordism map. There exists a map F̃ : ĈFD(Y − K) → ĈFD((Y −
K)(S0), t0|(Y−K)(S0)) making the following diagram commute up to homotopy:
CFA−(HP )⊠ ĈFD(Y −K) ≃ CFK−(Y,KP )
IH ⊠F̃P F
CFA−(HP )⊠ ĈFD((Y −K)(S0), t0| ≃ − 0(Y−K)(S0)) CFK (Y (S ), KP , t0),
67
where t is the torsion SpinC-structure on Y (S00 ), i.e. c1(t0) = 0.
Proof. Fix a nice Heegaard diagram H = (Σ , αc , . . . , αcg 1 g−1, β1, . . . , βg) for Y − K.
Choose curves λ and µ in Σ isotopic to a longitude and meridian of K respectively,
which intersect in a single point and avoid the α circles. A bordered Heegaard
diagram HB for Y − K is obtained from H by deleting a neighborhood of p, and
defining αa1 = λ − p and αa2 = µ − p to be the α-arcs parametrizing the boundary,
i.e.
H = (Σ− p, αaB 1, αa c2, α1, . . . , αcg−1, β1, . . . , βg).
Let HP be a nice doubly pointed bordered Heegaard diagram for the pattern knot
embedded in the solid torus. A doubly pointed Heegaard diagram HK for (Y,KP P )
is obtained by gluing HB and HP along their common boundary.
Recall the definition of the 1-handle map F : CFK−(Y,KP ) →
CFK−(Y (S0), K ′P , t0). Choose a pair of points p1 and p2 in HKp away from HP .
Moreover, assume p1 and p2 lie in the same connected component of Σ − (∪iαi) −
(∪iβi) as the basepoint z. Remove neighborhoods of p1 and p2, and attach an
annulus. Add two new curves α0 and β0 which are homologically essential in the
annulus and intersect transversely in a pair of points, which we denote θ+ and θ−.
There are two bigons from θ+ to θ−. The 1-handle map is simply
x →7 x⊗ θ+.
In exactly the same way, S0-surgery on the bordered Heegaard diagram for Y − K
gives rise to a map F̃ : ĈFD(Y −K) → ĈFD((Y − K)(S0)), t0|(Y−K)(S0)) defined
x′ 7→ x′ ⊗ θ+. Since we chose nice diagrams, the identification of CFA−(HP ) ⊠
≃
ĈFD(Y −K) −→ CFK−(Y,K) is simply the map which takes a pair of intersection
68
points in HP and HK and views them as a single intersection point in HP ∪ HK .
Tautologically, then, the desired diagram commutes.
The case of the 3-handles is dual to this case, and follows similarly.
Lemma 4.1.2. Let H : CFK−(Y (S0), KP , t0) → CFK−(Y,K ′P ) be the 3-handle
cobordism map. There exists a map H̃ : ĈFD((Y − K)(S0), t0|(Y−K)(S0)) →
ĈFD(Y −K ′) making the following diagram commute up to homotopy:
CFA−(HP )⊠ ĈFD((Y −K)(S0), t ≃ − 00|(Y−K)(S0)) CFK (Y (S ), KP , t0)
IH ⊠H̃P H
CFA−(HP )⊠ ĈFD(Y −K ′) ≃ CFK−(Y,K ′P ).
2-handle map:
The 2-handle cobordism is the only interesting case, and follows from the
pairing theorem for triangles [LOT16, Proposition 5.35]. Before we define the 2-
handle cobordism maps on ĈFD, we recall some facts about bordered Heegaard
triple diagrams.
Given a doubly pointed bordered Heegaard diagram Hα,β0 , we obtain
a doubly pointed bordered Heegaard triple diagram Hα,β0,β1 by performing a
Hamiltonian translation on each of the β0-curves. By removing the β0-curves, we
obtain an ordinary doubly pointed bordered Heegaard diagram Hα,β1 . Counting
holomorphic triangles defines a map
m : CFA−2 (H − −α,β0)⊗ CFA (Hβ0,β1) → CFA (Hα,β1).
69
Taking Θβ0,β1 to be the top graded generator of the homology of ĈF (Hβ0,β1), we
can consider the map
m2(−,Θ −β0,β1) : CFA (Hα,β0) → CFA−(Hα,β1).
We will make use of the fact that this map is homotopic to the map ΨHα,β1←Hα,β0
induced by the isotopy of β0 to β1, and is just the “nearest point map”, taking an
intersection point in α∩β0 to the closest intersection point in α∩β1. For a proof in
the classical case, see [Lip06, Proposition 11.4] or [JTZ21, Lemma 9.7].
FIGURE 16 By pinching along the dotted line, we see a dynamic bigon map is
homotopic to the composition of a monogon map with a triangle map.
Lemma 4.2.1. Let Hα,β0,β1 be a bordered Heegaard triple diagram where the β1-
curves are small Hamiltonian translates of the β0-curves. Let Θβ0,β1 be the top
graded generator of H 0 1∗(ĈF (β , β )). Then,
m2(−,Θ) ∼ ΨHα,β1←H .α,β0
Moreover, ΨH 1←H 0 is the nearest point map.α,β α,β
Proof. The holomorphic polygon maps are defined by counting certain holomorphic
maps
u : (S, ∂S) → (Σ×∆, (α× e0) ∪ (β1 × e1) ∪ . . . ∪ (βn × en),
70
where ∆ is a disk with n boundary punctures, and edges labeled e0, . . . , en. If B ∈
π2(x0, . . . , xn; ρ1, . . . , ρm) then denote by MB(x0, . . . , xn; ρ1, . . . , ρm) the moduli
space of embedded holomorphic maps in the homotopy class B with asymptotics
x1, . . . , xn, ρ1, . . . , ρm. The map ΨH is defined by counting bigons withα,β1←Hα,β0
dynamic boundary conditions, i.e. maps
⋃
(S, ∂S) → (Σ× [0, 1]× R, (α× 1× R) ∪ (βt × 0× {t}))
t
where βt = β0 for t ≤ 0 and βt = β1 for t ≥ 1. By a neck stretching argument,
the moduli space of such maps splits into a product of MB(x, y, z; ρ1, . . . , ρm) and
MA(x; ρ1, . . . , ρm). In other words, ΨH 1←H 0 is the composition of the monogonα,β α,β
map θ with the triangle map m2(−,−). See Figure 16.
Claim: θ(1) = Θβ0,β1 . In particular, m2(−, θ(1)) = m2(−,Θβ0,β1), as desired.
If θ(1) was not equal to Θβ0,β1 , then take the standard genus g Heegaard
diagram for S1 × S2 and consider the two maps
m2(−, θ(1)),Ψ − −Hα,β1←H 0 0α,β0 : CFA (Hα,β ) → CFA (Hα,β )
from above. The map ΨH 1←H 0 is a homotopy equivalence, but is homotopic toα,β α,β
m2(−, θ(1)). If θ(1) were not the highest graded generator, the composition would
be zero, a contradiction.
Finally, as t → 0, and βt approaches β0, holomorphic disks with Maslov index
0 limit to bigons for the pair (α, β0). But, any bigon for (α, β0) with Maslov index
0 must be constant, since the R-action on moduli space of homotopy classes of non-
constant bigons is free, implying the dimension of the moduli space is nonzero.
71
We are now ready to define the 2-handle cobordism maps. Let L be a framed
link in Y . Let W (L) be the cobordism corresponding to attaching 2-handles along
L. Let Y (L) be the 3-manifold obtained by surgery on L. As usual, the 2-handle
map is defined by counting holomorphic triangles.
Lemma 4.2.2. Fix a SpinC-structure s on W (L). Let G −s : CFK (Y,KP , s|Y ) →
CFK−(Y ′, K ′P , s|Y ′) be the 2-handle cobordism map. There exists a map G̃s :
ĈFD(Y − K, s|Y−K) → ĈFD(Y ′ − K ′, s|Y ′−K′) making the following diagram
commute up to homotopy:
CFA−(H ≃ −p)⊠ ĈFD(Y −K, s|Y−K) CFK (Y,KP , s|Y )
ICFA−(H )⊠G̃sP Gs
CFA−(HP )⊠ ĈFD(Y ′ −K ′, s| ≃ −Y ′−K′) CFK (Y ′, K ′P , s|Y ′).
Proof. To define the map for 2-handles, let L be a framed, k-component link in
Y − K, and let B be a bouquet for L. Choose a Heegaard triple diagram Hα,β,β′
which is subordinate to this bouquet in the sense of [OS06] and also admits a
′ ′ ′
decomposition into two bordered Heegaard diagrams Hα,β,β ∪ Hα,β,βP B , where H
α,β,β
P
is the bordered Heegaard triple diagram obtained from HP by adding Hamiltonian
translates of the β-curves. Such a diagram can be produced as follows.
1. Start with a Heegaard diagram H = (Σg, α1, . . . , αg−1, βk+1, . . . , βg) for Y −
K−B. For each β-curve in H, add a β′-curve which is a Hamiltonian translate
so |βi ∩ β′j| = 2δij.
2. Add a collection {β1, ..., βk} which are meridians of the components of L.
Attaching 1- and 2-handles along the α and β curves yields Y −K.
3. For each component of L, add a curve β′i according to its framing. Attaching
1- and 2-handles along the α and β′ curves produces (Y −K)(L) = Y ′ −K ′.
72
4. Choose a longitude and meridian of K in Σ disjoint from the α-curves which
intersect in a single point, p. Define arcs αa1 = λ− p and αa2 = µ− p.
Altogether, this defines a bordered Heegaard triple diagram
Hα,β,β
′
B = (Σ
a a c c ′
g − p, α1, α2, α1, . . . , αg−1, β1, . . . , βg, β1, . . . , β′g).
′ ′
By construction, Hα,β,βP ∪ H
α,β,β
B is a Heegaard triple subordinate to our chosen
bouquet B.
For δ, ε ∈ {α, β, β′}, let Hδ,εP and H
δ,ε
B be the various standard bordered
Heegaard diagrams associated to these triples. Let Θβ,β′ and Θ be the top
′ ′
dimensional generators of H∗(ĈF (Hβ,βB )) and H∗(ĈF (H
β,β
P )) respectively. The
pairing theorem for triangles [LOT16, Proposition 5.35] gives the following
homotopy-commutative square:
CFA−(Hα,β)⊠ ĈFD(Hα,β) ≃ CFK−(Hα,β ∪Hα,βP B P B )
m2(−,Θ)⊠m2(−,Θβ,β′ ) m2(−,Θ⊗Θβ,β′ )
− α,β′ α,β′ ≃ − α,β′ ′CFA (H α,βP )⊠ ĈFD(HB ) CFK (HP ∪HB ).
The right vertical arrow is by definition the 2-handle cobordism map G on
CFK−. After identifying CFA−(Hα,β) with CFA−(Hα,β′∞ ∞ ), the map m2(−,Θ) is the
identity. Define m2(−,Θβ,β′) to be the 2-handle cobordism map G̃s on ĈFD.
Since each handle attaching cobordism map was defined with respect to
a particular Heegaard diagram, the last step is to ensure maps induced by the
Heegaard moves relating two diagrams induce homotopy equivalences which are
compatible with the bordered Floer homology pairing theorem. This is shown by
Hendricks-Lipshitz in [HL19].
73
Lemma 4.2.3. [HL19, Lemma 5.6] Suppose H1 and H2 are a pair of bordered
Heegaard diagrams related by a bordered Heegaard move and H0 is another bordered
Heegaard diagram with ∂H0 = −∂Hi, i ∈ {1, 2}. Then, the diagram
CFA−(H1)⊠ ĈFD(H0) ≃ CFK−(H1 ∪H0)
CFA−(H2)⊠ ĈFD(H0) ≃ CFK−(H2 ∪H0)
commutes up to homotopy. The vertical maps come from the proof of invariance of
bordered and classical Floer homology.
Proof of Proposition 4.0.1: We have a decomposition of W − C into handle-
attachment cobordisms, W1 ∪W2 ∪W3. Define FWC to be the composition
FW−C = H̃ ◦ΨH3←H2 ◦ G̃s ◦ΨH2←H1 ◦ F̃ ,
where ΨH ←H i ∈ {1, 2} are the change of diagram homotopy equivalences andi+1 i
s is the restriction of the unique SpinC-structure on W to W2. By stacking the
diagrams from Lemma 4.1.1, 4.1.2, 4.2.2, and 4.2.3 the result follows.
No Cancelation Lemma
In Chapter 5, we will be interested in distinguishing maps induced by satellite
concordances. Let C and C ′ be two concordances from K0 to K1. Assuming FC ̸=
FC′ , we would like a sufficient condition to guarantee that FP (C) ̸= FP (C′).
Given a concordance C : U → K, Theorem 13 guarantees the existence of a
map
F : ĈFD(S3 − U) → ĈFD(S3 −K),
74
with the property that for any satellite pattern P , the map induced by the
concordance P (C) can be computed as
I 1P ⊠ F : ĈFA(S ×D2, P )⊠ ĈFD(S3 − U) → ĈFA(S1 ×D2, P )⊠ ĈFD(S3 −K),
where IP is the identity map on ĈFA(S1 × D2, P ). By definition, the map IP ⊠ F
can be written ∑∞
IP ⊠ F (x⊗ v) = (mk+1 ⊗ IP )(x⊗ F k(v)),
k=1
or, diagrammatically as
δN1
F
IP ⊠ F = .
δN2
m
As described in Chapter 11 of [LOT18], a model for CFK−(K) gives rise to
a model for ĈFD(S3 − K): roughly a basis for ĤFK (K) forms a basis for ι0 ·
ĈFD(S3 −K) and the structure of the differential of CFK−(K) determines a basis
for ι1 · ĈFD(S3 −K) as well as the differentials.
Since we will be primarily interested in slice disks, we will restrict ourselves to
unknotted patterns. If P is an unknotted pattern, then
ĈFA(S1 ×D2, P )⊠ ĈFD(S3 − U) ≃ ĈFK(S3, U).
75
Therefore, there is some element a in ĈFA(S1 × D2, P ) such that a ⊗ v generates
homology of ĈFK(S3, U), where v is the unique element in ĈFD(S3 − U). On
homology, IP ⊠F is therefore determined by the image of a⊗v. We show that given
some seemingly restrictive assumptions on the structure of ĈFA(S1×D2, P ), IP⊠F
is guaranteed to be nontrivial on homology.
Lemma 4.3.1. Let
F : ĈFD(S3 − U) → ĈFD(S3 −K),
be a morphism of type-D structures which tensors with the identity map of
ĈFA(S1 × D2, λ) to give a nontrivial map, where λ is the knot S1 × {pt} ⊂
S1 × D2. Let a be an element of ĈFA(S1 × D2, P ) such that a ⊗ v generates
the homology of ĈFA(S1 × D2, P ) ⊠ ĈFD(S3 − U) and extend {a} to a basis
for ĈFA(S1 × D2, P ). If the coefficient of a is zero in every A∞ operation
m (b, ρ , . . . , ρ ) of ĈFA(S1 × D2k i1 i − , P ) which preserves the filtration, then thek 1
map
I ⊠F : H (ĈFA(S1×D2, P )⊠ĈFD(S3−U)) → H (ĈFA(S1×D2, P )⊠ĈFD(S3P ∗ ∗ −K))
is nontrivial.
Proof. Let x = F (v), where F is the map of type-D structures induced by FC . We
can write ∑
x = 1 · θ + ρIθI ,
I∈{1,2,3,12,23,123}
for some θI ∈ ĈFD(S3 − K). The term θ must be nontrivial, since we have
assumed that I 1× 2 ⊠ F has nontrivial image (this follows from the factĈFA(S D ,λ)
76
that ĈFA(S1 × D2, λ) has no nontrivial A∞-operations and so all other terms in
F (x) are annihilated after taking the box tensor product.) (IP ⊠F )(a⊗v) is defined
to be
∑∞
(IP ⊠ F )(a⊗ v) = (m kk+1 ⊗ IP ) ◦ (a⊗ F (v))
k=1
= a⊗ θ + other terms.
The term a⊗ θ could be canceled if, for some k, F k(v) = ρi1 ⊗ . . .⊗ρi ⊗ θ and therek
is an operation of the form mk+1(a, ρi1 , . . . , ρi ) in which a appears with nonzerok
coefficient. However, we have assumed that no such operations exist.
Moreover, when we pass to homology, there are no relation between a ⊗ θ
and any other elements of H∗(ĈFA(S
1 × D2, P ) ⊠ ĈFD(S3 − K)), since a ⊗ θ
could only appear as a term in the boundary of another element if there were a
filtration preserving operation of the form mk+1(b, ρi1 , . . . , ρi ) in which a appearedk
with nonzero coefficient. Again, no such operations exist.
Therefore, a ⊗ θ appears as a non-canceling term in the expansion of (IP ⊠
F )(a ⊗ v) ∈ H (ĈFA(S1 × D2∗ , P ) ⊠ ĈFD(S3 − K)), from which it follows that
IP ⊠ F has nontrivial image.
77
CHAPTER V
INJECTIVE SATELLITE OPERATORS
The material in this section draws from forthcoming work joint with Hayden-
Kang-Park.
We turn now to applications of Theorem 13. We begin by showing that if
knot Floer homology distinguishes a pair of slice disks, it will also distinguish their
positive Whitehead doubles. Given a slice disk D of K, we denote the induced
element FD(1) in ĤFK(S
3, K) by tD.
Theorem 10. Let K be a knot in S3 with slice disks D1, D2. If tD1 ≠ tD2, then
tWh+(D ̸= t + as well.1) Wh (D2)
Proof. Let F1, F2 be the type-D morphisms determined by D1 and D2 respectively.
Since tD1 ̸= tD2 , we have that I 1× 2 ⊠ (F1 + F2) is nontrivial. The type-AĈFA(S D ,λ)
structure for the positive Whitehead double is computed by Levine [Lev12]:
c c′
ρ3ρ2ρ1 ρ3ρ2ρ1
b b′
ρ1 ρ1
ρ123
a a′ ρ12
1+ρ23
ρ2
ρ3
d.
ρi ...ρ1
In the diagram above, an arrow of the form x −−−−→ik y indicates that
mk+1(x, ρi1 , . . . , ρi ) = y. Arrows pointing left lower the filtration. A shortk
computation illustrates that H∗(ĈFA(S
1 ×D2,Wh+)⊠ ĈFD(S3 − U)) = F⟨b⊗ v⟩.
78
There is a single arrow into b, but it lowers the filtration level. Therefore, by
Lemma 4.3.1, IWh+ ⊠ (F1 + F2) is nontrivial. Therefore, by Theorem 13, tWh+(D ̸=1)
tWh+(D2).
We now prove that for any slice disks D1 and D2 for K, their positive
Whitehead doubles are topologically isotopic.
A slice disk D is called a Z-disk if the fundamental group of its complement
is isomorphic to Z. By the work of Conway and Powell, any two Z-disks with
common boundary are topologically isotopic rel. boundary [CP21, Theorem 1.2].
We can arrange to work in this situation by choosing appropriate satellite patterns.
Recall that the winding number of a pattern P is the algebraic intersection of P
with a generic meridional disk of the solid torus containing P .
Proposition 5.0.1. If P is a winding number zero pattern with P (U) = U and D
is a slice disk, then the satellite disk P (D) is a Z-disk.
Proof. Choose a tubular neighborhood ν(D) of D. The satellite disk P (D) is
contained in ν(D), so we have a splitting
B4 ∖ P (D) = (B4 ∖ ν(D)) ∪ (ν(D)∖ P (D)).
Since the homeomorphism class of ν(D) ∖ P (D) does not depend on the choice of
D,
ν(D)∖ P (D) ∼= ν(D0)∖ P (D0),
where D0 denotes the trivial slice disk of an unknot U . Since P is unknotted P (D0)
is also a trivial slice disk of P (U) = U so we may take ν(D0) to be the whole of B
4.
79
Therefore,
ν(D0)∖ P (D0) ∼= B4 ∖D ∼0 = D2 × (D2 ∖ {pt}),
implying π1(ν(D)∖ P (D)) ∼= Z.
The fundamental group of the intersection
(B4 ∖ νD) ∩ (νD ∖ P (D)) ∼= S1 ×D2
is also Z, and the natural maps
( ) ( )
π (B4 ∖ νD) ∩ (νD ∖ P (D) → π B41 ( ) 1 ∖ νD ,
π (B41 ∖ νD) ∩ (νD ∖ P (D) → π1 (νD ∖ P (D)) ,
are given by the inclusion of a meridional class of π1(B
4 ∖ νD) and the
×w(P )
multiplication map Z −−−−→ Z, respectively, where w(P ) denotes the winding
number of P . Since P has winding number 0 and π1(B
4 ∖ νD) is normally
generated by a meridian of D, we see
π1(B
4 ∖ P (D)) ∼= π1(νD ∖ P (D)) ∼= Z.
Therefore P (D) is a Z-disk.
As the positive Whitehead double pattern satisfies all the hypotheses of
Proposition 5.0.1, we have the following corollary.
Corollary 5.0.2. Let D1 and D2 be any slice disks for K which are distinguished
by their induced maps on ĤFK . Then, Wh+(D ) and Wh+1 (D2) are exotic disks.
80
By considering deform spun disks, in forthcoming work, we prove the
following.
Theorem 11. For any nontrivial knot K, the knot Wh+(K#K#−K#−K) bounds
a pair of exotic disks.
81
CHAPTER VI
STABLY EXOTIC SURFACES
In this section, we produce a pair of exotic disks which remain exotic after
many stabilizations.
Theorem 12. For any p, there exists a knot Jp which bounds a pair of exotic disks
D and D′p p which remains exotic after p− 1 internal stabilizations.
We distinguish our surfaces by comparing their induced maps on knot Floer
homology. Stabilization has a simple effect on the induced map; attaching a tube
simply corresponds to multiplication by U (or V ) [JMZ20, JZ21]. Juhász-Zemke
make use of this fact to define their suite of secondary Heegaard Floer invariants
which provide lower bounds for the stabilization distance of two surfaces. See
Chapter 2 for details.
Recall that the torsion order OrdU(K) of a knot K is the smallest power
of U which annihilates the torsion submodule of HFK−(K). Since a stabilization
corresponds to multiplication by U , any two maps induced by disks with boundary
K become indistinguishable after multiplication by UOrdU (K). Therefore, these lower
bounds cannot be used to show disks bounding knots with torsion order 1 have
large stabilization distance. OrdU(K) is bound above by the fusion number of K,
which is the minimal number of bands occurring in ribbon disk for K [JMZ20].
Therefore, to have any hope in finding disks with large stabilization distance, it is
necessary to work with knots with large fusion number.
82
Cabled concordances
Recent work of Hom-Kang-Park [HKP20] and Hom-Lidman-Park [HLP22]
studies how cabling is related to the torsion order and fusion number of a knot.
If K is ribbon with fusion number 1, then the knot Floer homology of the (p, 1)-
cable of K has torsion order p [HKP20, Lemma 3.3]. Cabling has a natural four-
dimensional extension: given a concordance C : K → K ′, there is a cabled
concordance between the cables of K and K ′. In particular, given a ribbon knot K
with fusion number 1, K(p,1) bounds a “cabled” ribbon disk, and has fusion number
p.
The knot J shown in Figure ??, bounds an exotic pair of disks D and D′ by
the work of Hayden [Hay21]. We will refer to these disks as the “positron disks”
since their double branched cover is the positron quark of [AM98].These disks are
distinguished by their induced maps on knot Floer homology [DMS22]. But, J
has fusion number 1, so the two maps become equal after a single stabilization. In
fact, we will show directly that these two disks are smoothly isotopic after a single
stabilization. However, the cabled disks Dp and D
′
p have fusion number p, and as
we show, have stabilization distance p as well.
We begin by reviewing the two definitions of stabilization distance and
illustrate that the two need not agree. We then review the construction of cabled
concordances and show how this operation can be used to produce disks which are
topologically isotopic. We conclude this section by giving an upper bound on the
stabilization distance of Dp and D
′
p by explicitly showing they become isotopic after
p stabilizations.
83
Two notions of stabilization distance
The most general notion of internal stabilization is due to [JZ21].
Definition 6.2.1. Let Σ be an oriented surface with boundary, smoothly
embedded in a 4-manifold W . Let B be a 4-ball in the interior of W whose
boundary intersects Σ in an n-component unlink L. Moreover, suppose Σ ∩ B
is a collection of disks D1, ..., Dn which can be isotoped into ∂B relative to their
boundaries. Let S0 be a connected genus g surface in B with boundary L. The
surface Σ′ = (Σ − B) ∪L S0 is called the (g, n)-stabilization of Σ along (B, S0). We
call Σ the (g, n)-destabilization of Σ′ along (B, S0). See Figure 17.
FIGURE 17 A (g, n)-stabilization along (B4, S0). The case (g, n) = (2, 2) is shown.
Definition 6.2.2. Let Σ and W be as above, and let Σ′ be a (g, n)-stabilization
of Σ. When (g, n) = (0, 2) and S0 ∪ D1 ∪ D2 bounds a 3-dimensional 1-handle
embedded in W , we say Σ′ is a 1-handle stabilization of Σ.
We will simply write “stabilization” instead of (g, n)-stabilization, and state
explicitly when we mean 1-handle stabilization. We now formally define the two
notions of stabilization distance. For simplicity, we will only define the stabilization
distance for disks.
84
Definition 6.2.3. Let Σ and Σ′ be disks in W such that ∂Σ = ∂Σ′ and [Σ] =
[Σ′] ∈ H2(W,∂W ;Z). The 1-handle stabilization distance d(Σ,Σ′) between Σ and
Σ′ is the minimal number k such that Σ and Σ′ become isotopic rel boundary after
each is stabilized k times.
Definition 6.2.4. Let Σ and Σ′ be disks in W such that ∂Σ = ∂Σ′ and [Σ] = [Σ′] ∈
H2(W,∂W ;Z). The stabilization distance µ(Σ,Σ′) between Σ and Σ′ is defined to
be the minimum of
max{g(Σ1), . . . , g(Σk)}
over all sequences of connected surfaces from Σ = Σ1 to Σ
′ = Σk in W such that
∂Σi = K for all i and Σi and Σi+1 are related by a stabilization or a destabilization.
Note that necessarily d(Σ,Σ′) ≥ µ(Σ,Σ′). However, as the next example
illustrates, the two notions are distinct.
Example 6.2.5. The knot K = 946 is shown in Figure 19. K bounds an obvious
torus in S3. Moreover, by compressing the two circles which generate the first
homology of this torus, we obtain two slice disks D and D′ with boundary K.
Both disks can be described as banded unlinks with a single band [Swe01, HKM20]
(Figure 19). It is shown in [MP19] that d(D,D′) = 1. This can be seen as follows.
FIGURE 18 Swimming one band through another.
To show that the two disks become isotopic after a single 1-handle
stabilization, it suffices to show that by attaching a tube, the relative positions
85
FIGURE 19 An isotopy taking a 1-handle stabilization of D to a 1-handle
stabilization of D′.
A swim move occurs in frame 6. (Continued in Figure 20)
.
of the two bands can be swapped, and then that the tube can be isotoped until it is
once again clearly the result of a 1-handle stabilization.
By strategically attaching a tube, we can slide the right-hand band, b, to the
left (Figure 19.) However, this band slide separates the tube into two bands, v and
v∗. Next, slide v into the position originally occupied by b. Now, we can drag v∗
around K until it once again forms a tube with v by performing swim moves as
necessary to change crossings of v∗ with the diagram for K. See Figure 18 for an
example of a swim move, and see 20 for the remainder of the isotopy taking the
stabilization of D to the stabilization of D′.
86
FIGURE 20 The remainder of the isotopy between the stabilizations of D and D′.
Swim moves occur in frames 2 and 4.
Miller-Powell use Alexander modules to show that by taking boundary
connected sums of these disks, they can produce disks with arbitrarily large 1-
handle stabilization distance: d(♮mD, ♮mD′) = m. However, it is clear from Figures
19 and 20, that µ(♮mD, ♮mD′) = 1; no band ever slid over the marked point on
the diagram, so we can take the connected sums at the marked points. Since
the more general stabilization distance allows us to stabilize and destabilize, we
can attaching a single tube in order to isotope the first D summand to D′, then
destabilize, and then repeat the strategy on the next copy of D, until we are left
with ♮mD′.
It is worth noting that even though the obvious ribbon disk for ♮mD has m
bands, we can in some sense “reuse” (stabilize then destabilize) the same tube
m times to make ♮D and ♮D′ isotopic. K has fusion number 1, so necessarily
87
OrdU(K) = 1. By the knot Floer homology Künneth formula,
Ord (#mU K) = OrdU(K) = 1.
This implies that τ(♮mD, ♮mD′) ≤ 1 (see Definition 2.3.6), and therefore τ cannot
detect the large 1-handle stabilization distance of these disks.
Concordances induced by cables
Let C : K → K ′ be a concordance. Given a pattern knot P ⊂ S1 × D2,
we obtain a concordance between the satellites of K and K ′ as follows. Remove a
neighborhood of C in S3× I . The Seifert framing of K determines an identification
φ : ∂ν(K) → S1 × ∂D2. Define the satellite concordance CP to be (S3 × I −
C,∅) ∪φ×id (S1 ×D2 × I, P × I).
The (p, q)-cable of a knot is a satellite with pattern P = Tp,q, the (p, q)-torus
knot. Since the (p, 1)-torus knot is the unknot, it is clear that if a knot K is slice,
so is its (p, 1)-cable. Moreover, by capping off the (p, 1)-cable of the unknot, we
obtain a cabled disk for K.
By the work of Freedman and Quinn, locally flat proper submanifolds have
topological normal bundles which are unique up to ambient isotopy [FQ90, Section
9.3]. Therefore, by the nature of the construction, topological isotopy is preserved
by the cabling operation. Therefore, since the disks D and D′ which bound
the knot J are topologically isotopic by the work of Conway and Powell [CP21,
Theorem 1.2], this topological isotopy also produces a topological isotopy between
the (p, 1)-cables of D and D′. This gives us:
88
Lemma 6.3.1. The cabled disks Dp and D
′
p which bound Kp are topologically
isotopic.
An upper bound for the stabilization distance
We now turn to the proof that the stabilization distance between D ′p and Dp
is at most p.
As a warm up case, consider the disks D and D′ with boundary J . Let b be
the left-hand band which defines D and let b′ be the right-hand band which defines
D′. As in Example 6.2.5, attach a tube v ∪ v∗ to J , so that the band b can be slid
until it becomes isotopic to the band b′. Next, slide the band v into the position
originally occupied by b. We have exchanged the roles of b and b′ at the cost of
tangling the bands v and v∗ which made up the stabilization; it is not clear whether
the resulting surface is isotopic to a stabilization of D′. If can isotope v∗ away from
J ∪ b and back onto v, we will be done.
FIGURE 21 An isotopy of D ∪ (v ∪ v∗).
89
As in Example 6.2.5, we can use the bands b and v to pull v∗ into the correct
position by a sequence of swim moves. Recall that a swim move of v∗ through b
corresponds to pushing v∗ below the critical point for b, and performing an isotopy
of v∗ in S3 − J(b), i.e. in the complement of link obtained by band surgery on b.
Performing the entire isotopy at this level will turn out to be easier to visualize,
especially once we progress to the cabled case.
Push the band v∗ into the interior of B4, (say to radius r = 2/3) below the
critical points for the bands b and v. Here, the level sets of the surface are isotopic
to J(b)(v) = J(b)(b′) (the result of band surgery on both b and b′) which is the
unknot. Moreover, from Figure 21, we see that the band v∗ is attached trivially.
At this point, the diagram is symmetric. Hence, this argument can be
repeated with the disk D′ stabilized with tube u∪ u∗: we isotope u∗ into B4 as well,
until we see u∗ attached to the unknot. U ∪ v∗ and U ∪ u∗ are clearly isotopic, so
by composing the first isotopy with the inverse of the second we obtain the desired
isotopy from D ∪ (v ∪ v∗) to D′ ∪ (u ∪ u∗).
The cabled case is similar.
Proposition 6.4.1. The disks Dp and D
′
p become isotopic after p 1-handle
stabilizations.
Proof. Figure 22 gives a band presentation for Dp. Let b1, . . . , bp and b
′
1, . . . , b
′
p be
the bands of D and D′p p. As in the case p = 1, attach tubes v
∗ ∗
1 ∪ v1, . . . , vp ∪ vp in
order to move the band bi into the position of b
′
i, and then slide vi into the original
position of bi (Figure 22). The result of band surgery on all the bi and vi bands
is again the unknot, and as shown in Figure 23 this unknot is naturally identified
with the (p, 1)-cable of the unknot. Moreover, at this level set, the v∗i bands are
attached trivially. So, just as before, perform the symmetric isotopy of D′p, and
90
FIGURE 22 Part 1 of an isotopy between p-fold stabilizations of D and D′p p
concatenate to obtain an isotopy between the p-fold stabilizations of Dp and D
′
p.
A lower bound for the stabilization distance
We now make use of the secondary invariants of Juhász-Zemke to provide a
lower bound on the stabilization distance of the disks D ′p and Dp.
The positron disks D and D′ are distinguished by their induced maps on
knot Floer homology. In [DMS22], Dai-Mallick-Stoffregen compute HFK−(J). The
summand which contains the image of this disk has the form F[U ]⟨x⟩ ⊕ F⟨e1, e2⟩.
91
FIGURE 23 Part 2 of an isotopy between p-fold stabilizations of D ′p and Dp
The formal variable U acts trivially on the F-summands. Using their equivariant
knot Floer homology program, they prove the following.
Theorem 13. Let D and D′ be the exotic positron disks. Then, the maps FD and
FD′ satisfy:
(FD + FD′)(1) = e1 + e2.
As a sanity check, notice that
(FD#T 2 + FD′#T 2)(1) = U(FD + FD′)(1)
= U(e1 + e2)
= 0,
since D and D′ become isotopic after a single stabilization.
92
Theorem 14. Let D and D′ be a pair of disks distinguished by their induced maps
on HFK−. Let D ′p and Dp denote their (p, 1)-cables. Then, the the stabilization
distance between D and D′ is at least p.
Proof. Let Ap = CFA
−(S1 × D2, Cp,1) be the type-A structure associated to
the (p, 1)-cabling pattern. Ap is generated by α, β1, . . . , β2p−2. Since we are only
interested in computing maps CFA−(S1 ×D2, Cp,1)⊠ ĈFD(S3 − U) → CFA−(S1 ×
D2, Cp,1) ⊠ ĈFD(S3 − J) and the homology of CFA−(Hp) ⊠ ĈFD(S3 − U) ≃
CFK−(U) is generated by α ⊗ v, it is enough to consider the A∞-operations
involving α.
︷ ︸i︸ ︷
m2+i(α, ρ12, . . . , ρ12, ρ1) = β2p−i−2 0 ≤ i ≤ p− 2
︷ ︸j︸ ︷ ︷ ︸i︸ ︷
m4+i+j(α, ρ3, ρ23, . . . , ρ23, ρ2, ρ12, . . . , ρ12, ρ ) = U
pj+i+1
1 βi+1 0 ≤ i ≤ p− 2, 0 ≤ j
︷ ︸j︸ ︷
m3+i(α, ρ3, ρ23, ..., ρ23, ρ2) = U
p(j+1)α, 0 ≤ j.
For the full collection of A∞-operations, see [Pet13, Section 4]. Let F1 and
F2 be the type-D morphisms determined by FD and FD′ . An argument identical
to that of L︷emm︸︸a 4.︷3.1 shows that, since the only arrows into α are of the formj
m (α, ρ , ρ , ..., ρ , ρ ) = Up(j+1)3+i 3 23 23 2 α for 0 ≤ j we must have an element θ ∈
ĈFD(S3 − J) such that α ⊗ θ appears in ICp,1 ⊠ (F1 + F2)(α ⊗ v) which cannot
be cancelled. Moreover, Ukα ⊗ v is nontrivial in homology for k < p. Therefore, we
have that
Uk(ICp,1 ⊠ (F1 + F2)(α⊗ v)) ̸= 0,
93
for k < p. Therefore,
p ≤ µ(Dp, D′p),
as claimed.
In particular, the (p, 1)-cables of the positron disks have stabilization distance
exactly p.
94
APPENDIX
DIRECT COMPUTATIONS
In this section, we illustrate the how Theorem 13 can be used to carry out
explicit computations. Again, let FDp and FD′ be the (p, 1)-cables of the exoticp
positron disks D and D′. It is more convenient to view these disks as concordances
Cp and C
′
p from the unknot. FCp and FC′ will be computed in terms of FS3p ×I−C
and FS3×I−C′ , which as we will see, are determined by FC and FC′ (up to some
indeterminacy).
The complex CFK−(J) consists of a singleton generator x as well as four
boxes.
bi ai gi fU iU
x
V V V V
ei cU i ji hU i
FIGURE 24 The complex CFK−(J), i ∈ {1, 2}.
Generator grU grV
x 0 0
ai 0 0
bi 1 −1
ci −1 1
ei 0 0
fi −1 −1
gi 0 −2
hi −2 0
ji −1 −1
TABLE 1 Bi-gradings of generators of CFK−(J).
95
The summands generated by the two boxes generated by {fi, gi, hi, ji} contain
no elements of bigrading (0, 0), and therefore do not intersect the images of the
maps FC and FC′ . For this reason, we will work primarily with the subcomplex of
CFK−(J) generated by {x, ai, bi, ci, ei} to simplify the notation.
[LOT18] gives an algorithm for determining ĈFD(S3 − K) in terms of
CFK−(K). Figure 25 shows how a box and a singleton generator give rise to
summands of ĈFD(S3 − J).
b 1i ρ y2 i ρ a3 i
ρ1 ρ1
bi aU i
x ⇝ y2 y4V V i i x ρ12
ei cU i ρ123 ρ123
e 3i ρ y c2 i ρ3 i
FIGURE 25 On the left is the summand of CFK−(J) containing FD(1) and FD′(1).
On the right is a model for the corresponding summand of ĈFD(S3 − J).
Computing the morphism complex
By Theorem 13, there exists a map F : ĈFD(S3 − J) → ĈFD(S3 − J)
with the property that for any pattern P in the solid torus, ICFA−(P ) ⊠ F computes
the concordance map induced by the pattern. In particular, if we take P to be
the longitudinal unknot in the solid torus, which we denote (T∞, λ) this map also
computes FC (respectively FC′). Since we did not show this map F is unique, we
will try to pin it down by computing the morphism space MorA(T 2)(ĈFD(S
3 −
U), ĈFD(S3 − J)) and considering which maps f ∈ Mor 3A(T 2)(ĈFD(S −
96
U), ĈFD(S3 − J)) have the property that ICFA−(T∞,λ) ⊠ f ≃ FC (respectively
FC′).
We begin by computing the dimension of the homology of the morphism
space MorA(T 2)(ĈFD(S
3 − U), ĈFD(S3 − J)).
Lemma A.1.1. The space of homotopy classes of maps from ĈFD(S3 − U) to
ĈFD(S3 − J) is 10 dimensional.
Proof. By [LOT11], there is a homotopy equivalence:
MorA(T 2)(ĈFD(S
3 − U), ĈFD(S3 − J)) ≃ ĈF (−(S3 − U) ∪ (S3 − J))
= ĈF (S30(J)).
The mapping cone formula [OS08b] shows that HF+(S30(J), [0]) is the homology of
the mapping cone H∗(Cone(A+ −
v−0+−h00 → B+0 )). We illustrate part of the complex.
A+0 B+0
(UV )−1x (UV )−1i x̃i
bi (UV )
−1ai b̃i (UV )
−1ãi
ei ci ẽi c̃i
xi x̃i
Ubi ai v0 + h
ãi
0
V ci V c̃i
The homology of the portion of the complex shown is T +⟨xi⟩ ⊕ T +⟨x̃i⟩ ⊕
F⟨Ubi = V ci⟩, where T + = F[U,U−1]/(U · F[U ]). The homology of the summand
97
which is not shown (the remaining two boxes) is F⟨Ugi = V fi⟩. From HF+(S30(J)),
ĤF (S30(J)) is obtained via the exact triangle
HF+(S30(J))
U HF+(S30(J))
ĤF (S30(J)).
A straightforward computation shows that ĤF (S30(J)) = ker(U)⊕ coker(U) ∼=
F⊕10.
Having computed the dimension of H∗(MorA(T 2)(ĈFD(S
3 − U), ĈFD(S3 −
J))), our next task is to find a basis. To simplify the exposition, we introduce some
notation. The summand of CFK−(J) generated by x gives rise to a summand of
ĈFD(S3 − J) generated by an element we will also denote x, with differential
δ1(x) = ρ12x. Call this type-D structure B. The unit boxes in CFK−(J) correspond
to boxes in ĈFD(S3 − J) as in Figure 26. Let C be such a type-D structure.
b y1ρ ρ a2 3
ρ1 ρ1
y2 y4
ρ123 ρ123
e ρ y
3
ρ c.2 3
FIGURE 26 The type-D structure C associated to a unit box.
If we ignore the gradings, ĈFD(S3−U) is isomorphic to B, and ĈFD(S3−J)
is isomorphic to B ⊕ C⊕4. Therefore, it suffices to find bases for the homologies of
MorA(T 2)(B,B) and MorA(T 2)(B,C).
The space of homotopy classes of maps B → B has a basis given by:
ϕ = (x 7→ x)
98
ψ = (x 7→ ρ12x).
Since H∗(MorA(T 2)(B,B ⊕ C⊕4) is ten dimensional, H∗(MorA(T 2)(B,C)) must be
two dimensional. By inspection, a basis is given by:
g = (x 7→ e+ ρ3y2 + ρ1y3)
h = (x 7→ ρ1y4).
This is confirmed by Zhan’s bordered Floer homology calculator [Zha].
From concordance maps to complement maps
The maps FS3×I−C and FS3×I−C′ satisfy the property that
ICFA−(T∞,λ) ⊠ FS3×I−C ≃ FC
and
ICFA−(T∞,λ) ⊠ FS3×I−C′ ≃ FC′ .
Therefore, to determine these maps, we will compute ICFA−(T∞,λ) ⊠ f for all basis
elements, f , of the morphism space.
CFA−(T∞, λ) is a right A∞-module over A(T 2), whose A∞-operations can be
computed by counting holomorphic disks in the doubly pointed bordered Heegaard
diagram shown in Figure 27.
Let α be the single intersection point in α ∩ β. The only non-trivial A∞-
operations are given by
︷ ︸j︸ ︷
m3+j(α, ρ
j+1
3, ρ23, . . . , ρ23, ρ2) = U α.
99
FIGURE 27 A doubly pointed bordered Heegaard diagram for the longitudinal
unknot in the solid torus, (T∞, λ).
Call this module A. Since A⊠B has a single generator, α⊗x, the maps IA⊠f
are determined by the image of this element.
Lemma A.2.1. Let ϕ, ψ, g, h be the basis for H∗(MorA(T 2)(B,B ⊕ C)) described
above. Then,
IA ⊠ ϕ = (α⊗ x →7 α⊗ x)
IA ⊠ ψ = 0
IA ⊠ g = (α⊗ x 7→ α⊗ e)
IA ⊠ h = 0.
Proof. The differentials of C are shown again, as they are needed below.
b y1ρ2 ρ a3
ρ1 ρ1
y2 y4
ρ123 ρ123
e ρ y
3
ρ c.2 3
Since many of the differentials on the tensor product are trivial, many terms will be
forced to be zero. In particular, δ1(x) = ρ12x, and since there are no A∞-operations
100
involving ρ12, any terms involving δ
1(x) will be zero. IA ⊠ ψ is zero for this reason.
We make use of the graphical notion of [LOT18, Chapter 2].
By strict unitality, IA ⊠ φ(α⊗ x) has a single term, α⊗ x.
α x
ϕ
IA ⊠ ϕ = = (α⊗ x →7 α⊗ x),
1
m x
α
α x
To compute IA ⊠ g = IA ⊠ (x →7 e + ρ y23 + ρ y31 ), we first note that,
again, by strict unitality, the only term e could cont︷ribu︸t︸e is α︷ ⊗ e. Secondly, anyi
term contributed by ρ 31y will be of the form mℓ(α, ρ12, . . . , ρ12, ρ1, . . .) × ξ. But,
since none of the A∞-operations involve ρ1, any term of this form must be zero.
Therefore, all that remains is to check whether ρ y23 contributes any nonzero terms
to IA ⊠ g. A nonzero term could appear, since all the A∞-operations involve ρ3, but
δ1(y2) = 0, so no ρ2 coefficient will appear. Therefore,
α x
α x α x
g
g y
3
g
IA ⊠ g = + + ρ .ρ 13 .. = (α⊗ x →7 α⊗ e)
1
2
m e m y
α m δ1
α e 0 y2
0 ξ
101
Consider IA ⊠ h = IA ⊠ (x 7→ ρ1y4). Much like the previous case, ρ1y4 can
contribute no nonzero terms, since ρ1 does not appear in any of the A∞-operations.
α v
h
IA ⊠ h = = (α⊗ v 7→ 0).ρ1
m y4
0
α y4
This concludes this computation.
We can now turn to the maps FS3×I−C and FS3×I−C′ . Recall from that Dai-
Mallick-Stoffregen show that:
FC(v) + FC′(v) = e1 + e2,
where v is the generator for HFK−(U). A basis for H∗(Mor
3
A(T 2)(ĈFD(S −
U), ĈFD(S3 − J))) is given by {ϕ, ψ, gi, hi}, where gi and hi are maps from
ĈFD(S3 − U) to the ith box in ĈFD(S3 − J)) which agree with the maps g and h
above. The complex ĈFD(S3 − J) is shown in full in Figure 28.
b y1 a g 1i ρ2 i ρ3 i ρ z2 i ρ f3 i
ρ1 ρ1 ρ1 ρ1
y2 y4i i z
2 z4 x ρ12i i
ρ123 ρ123 ρ123 ρ123
ei ρ y
3
i ρ c
3
2 3 i
ji ρ z2 i ρ h3 i
FIGURE 28 The full complex ĈFD(S3 − J).
102
Let v be the single generator for ĈFD(S3 − U). By Lemma A.2.1, we can
identify which maps f : ĈFD(S3 − U) → ĈFD(S3 − J) have the property that
IA ⊠ f is homotopic to either FC or FC′ .
Again, FC(v) + FC′(v) = e1 + e2. Since the maps ψ, h1, . . . , h4 satisfy IA ⊠ ψ =
IA ⊠ hi = 0, we cannot immediately rule out the possibility that they appear as
terms in FS3×I−C(v) and FS3×I−C′(v). Therefore, we can deduce that
∑4
FS3×I−C + FS3×I−C′ = g1 + g2 + ε1ψ + ε2 · hi
i=1
where εi ∈ {0, 1}. Surprisingly, despite the indeterminacy of these maps, this will
be sufficient information to compute the sum of the maps associated to the cabled
disks.
In summary, we have the following:
Proposition A.2.2. Let C and C ′ be the exotic concordances from the unknot to
J . Then, the maps FS3×I−C and FS3×I−C′ satisfy:
∑4
FS3×I−C + FS3×I−C′ = g1 + g2 + ε1ψ + ε2 · hi
i=1
where εi ∈ {0, 1}.
Remark A.2.3. We will see that ϵ2 can be taken to be zero. Once we compute
CFA−(Hp) ⊠ ĈFD(S3 − K), we will see that the terms contributed by hi are in
the wrong grading. However, this fact has no effect on the final result, so we do not
emphasize this.
103
From complement maps to cabled concordance maps
Having found our candidates for the maps FS3×I−C and FS3×I−C′ , all that
remains is to compute the tensor products of the candidates with the identity map
for the A∞-module associated to the (p, 1)-cable pattern in the solid torus.
A doubly pointed Heegaard diagram HP for the (p, 1)-cable in the solid torus
is shown in Figure 29. Let A −p = CFA (Hp). Ap is generated by α, β1, . . . , β2p−2.
FIGURE 29 A doubly pointed bordered Heegaard diagram for the (p, 1)-cable in
the solid torus.
Since we are only interested in computing maps CFA−(Hp) ⊠ ĈFD(S3 − U) →
CFA−(Hp) ⊠ ĈFD(S3 − J) and the homology of CFA−(Hp) ⊠ ĈFD(S3 − U) ≃
CFK−(U) is generated by α ⊗ v, it is enough to consider the A∞-operations
involving α.
︷ ︸i︸ ︷
m2+i(α, ρ12, . . . , ρ12, ρ1) = β2p−i−2 0 ≤ i ≤ p− 2
︷ ︸j︸ ︷ ︷ ︸i︸ ︷
m4+i+j(α, ρ3, ρ23, . . . , ρ , ρ , ρ , . . . , ρ , ρ ) = U
pj+i+1
23 2 12 12 1 βi+1 0 ≤ i ≤ p− 2, 0 ≤ j
︷ ︸j︸ ︷
m p(j+1)3+i(α, ρ3, ρ23, ..., ρ23, ρ2) = U α, 0 ≤ j.
104
For the full collection of A∞-operations, see [Pet13, Section 4]. As before, we
will start by computing IA 2p ⊠ f for the basis elements of H∗(MorA(T )(B,B ⊕ C)),
and then use the fact that ĈFD(S3 − J) is isomorphic to B ⊕ C⊕4 to compute
FS3×I−C and Fp S3×I−C′ .p
Lemma A.3.1. Let ϕ, ψ, h, and h be the basis of H∗(MorA(T 2)(B,B ⊕ C)) as
computed in Section A.2. Then,
IAp ⊠ ϕ = (α⊗ x 7→ α⊗ x)
IAp ⊠ ψ = ((α⊗ x 7→ 0) ∑ )p−2
IAp ⊠ g = (α⊗ x →7 α⊗ e+ β
3
2p−i−2 ⊗ y
∑ i=0 )p−2
IAp ⊠ h = α⊗ x →7 β 42p−i−2 ⊗ y
i=0
Proof. We proceed as in Lemma A.2.1. This computation is slightly more involved,
as there are more A∞-operations.
First, for the map ψ = (x 7→ ϕ12x), it must be that IAp ⊠ ψ = 0. Since
δ1(x) = ρ12x, any term coming from ρ12x will be of the form mk(ρ12, . . . , ρ12), which
must be zero, as there are no A∞-operations only involving ρ12.
For the map ϕ = (x →7 x), the map IAp ⊠ ϕ has a single term by strict
unitality:
α v
f
IAp ⊠ ϕ = = (α⊗ v 7→ α⊗ v).
1
m v
α
α v
105
The map h = (x 7→ ρ 41y ) has many nonzero terms,︷sinc︸e︸ther︷e are nontrivialiA∞-operations involving ρ12 and ρ1, namely m2+i(α, ρ12, . . . , ρ12, ρ1) = β2p−i−2 for
0 ≤ i ≤ p − 2. But these are the only terms that can appear, since δ1(y4) = 0.
Therefore,
α v
δ1
v
. ( )
.. ∑p−2
IAp ⊠ h = 4ρ12 v = α⊗ v →7 β2p−i−2 ⊗ y .
i=0
ρ12 g
ρ1
m y4
β2p−i−2
β 42p−i−2 y
The map g = (x 7→ e + ρ3y2 + ρ y31 ) is the most complicated. Once again, by
strict unitality, it must be that α⊗e appears as a term in IB⊠g(α⊗x), and no other
terms will come from e. Th︷e di︸ff︸eren︷tial of y2 is zero, so ρ y22 can only contributei
terms of the form m2+i(α, ρ12, . . . , ρ12, ρ ) ⊗ y22 , but these are all zero. Finally, the
differential of y3 is ρ2e, and th︷e di︸ff︸eren︷tial of e is ρ y2123 , so︷ther︸e︸ cou︷ld potentiallyi i
be terms of︷the︸f︸orm︷m2+i(α, ρ , . . . , ρ , ρ ) ⊗ y312 12 1 , m3+i(α, ρ12, . . . , ρ12, ρ1, ρ2) ⊗ ei
or m (α, ρ , . . . , ρ , ρ , ρ , ρ 24+i 12 12 1 2 123) ⊗ y , but only the first case introduces non-zero
106
terms. Therefore,
α v
δ1
α v
v
g .. (. ∑p− )2
IA ρp ⊠ g = + 12 v = α⊗ v 7→ α⊗ e+ β2p−i−2 ⊗ y
3 .
1
i=0
m e ρ12 g
α
ρ1
α e m y3
β2p−i−2
β2p−i−2 y
3
This concludes the computation.
In Proposition A.2.2, we found the sum of the maps FS3×I−C and FS3×I−C′ .
By Theorem 13, the maps FCp and F
′
C′ induced by the (p, 1)-cables of C and Cp
can be computed as IAp ⊠ FS3×I−C and IAp ⊠ FS3×I−C′ respectively. Lemma A.3.1
can now be applied to give us the candidates for the maps FCp and FC′ . As before,p
let {ϕ, ψ, gi, hi} be the basis for H∗(MorA(T 2)(ĈFD(S3 − U), ĈFD(S3 − J))) from
Section A.2.
Since ∑4
FS3×I−C + FS3×I−C′ = g1 + g2 + ε1ψ + ε2 · hi
i=1
for εi ∈ {0, 1}, from Lemma A.3.1 it follows that
(IAp ⊠ FS3×I−C + IAp ⊠ FS3×I−C′)(α⊗ v) =
107
(∑∑− )2 p 2
α⊗ (e + e ) + β ⊗ y31 2 2p−i−2 k + ε2 · β2p−i−2 ⊗ (y4 4k + zk)
k=1 i=0
Summarizing our results, we have the following.
Proposition A.3.2. Let C ′p and Cp be the (p, 1)-cables of the exotic concordances
C and C ′. Let Hp be the doubly pointed bordered Heegaard diagram for the (p, 1)-
cable pattern shown in Figure 29. Then, the maps FCp and FC′ satisfy:p
(IAp ⊠ FS3×I−C + IA ⊠ FS3p ×I−C′)(α⊗ v) =
(∑2 ∑− )p 2
α⊗ (e 3 4 41 + e2) + β2p−i−2 ⊗ yk + ε2 · β2p−i−2 ⊗ (yk + zk)
k=1 i=0
where ε is either 0 or 1.
108
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