THE REDUCED KNOT FLOER COMPLEX
By
David Thaddeus Krcatovich

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics - Doctor of Philosophy
2014

ABSTRACT
THE REDUCED KNOT FLOER COMPLEX
By
David Thaddeus Krcatovich
We define a “reduced” version of the knot Floer complex CF K − (K), and show that it
behaves well under connected sums and retains enough information to compute Heegaard
Floer d-invariants of manifolds arising as surgeries on the knot K. As an application to
connected sums, we prove that if a knot in the three-sphere admits an L-space surgery, it
must be a prime knot. As an application to the computation of d-invariants, we show that
the Alexander polynomial is a concordance invariant within the class of L-space knots, and
show the four-genus bound given by the d-invariant of +1-surgery is independent of the genus
bounds given by the Ozsv´ath-Szab´o τ invariant, the knot signature and the Rasmussen s
invariant.

To my wife Erin and my parents, Rocky and Patty – for their unending patience and
support.

iii

ACKNOWLEDGMENTS

This dissertation is the culmination of six years of work, and it is with sincere gratitude
that I would like to recognize those who have helped me along the way. I benefited extensively from discussions with other graduate students, most notably Faramarz Vafaee, Luke
Williams, Kyungbae Park, Adam Giambrone and Chris Hays. In addition to several other
outstanding faculty members, I would like to thank my dissertation committee members:
Effie Kalfagianni, Jon Wolfson, Ben Schmidt and especially Ron Fintushel, an inspiring
teacher who sparked my interest in topology. I am greatly indebted to my advisor, Matt
Hedden. His patience, direction, commitment of time and energy, and encyclopedic mathematical knowledge have all been vital to my progress as a student, and to the development
of the work contained in this dissertation. Most importantly, I found great motivation in
the encouragement and support of my wife Erin, and of all of my family.

iv

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Chapter 1 Introduction . . . . . . . . . . . .
1.1 Heegaard Floer homology . . . . . . . .
1.2 Heegaard Floer homology for knots . . .
1.3 Dehn surgeries on knots . . . . . . . . .
1.4 Correction terms . . . . . . . . . . . . .
1.5 Knot concordance and four-genus bounds

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1
3
6
8
10
13

Chapter 2 Algebraic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Filtered complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 (Z, U )-filtered complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18
18
22

Chapter 3 Reducing the knot Floer complex . . . . . . . . . . . . . . . . . .
3.1 The reduced complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Sums of knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
37

Chapter 4 Applications to L-space knots
4.1 Reduced complexes of L-space knots .
4.2 Correction terms and concordance . . .
4.3 Examples . . . . . . . . . . . . . . . .

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42
42
50
65

-BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

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LIST OF FIGURES

Figure 2.1:

A filtered chain deformation retraction. . . . . . . . . . . . . . . . . . .

21

Figure 3.1:

The knot Floer complex CF K − (T (2, 3)). . . . . . . . . . . . . . . . . .

29

Figure 3.2:

Reducing a knot Floer complex. . . . . . . . . . . . . . . . . . . . . . .

36

Figure 3.3:

A connected sum with a reduced complex. . . . . . . . . . . . . . . . .

40

Figure 3.4:

A tensor product without using the reduced complex. . . . . . . . . . .

41

Figure 4.1:

Reducing an L-space knot complex. . . . . . . . . . . . . . . . . . . . .

43

Figure 4.2:

Computing d1 from the knot Floer complex. . . . . . . . . . . . . . . .

53

Figure 4.3:

Labeling staircase complexes. . . . . . . . . . . . . . . . . . . . . . . . .

56

Figure 4.4:

An upside-down staircase. . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 4.5:

The difference of two L-space knots. . . . . . . . . . . . . . . . . . . . .

58

Figure 4.6:

Computing d1 of the difference of two L-space knots. . . . . . . . . . .

63

Figure 4.7:

Complexes for the knot Kp . . . . . . . . . . . . . . . . . . . . . . . . .

66

Figure 4.8:

A representative staircase. . . . . . . . . . . . . . . . . . . . . . . . . .

68

Figure 4.9:

Complexes for the knot Jp . . . . . . . . . . . . . . . . . . . . . . . . . .

69

vi

Chapter 1
Introduction
In the 1980’s, Donaldson applied gauge theory to smooth four-dimensional topology, with
remarkable results. Combining his results with those previously known in the topological
case [16], Donaldson showed in particular that many simply connected, closed, topological
four-manifolds admit no smooth structures [8]. In short, understanding the gap between
topological and smooth four-manifolds required genuinely different techniques than those
which had been useful in higher dimensions. By counting solutions to a PDE, up to gauge
equivalence, Donaldson introduced invariants which were sensitive enough to distinguish
different smooth structures on a topological four-manifold [9].
In [12], Floer defined the “instanton homology” of an integral homology three-sphere,
and showed how in certain cases, the Donaldson invariants of a compact four-manifold could
be computed from the instanton homology of a separating three-manifold. While counting
instantons is still difficult in general, Floer further showed that the instanton homology
groups of Y , Y and Y

fit into a “surgery exact triangle” [14], where Y and Y

are the

manifolds obtained by doing 0− and 1−framed surgery, respectively, on a knot in Y (this
required enlarging the class of “admissable” three-manifolds on which instanton homology
could be defined). This allows some computations to be determined algebraically, rather
than analytically.
Still, one of the primary difficulties in computing Donaldson invariants in general is the
fact that the moduli space of solutions is non-compact. A further revolution in topology us1

ing gauge theory came with the introduction of the Seiberg-Witten equations [69]. While it is
conjectured that the smooth four-manifold invariants defined using the Seiberg-Witten equations are equivalent to the Donaldson invariants (for an appropriate class of four-manifolds;
see [11] for partial results), the moduli space of solutions was shown to be compact, making
the computation of invariants considerably simpler. It was tempting to then try to emulate
Floer’s approach, and define a homology theory of three-manifolds which could be used to
compute the Seiberg-Witten invariants of smooth four-manifolds. This was the underlying
motivation for the development of Heegaard Floer homology, which provides the context for
the rest of this thesis. 1
In [54], Ozsv´ath and Szab´o define a collection of invariants of closed, oriented threemanifolds, known collectively as Heegaard Floer homology, which come in the form of various
graded abelian groups. While one of the goals was to compute four-manifold invariants
from invariants of separating three-manifolds, these three-manifold invariants themselves are
defined by first considering a separating two-manifold. That is, the basic input is a Heegaard
diagram, corresponding to a Heegaard splitting of a three-manifold [24]. In contrast to
counting instantons, which involves solving PDE, enumerating the generators of the Heegaard
Floer chain complexes becomes a simple combinatorial procedure. Defining the boundary
map on this complex, however, still requires solving PDE, in the form of finding pseudoholomorphic disks satisfying certain boundary conditions. Nevertheless, versions of Heegaard
Floer homology have been shown in many cases to be algorithmically computable from just
the combinatorics of the Heegaard diagram [36, 39, 40, 41, 66].
As in instanton homology, the Heegaard Floer groups satisfy surgery exact triangles,
1 This was also the motivation for the development of Seiberg-Witten Floer homology, which is explained in
detail in [30]. It has been shown in a series of papers [31] that the groups associated to a spinc three-manifold
by these two theories are isomorphic.

2

providing algebraic means of computation in some cases. In contrast, the invariants are
defined for a manifold resulting from surgery with any rational slope. To further simplify
computations, the theory can be extended to define an invariant of a knot [52, 64], and
this knot invariant can be used to compute the three-manifold invariants of surgery along
the knot, as explained in Section 1.3. The theory also extends to give invariants of links,
four-manifolds and three-manifolds with boundary [58, 57, 28, 34, 35]. A partial summary
of results in knot theory and three- and four-dimensional topology which have been proven
using Heegaard Floer homology is given in Sections 1.1 and 1.2.
The aim of this thesis is to provide an efficient use of some of the algebraic structure
provided by the theory. We will develop a reduced form of Heegaard Floer homology applied
to knots, called the reduced knot Floer complex, whose primary objective is to simplify
computations involving connected sums of knots. We will then show how it can be used to
aid explicit computations of invariants, or to prove general results.

1.1

Heegaard Floer homology

Let Σ be a closed, connected oriented surface of genus g. Let α = {α1 , . . . , αg } be a collection
of g mutually disjoint simple closed curves on Σ, which are linearly independent when viewed
as elements of H1 (Σ), and β = {β1 , . . . , βg } a collection with the same properties. This
specifies a closed, connected oriented three-manifold in the following way.
Take Σ × [0, 1], and attach a copy of D2 × [0, 1] along each αi by identifying ∂(D2 ) × [0, 1]
with a neighborhood of αi × {0}. Similarly, attach a copy of D2 × [0, 1] by idenifying
∂(D2 ) × [0, 1] with a neighborhood of βi × {1}. Because of the linear independence of each
family of curves, each boundary component of the resulting manifold is homeomorphic to

3

S 2 , and up to diffeomorphism there is a unique way to attach two three-balls to get a closed
manifold Y .
If z is any point in Σ \ α ∪ β , and α and β intersect transversely, then the collection
Σ, α, β, z

is called a pointed Heegaard diagram for Y . In [54], given a pointed Heegaard

diagram for Y , Ozsv´ath and Szab´o consider the symmetric product Symg (Σ), which is the
quotient of the product Σ×g by the obvious action of the symmetric group Sg . When
endowed with a symplectic form (which agrees with the product symplectic form away from
the diagonal; see [60]), this manifold contains two Lagrangian submanifolds,

Tα = α1 × · · · × αg

and Tβ = β1 × · · · × βg .

Heegaard Floer homology is then a filtered version of the Lagrangian Floer homology [13] of
Symg (Σ), Tα , Tβ .
More precisely, assuming Tα and Tβ are transverse, their intersection is a finite set,
written as G. An element in G can be thought of as a g-tuple of intersection points on Σ,
using each α and β curve exactly once. Ozsv´ath and Szab´o define a chain complex freely
generated over F[U, U −1 ] by the elements of G, written as CF ∞ (Y ) (the original definition
was over Z[U, U −1 ], but in this thesis we will always take coefficients in F, the field with two
elements). The differential counts pseudo-holomorphic disks in Symg (Σ) of Maslov index
one connecting two intersection points2 .
The basepoint z plays two roles in this construction. First, it partitions G into subsets
2 See [33] for a reformulation of the theory which involves counting disks in Σ × [0, 1] × R rather than
Symg (Σ).

4

corresponding with spinc structures on Y , so that we get a splitting

CF ∞ (Y ) =

CF ∞ (Y, t).
t∈Spinc (Y )

Second, it distinguishes a subvariety {z} × Symg−1 (Σ) ⊂ Symg (Σ), and the intersection
number of a pseudo-holomorphic disk with this subvariety is what determines the exponent
of the variable U in the differential. Since transverse holomorphic submanifolds of complimentary dimension can only intersect positively, the exponent of U can never be decreased
by the differential of a homogeneous element. That is to say, the basepoint provides a filtration of CF ∞ , by the exponent of the variable U . This extra structure provided by this
filtration makes Heegaard Floer homology a powerful three-manifold invariant.
As a result, complexes CF + , CF − and CF can be defined by taking coefficients in
F[U ]
F[U, U −1 ]
, F[U ] and
, respectively (alternatively, these are quotient-, sub- and subquoU F[U ]
U
tient complexes of CF ∞ , respectively). We will denote this collection of chain complexes as
CF ◦ for brevity. The construction of these complexes depends on several choices, but it is
shown in [54, Theorem 1.1] that their filtered chain homotopy types – and in particular, the
homology groups, which are denoted HF ∞ , HF + , HF − and HF – are topological invariants
of (Y, t).
In [51], it was shown that Heegaard Floer homology detects the Thurston norm, and in
particular the Seifert genus of a knot. There is also an invariant of a contact structure on a
three-manifold [55] which vanishes on overtwisted contact structures.

5

1.2

Heegaard Floer homology for knots

Suppose (Σ, α, β, z) is a pointed Heegaard diagram for S 3 . Placing a second basepoint w
on Σ specifies a knot K ⊂ S 3 in the following way: let Kα be an oriented embedded arc
in Σ \ α from z to w and Kβ be an oriented embedded arc in Σ \ β from w to z. After
pushing the interior of Kα into Uα and the interior of Kβ into Uβ , K := Kα ∪ Kβ is a knot
in S 3 . It was observed independently by Ozsv´ath-Szab´o [52] and Rasmussen [64] that this
additional basepoint can be used to put a second Z-filtration on CF ∞ (S 3 ), and that the
Z ⊕ Z-filtered chain homotopy type of this complex, denoted CF K ∞ (K), is an invariant
of K ⊂ S 3 . Similarly, changing the coefficient module as in Section 1.1, we get filtered
complexes CF K + , CF K − and CF K. Again, we may refer to this collection as CF K ◦
when we do not wish to specify.
In particular, the filtration on CF K(K) gives a sequence of subcomplexes

· · · ⊆ F(K, i − 1) ⊆ F(K, i) ⊆ F(K, i + 1) ⊆ · · · ,

with associated graded complexes

CF K(K, i) :=

F(K, i)
F(K, i − 1)

.

The homology groups of the associated graded object,

HF K(K, i) := H∗ CF K(K, i)

are called the knot Floer homology groups of K. These groups “categorify” the symmetrized

6

Alexander polynomial of K, in the sense that

χ HF K(K, i) · T i = ∆K (T ).

(1.1)

i

Strictly stronger information can be obtained from knot Floer homology than from its Euler
characteristic, however. For example, the lower bound on the Seifert genus

deg ∆K (T ) ≤ g(K)

is strengthened to the result

g(K) = max{i|HF K(K, i) = 0}.

(1.2)

Similarly, the fact that a fibered knot must have monic Alexander polynomial is strengthened
to the statement [45]

K is fibered if and only if HF K(K, g(K)) ∼
= F.

(1.3)

Ignoring the knot filtration on CF K − (K) gives the Heegaard Floer complex CF − (S 3 ).
Our method in this thesis will be to ignore the other filtration, the one which measures the
exponent of the variable U , and obtain a simplified version of the knot Floer complex, called
the reduced knot Floer complex, which we will denote CF K − (K). This will be Z-filtered
chain homotopy equivalent to CF K − (K), and we will further require it to keep track of
“multiplication by U ”, in a sense to be made precise in Chapter 3.
In [52, Theorem 7.1], it was shown that there is a tensor product formula for the knot

7

Floer complexes of connected sums,

CF K ∞ (K1 #K2 ) ∼
= CF K ∞ (K1 ) ⊗F[U,U −1 ] CF K ∞ (K2 ).

A corresponding statement holds for other versions of CF K, tensoring over the appropriate
ring. These tensor product complexes become difficult to work with, however, even for simple
sums. The essence of the following theorem is that, while the reduced complex is smaller, is
still has a simple tensor product formula under connected sums.
Theorem 1.2.1. If K1 and K2 are knots in S 3 , then

CF K − (K1 ) ⊗F[U ] CF K − (K2 )

is a (Z, U )-filtered chain deformation retract of CF K − (K1 #K2 ).
Much of Chapter 2 is devoted to making precise the notion of equivalence mentioned in
Theorem 1.2.1. If we wish to connect sum a third knot, we can now reduce CF K − (K1 #K2 )
and iterate Theorem 1.2.1. As a result, this object can greatly simplify computations for
sums of knots, and, as we will show, still retains enough information for these computations
to be useful.

1.3

Dehn surgeries on knots

In addition to providing knot invariants, a crucial property of knot Floer complexes is their
relation to the Heegaard Floer complexes of the three-manifolds obtained by doing Dehn
surgery along K. It was observed in [52, 64] – roughly, due to the similarity between the

8

Heegaard diagrams for a knot and for manifolds obtained by integer surgery on the knot –
that the Heegaard Floer homology groups of sufficiently large surgeries are the homology of
certain subsets of CF K ∞ (K). This idea was used in [63] to explicitly compute HF + for
maniolfds obtained by large surgery on two-bridge knots, and in [52] for manifolds obtained
by large surgery on a knot resulting from blowing down a component of a two-bridge link.
On the other hand, it had been shown in [53, Section 9] that, as expected in analogy with
instanton Floer homology, the Heegaard Floer homology of manifolds differing by surgeries
on the same knot fit into various surgery exact triangles. By using these exact sequences,
in [59, 48], it was shown that the Heegaard Floer homology of any integer or rational Dehn
surgery is the homology of the mapping cone of a map between two subsets of CF K ∞ (K).
As a result, there is a deep connection between CF K ◦ (K) and Dehn surgeries along K.
This connection was used in [56], for example, to place strong restrictions on which knots in
S 3 admit lens space surgeries. Because a lens space admits a genus one Heegaard diagram in
which each intersection point represents a unique spinc structure, it follows that lens spaces
have the smallest possible Heegaard Floer homology; namely, if Y = L(p, q),

HF (Y, s) ∼
= F for all s ∈ Spinc (Y ).

(1.4)

Ozsv´ath and Szab´o define an L-space as any rational homology sphere satisfying (1.4), and
so their restrictions are more generally on which knots in S 3 admit L-space surgeries.
Roughly speaking, the only way to get such a simple result from the surgery formula is
to start with a knot which has a simple knot Floer complex. Ozsv´ath and Szab´o define an
L-space knot as any knot which has a positive integral surgery which results in an L-space,
and show that a knot K ⊂ S 3 is an L-space knot if and only if CF K ◦ (K) has a specific form,

9

described in Proposition (4.1.1). In particular, in light of (1.1), there are strong restrictions
on the Alexander polynomial of such a knot.
Examples of L-space knots include positive torus knots (or any knot with a positive lens
space surgery) and, for n > 0, the P (−2, 3, 2n + 1) pretzel knots [56], and more generally,
a family of twisted torus knots [68]. By combining work of Hedden and Hom [21, 26], the
(p, q)-cable of a knot K is an L-space knot if and only if K is an L-space knot and

q
≥ 2g(K) − 1,
p

where g is the Seifert genus. In [68], Vafaee asks if there are any other satellite operations
which can produce L-space knots. In this thesis, we give a negative answer for the simplest
satellite operation, connected sums. After describing the reduced complexes of L-space
knots, we use Theorem 1.2.1 to prove
Theorem 1.3.1. A knot in S 3 which admits an L-space surgery must be a prime knot.
We should remark here that it is easy to see that the sum of two non-trivial L-space knots
is not an L-space knot; for example, by observing that the characterization of knot Floer
complexes of L-space knots given in [56, Theorem 1.2] is not preserved under tensor products.
However, our reduced complex will make this statement just as apparent for the sum of any
non-trivial knots.

1.4

Correction terms

Suppose W is a cobordism from one three-manifold Y1 to another Y2 , and s a spinc structure
on W which restricts to ti on Yi . In [57], given a handle decomposition of W , Ozsv´ath and
10

Szab´o define chain maps from CF ◦ (Y1 , t1 ) to CF ◦ (Y2 , t2 ), which induce maps

◦ : HF ◦ (Y , t ) → HF ◦ (Y , t ).
FW,s
1 1
2 2

(1.5)

These maps are independent of handle decomposition, and of all choices involved in constructing CF ◦ (Yi ). 3
One result of this construction is a Z-valued invariant for closed spinc four-manifolds
(X, s) with b+
2 (X) > 1, which is conjecturally equal to the Seiberg-Witten invariant. This
is obtained by removing two four-balls from X and viewing the resulting manifold as a
cobordism from S 3 to S 3 , and using a “mixed” version of the cobordism invariant.
A second result, which is of greater interest in this thesis, is that if t is a torsion spinc
structure on Y (meaning c1 (t) is a torsion cohomology class), the relative Z-valued homological grading on HF ◦ (Y, t) can be lifted to an absolute Q-valued grading. Roughly, starting
with a surgery diagram for a three-manifold Y , Ozsv´ath and Szab´o construct a Heegaard
triple diagram which specifies a four-manifold bounded by S 3 , #n S 2 × S 1 and Y . They then
define the absolute grading of a class in ξ ∈ HF (Y, t), denoted gr(ξ), by considering a Whitney triangle in the diagram connecting ξ to the generators of highest grading in HF (S 3 ) and
HF (#n S 2 × S 1 ) [57, Equation 12]. They then show [57, Theorem 7.1] that, for a cobordism
W from Y1 to Y2 , this grading satisfies
2
◦ (ξ) − gr(ξ) = c1 (s) − 2χ(W ) − 3σ(W )
gr FW,s
4

(1.6)

for any homogeneous class ξ ∈ HF ◦ (Y1 , t1 ), where ti is the restriction of s to Yi .
3 While the groups HF ◦ are originally only defined up to isomorphism, it is shown in [29] that they can
be defined as explicit groups, and so the map (1.5) can be explicitly defined.

11

The remaining applications of the reduced complex presented in this thesis will pertain
to the Heegaard Floer correction terms, or d-invariants. In [49, Definition 4.1], Ozsv´ath
and Szab´o use the absolute grading to define the d-invariant of a spinc rational homology
three-sphere (Y, t) as

d(Y, t) = min{gr(x)| x ∈ Im π∗ : HF ∞ (Y, t) → HF + (Y, t) }.

(1.7)

We will work with the equivalent definition

d(Y, t) = max{gr(x)| x ∈ HF − (Y, t), x is not U -torsion.}

(1.8)

The condition of x being non-torsion is equivalent to saying that x is not in the kernel of
the induced map HF − (Y, t) → HF ∞ (Y, t).
Remark 1.4.1. Our convention which makes these definitions agree is slightly different than
that of Ozsv´ath and Szab´o – we assume both CF + and CF − to contain the element 1 in
F[U, U −1 ]. This will be convenient for computing correction terms, but has the drawback that
CF + is not quite the quotient complex corresponding to the subcomplex CF − .
These invariants have been used to answer questions related to Dehn surgery [6, 7, 46],
the smooth knot concordance group [4, 22, 27, 38] and various notions of genera of knots
[1, 17, 47].
The property of “keeping track of multiplication by U ” mentioned in Section 1.2 is
essential, since it will allow us to compute d-invariants with the reduced complex.

12

1.5

Knot concordance and four-genus bounds

In [15], Fox and Milnor investigated knotted embeddings ι : S 2 → R4 . Of particular interest
here, they considered the intersection of ι(S 2 ) with a generic R3 -hyperplane slice of R4 . A
generic intersection would be a closed embedded one-manifold in R3 . In the case that this
intersection is a knot K, then because K is a simple closed curve on a manifold homeomorphic
to S 2 , it bounds a disk, which is embedded in R4 . Since S 2 is embedded as a submanifold, it
has a trivial normal bundle, and so the disk is in fact locally flatly embedded. This leads to
the definition of a slice knot as a knot in S 3 which bounds a locally flatly, properly embedded
disk in B 4 .
Of course, bounding a disk is equivalent to cobounding an annulus with the unknot.
So, the notion of a slice knot being “four-dimensionally trivial” can be generalized to an
equivalence relation on knots: K and J are called concordant if there is a locally flatly
embedded annulus in S 3 × [0, 1] which is bounded by K ⊂ S 3 × {0} and J ⊂ S 3 × {1}. If
we consider only smooth embeddings, we get notions of smoothly slice knots and smooth
concordance, which are distinct from the topological, locally flat case (see, for example,
[5, 18, 10, 22]). For the remainder of this thesis, unless explicitly stated otherwise, we will
always be working in the smooth category.
For any knot K ⊂ S 3 , consider the product annulus K × [0, 1] ⊂ S 3 × [0, 1]. If we let
“−K” denote the mirror image of K with reversed orientation, then, for x ∈ K, removing
a neighborhood of x × [0, 1] from the annulus yields a disk bounded by K#(−K), so it is
clear that K#(−K) is a slice knot. As a result, the set of knots up to concordance forms an
abelian group under the connected sum operation, called the knot concordance group.
If a knot is not slice – that is, it does not bound a smooth disk in B 4 – it is natural to ask:

13

what is the minimal genus of a smoothly, properly embedded surface in B 4 which is bounded
by K? This minimum is called the four-genus of K, denoted g4 (K). It is evident that g4
is an invariant of the concordance class [K], and also that g4 (K) ≤ g(K). Further, g4 (K)
gives a lower bound on the unknotting number u(K), the minimum number of crossings
which can be changed to turn a diagram of K into a diagram of the unknot. Both g4 and
u are hard to compute in general; using their definitions directly requires considering every
possible surface bounded by K, or every possible choice of crossing changes in every possible
diagram of K. For this reason, it is useful to have computable invariants which provide lower
bounds on g4 (and therefore u). In particular, such invariants also provide obstructions to a
knot being slice.
The tools of Heegaard Floer homology can be used to define and compute several such
invariants; two of which will be considered in this thesis. If K ⊂ S 3 is a knot, consider the
integral homology sphere S13 (K), the manifold resulting from Dehn surgery along K with
slope 1. This manifold has a unique spinc structure t, so we can define an invariant

d1 (K) := d(S13 (K), t).

(1.9)

This invariant was studied by Peters in [61], where he showed that it is in fact a concordance
invariant. This can be seen by surgering the annulus which provides the concordance between
K and J; the resulting four-manifold is a homology cobordism from S13 (K) to S13 (J), and the
boundary components of a homology cobordism have equal d-invariants (see, for example,
[61, Corollary 1.3]).
Further, by computing d-invariants of circle bundles over surfaces with Euler number ±1,

14

Peters is able to show that we get a four-genus bound,

0 ≤ −d1 (K) ≤ 2g4 (K).

Using the surgery formula provided by Ozsv´ath and Szab´o, Peters then gives an algorithm
to compute d1 (K) from the knot Floer complex CF K ∞ (K).
There is another concordance invariant which can be computed – and is in fact defined
– using the knot Floer complex; namely, the Ozsv´ath-Szab´o τ invariant. This is defined in
[50], where it is also shown that
|τ (K)| ≤ g4 (K).
In comparing the computations of these two invariants, Peters poses the question:
Question 1.5.1. What is the relation between d1 and τ ? Is it necessarily true that

|d1 (K)| ≤ 2|τ (K)|?

Of course, if the answer were “yes”, then d1 would be a rather ineffective four-genus
bound. At this point, however we should observe that τ defines a group homomorphism
from the concordance group to Z, while d1 does not (it is always non-positive). One could
hope to exploit this by finding a sum of knots for which τ is forced to be zero, but d1 is
not. Indeed, this will be the strategy to providing negative answers to Question 1.5.14 .
In some sense, the role which the reduced complex plays is that it simplifies computations
4 The first knot in a family of knots described in Chapter 4 which provides the negative answer was
actually alluded to by Peters later in his paper.

15

when dealing with sums of knots, by virtue of Theorem 1.2.1. This makes computations
manageable when dealing with an invariant which is not additive under connected sum.
Using the reduced complex, we first show
Theorem 1.5.2. Suppose that K1 and K2 are two knots in S 3 which admit L-space surgeries.
If
d1 (K1 # − K2 ) = d1 (−K1 #K2 ) = 0,
then
∆K1 (T ) = ∆K2 (T ).
In particular, the Alexander polynomial is a concordance invariant of L-space knots.
Following from this, we have the corollary
Corollary 1.5.3. If K1 and K2 are two L-space knots whose Alexander polynomials are
distinct but have the same degree, then

τ (K1 # − K2 ) = τ (−K1 #K2 ) = 0,

but either
d1 (K1 # − K2 ) = 0

or

d1 (−K1 #K2 ) = 0.

In particular, d1 gives a stronger four-genus bound than τ for K1 # − K2 and its mirror.
As another consequence, this yields an alternate proof of a result of Lˆe.
Corollary 1.5.4 (Lˆe). Algebraic knots are concordant if and only if they are isotopic.
In addition to τ , two other concordance invariants which have proven to give useful fourgenus bounds are the knot signature σ, defined from a bilinear form corresponding to a
16

Seifert surface [44], and the Rasmussen s invariant which comes from a spectral sequence
on Khovanov homology [65]. To strengthen the result of Corollary 1.5.3, and show the
effectiveness of d1 as a smooth four-genus bound, we give examples of knots for which
τ (K) = σ(K) = s(K) = 0, but |d1 (K)| is arbitrarily large. It should be noted that in
general the ω-signature, of which σ is a special case [67], gives a topological four-genus
bound for any unit complex number ω = 1. For the examples mentioned above, which
involve sums of torus knots, the collection of ω-signatures give a better bound than d1 .
The remainder of this thesis is organized as follows. In Chapter 2, we begin by introducing the algebraic framework which will be necessary. In Chapter 3, we review the
definition and properties of the knot Floer complex, and define its reduced form. Section
3.2 explains how the tensor product formula extends to the reduced complex. In Chapter 4,
we apply the theory to L-space knots, prove Theorems 1.3.1 and 1.5.2, and provide examples.

17

Chapter 2
Algebraic preliminaries

2.1

Filtered complexes

Throughout this thesis, we will be working with coefficients in the field with two elements,
which we will denote F. Given a chain complex (C, ∂), and a partially ordered set S, a
(decreasing) S-filtration on C is an exhaustive collection of subcomplexes indexed by S,

C i,

C=
i∈S

such that
Ci ⊆ Cj

if i ≤ j.

A filtration is said to be bounded if C i = C for some i ∈ S.
If (C, ∂) and (C , ∂ ) are S-filtered complexes, then a map f : C → C is a filtered map
if, for all i ∈ S,
f (C i ) ⊆ (C )i .
We will say that C and C are filtered chain homotopy equivalent if there exist filtered chain
maps f : C → C and g : C → C, and filtered chain homotopies h : C → C and h : C → C
such that
f ◦ g = IC + ∂ h + h ∂

and g ◦ f = IC + ∂h + h∂.

18

We will further say that C is a filtered chain deformation retract of C if the chain homotopy
h is trivial; i.e., if
f ◦ g = IC .
The complexes dealt with in this thesis will be filtered by Z or Z⊕Z, with partial ordering

(i, j) ≤ (i , j ) iff i ≤ i and j ≤ j .

In the case that (C, ∂) is Z-filtered, a convenient way to describe the filtration is as a function
F : C → Z ∪ {−∞}, where

F (x) =





min i ∈ Z x ∈ C i

if the minimum exists,




−∞

otherwise,

(2.1)

with the convention that −∞ ≤ i for all i ∈ Z. Note that this function satisfies

F (x + y) ≤ max{F (x), F (y)} and F (∂x) ≤ F (x)

for all x, y ∈ C. In every case which will be considered here, we will further have that
F −1 (−∞) = {0}. We will call F (x) the filtration level of x, and we will abuse terminology
and sometimes refer to F as the filtration.
A Z ⊕ Z-filtration, by complexes C i,j , gives rise to two Z-filtrations, by considering

C i :=

C i,j

and C j :=

j

C i,j .
i

19

On the other hand, if collections {C i } and {C j } are two Z-filtrations of C, then the complexes

C i,j := C i ∩ C j

are a Z ⊕ Z-filtration, so in fact the two notions – a Z ⊕ Z-filtration and a pair of Zfiltrations – are equivalent. Therefore, we can describe a Z ⊕ Z-filtration as a function
F (x) = (F1 (x), F2 (x)), where F1 and F2 are the functions describing the two corresponding
Z-filtrations.
We present here a prototypical example of what will follow. Figure 2.1 represents a Zfiltered complex C generated over F, where the vertical height of each generator corresponds
to its filtration level. We will denote by ∂(x, y) the coefficient of y in ∂x. If ∂(x, y) = 1, we
draw an arrow from x to y. Intuitively, we can “cancel” an arrow which is horizontal, while
preserving the filtered chain homotopy type of C. For example, canceling the arrow from b
to c gives a complex C in the following way. The generators are obtained by deleting the
generators b and c, and the differential on C is given by

∂ (x, y) = ∂(x, y) + ∂(x, c)∂(b, y).

In other words, if an arrow went from x to c, and another arrow went from b to y, we add
an arrow going from x to y. To put it precisely, we make the filtered change of basis c → ∂b,
then take the quotient of C by the acyclic subcomplex which is generated by b and ∂b. If we
define a homomorphism h : C → C by setting h(c) = b and h(x) = 0 for all other generators
(i.e., h is the inverse of the horizontal arrow we are canceling), then C is seen to be a filtered

20

F

d

w



co

a

1

b

0



e

F
a
G

d

~

0


−1

e

C

1

−1

C

Figure 2.1: A filtered chain deformation retraction. Canceling the horizontal arrow
from b to c yields the filtered chain deformation retract C . The additional arrow from a to
e is obtained by “traveling backward” through the canceled arrow. For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this thesis.
chain deformation retract of C, via the maps

f = π ◦ (I + ∂h),

g = (I + h∂) ◦ ι

and the chain homotopy h. Details are explained well in [23, Lemma 4.1] and [64, Section
5.1], and will be worked out in Chapter 3.
A vertical arrow could similarly be canceled, but the map h would not be filtered, and so
the result would be a chain homotopy equivalent, but not filtered chain homotopy equivalent,
complex. The idea of reduction, put simply, is that as long as we have horizontal arrows
(terms in the differential which preserve the filtration level), we can reduce the number of
generators of a chain complex, while maintaining its filtered chain homotopy type.

21

2.2

(Z, U )-filtered complexes

Suppose C is a Z-filtered complex which comes equipped with a specified filtered chain map
U . Then we will call the pair (C, U ) a (Z, U )-filtered chain complex. It will be convenient to
view such complexes as modules over F[U ], so that we will often refer to composition with the
map U as “multiplication by U ”. Two (Z, U )-filtered chain complexes (C, U ) and (C , U )
will be called (Z, U )-filtered chain homotopy equivalent if they are filtered chain homotopy
equivalent, and the maps f and g respect multiplication by U , in the sense that

gU ∼ U g

and

f U ∼ U f,

where ∼ signifies that the maps are filtered chain homotopic.
Remark 2.2.1. The notion of (Z, U )-filtered chain homotopy equivalence is an equivalence
relation.
Finally, we will also want to consider complexes which arise as products of other complexes. If (C1 , F1 ) and (C2 , F2 ) are two Z-filtered chain complexes which are freely generated
over F, we can define a filtration F × on the direct product C1 × C2 in the following way.
Suppose that {xi } and {yi } are generating sets for C1 and C2 , respectively, so that any
element in C1 × C2 can be written as

i,j εij xi yj ,

where εij ∈ F, and all but finitely many

of the ij are zero. Then,

F×

εij xi yj

= max{F1 (xi ) + F2 (yj )|εij = 1}

i,j

defines a filtration on C1 × C2 .
If each Ci comes with a filtered chain map Ui , they can naturally be thought of as modules
22

over F[U ], so that we will refer to the maps Ui as “multiplication by U ”. In this case, we
can also consider the complex
C1 ⊗F[U ] C2 .
An element in C1 ⊗F[U ] C2 is an equivalence class of elements in C1 × C2 , and we define
a filtration F on the tensor product by simply taking the minimum over each equivalence
class. That is, for any s ∈ C1 × C2 ,

F ([s]) = min F × (t) t ∈ [s] .

(2.2)

Actually, for the tensor products we will consider in this thesis, we can describe the product
filtration more concretely. In our case, we will consider a (Z, U )-filtered complex C1 for
which U is not necessarily homogeneous, but always decreases the filtration level by at least
1; that is,
F1 (U x) ≤ F1 (x) − 1 for all x ∈ C1 .

(2.3)

We will then consider a second (Z, U )-filtered complex, C2 , on which U is homogeneous of
degree 1, so
F2 (U x) = F2 (x) − 1 for all x ∈ C2 ,
and further, C2 is free when viewed as an F[U ]-module. In this case, for a homogeneous
element x ∈ C1 and a generator y ∈ C2 , the filtration on C1 ⊗F[U ] C2 given by (2.2) is

F (x ⊗ U n y) = F1 (U n x) + F2 (y),

(2.4)

for all n ≥ 0. In other words, to avoid ambiguity, we can think of U as always being applied

23

to the first component (the module on which U is not necessarily homogeneous). In this
case, we can prove the following lemma, which we will use in Section 3.2.
Lemma 2.2.2. Suppose that there is a (Z, U )-filtered chain homotopy equivalence between
C1 and C1 , and C2 is a (Z, U )-filtered chain complex which is freely generated over F[U ].
Suppose also that the maps U on C1 and C1 always decrease the filtration level by at least
k, and the map U on C2 is homogeneous of degree k. Then there is a (Z, U )-filtered chain
homotopy equivalence between C1 ⊗F[U ] C2 and C1 ⊗F[U ] C2 .
Proof. The idea is that, since C2 is freely generated, we can define a map on C1 ⊗ C2 , for
example, by extending a map defined on C1 . Further, because the map U on C1 decreases
the filtration level by at least as much as the map U on C2 , the extended map will still be
filtered.
More precisely, let f : C1 → C1 and g : C1 → C1 be the chain maps which give the
equivalence, and let h and h be the chain homotopies from g ◦ f to IC1 and from f ◦ g to
IC , respectively. We define chain maps f : C1 ⊗ C2 → C1 ⊗ C2 and g : C1 ⊗ C2 → C1 ⊗ C2
1
as follows. Suppose first that y is a generator of C2 as an F[U ]-module, then we set

f (x ⊗ U n y) = f (U n x) ⊗ y

and g(x ⊗ U n y) = g(U n x) ⊗ y,

and extend the maps bilinearly over F. Similarly, for y a generator of C2 , we define maps
h : C1 ⊗ C2 → C1 ⊗ C2 and h : C1 ⊗ C2 → C1 ⊗ C2 by setting

h(x ⊗ U n y) = h(U n x) ⊗ y

and h (x ⊗ U n y) = h (U n x) ⊗ y,

and extending bilinearly.
24

Now we have

g ◦ f (x ⊗ U n y) = g ◦ f (U n x) ⊗ y
= (I + ∂h + h∂)(U n x) ⊗ y
= U n x ⊗ y + ∂h(U n x) ⊗ y + h(∂U n x) ⊗ y
= U n x ⊗ y + ∂h(U n x) ⊗ y + h(∂U n x) ⊗ y
+ [h(U n x) ⊗ ∂y + h(U n x) ⊗ ∂y]
= U n x ⊗ y + (∂h(U n x) ⊗ y + h(U n x) ⊗ ∂y)
+ (h(∂U n x) ⊗ y + h(U n x) ⊗ ∂y)
= (I + ∂h + h∂)(x ⊗ U n y),

and, since the maps are bilinear, we see that g ◦ f is chain homotopic to IC1 ⊗C2 via the
chain homotopy h. By a symmetric argument, f ◦ g ∼ IC ⊗C via h .
2
1
Note also that, for example,

FC ⊗C (f (x ⊗ U n y)) = FC ⊗C (f (U n x) ⊗ y)
2
2
1
1
= FC (f (U n x)) + FC2 (y)
1

≤ FC1 (U n x) + FC2 (y)
= FC1 ⊗C2 (x ⊗ U n y).

In other words, because the maps f, g, h and h are filtered on C1 and C1 , the maps f, g, h
and h are filtered on C1 ⊗ C2 and C1 ⊗ C2 . So, the two tensor product complexes are
Z-filtered chain homotopy equivalent.

25

Finally, we verify that this equivalence is in fact (Z, U )-filtered. By assumption, f U ∼ U f
as maps from C1 to C1 , so there exists a map φ : C1 → C1 such that

f U = U f + ∂C φ + φ∂C1 .
1

In a similar fashion as before, we define

φ : C 1 ⊗ C2 → C1 ⊗ C2

by setting
φ(x ⊗ U n y) = φ(U n x) ⊗ y
when y is a generator of C2 , and extending bilinearly over F.
We then check that

f (U (x ⊗ U n y)) = f (U n+1 x) ⊗ y
= (U f + ∂φ + φ∂)(U n x) ⊗ y
= (U f + ∂φ + φ∂)(U n x) ⊗ y
+ [φ(U n x) ⊗ ∂y + φ(U n x) ⊗ ∂y]
= (U f + ∂φ + φ∂)(x ⊗ U n y).

That is, f U ∼ U f , and by the same reasoning, U g ∼ gU .
This lemma will be the key to simplifying computations for connected sums of knots, for
the complexes which we will define in Chapter 3.

26

Chapter 3
Reducing the knot Floer complex

3.1

The reduced complex

We now turn to the complexes of interest in this thesis. Given a pointed Heegaard diagram
for a three-manifold Y (with k α-curves and k β-curves), Ozsv´ath and Szab´o [54] define a
Z-filtered chain complex CF ∞ (Y, t) for each Spinc structure t on Y . The complex is freely
generated over F[U, U −1 ] by k-tuples of intersection points on the Heegaard diagram. We
will denote the set of generators by G. The filtration level of the homogeneous element U n x
is −n, for any x ∈ G and any n ∈ Z. Adding a second basepoint to the Heegaard diagram
specifies a knot K in Y . Using this additional basepoint (and fixing a Seifert surface for
K), each xi ∈ G can be assigned an integer A(xi ), called the Alexander grading. If we let yi
denote a homogeneous element, then after setting

A(U n xi ) = A(xi ) − n,

A

yi
i

= max{A(yi )},
i

(3.1)

A defines an additional filtration on CF ∞ (Y, t), discovered in [52], and independently by
Rasmussen [64]. The Z ⊕ Z-filtered chain homotopy type of this complex is an invariant of
K ⊂ Y , and in the case Y ∼
= S 3 , this invariant is denoted CF K ∞ (K).

27

We can write the Z ⊕ Z-filtration level of the homogeneous element U n x as

F (U n x) = (−n, A(x) − n)

for any x ∈ G and n ∈ Z. It will be convenient to represent these complexes graphically in
the (i, j)-plane, where U n x will be represented by a dot with coordinates (−n, A(x) − n). If
x and y are two homogeneous elements such that ∂(x, y) = 1, then we will draw an arrow
from the dot representing x to the dot representing y. We should also point out here that
CF K ∞ (K) comes with a homological Z-grading M , called the Maslov grading, and that
multiplication by U decreases M by 2; i.e.,

M (U x) = M (x) − 2.

(3.2)

The difference between the Maslov gradings of two generators can be read from the Heegaard
diagram, and to fix an absolute Maslov grading, we declare that the element 1 in

H∗ (CF K ∞ (K)) ∼
= HF ∞ (S 3 ) ∼
= F[U, U −1 ]

has Maslov grading zero.
Following convention, for a subset S ⊂ Z ⊕ Z, we will denote by C{S} the elements of C
whose (i, j)-coordinates are contained in S, along with the arrows between these elements.
We will often consider the subcomplex C{i ≤ 0}, which is written as CF K − (K) (using the
convention described in Remark 1.4.1). Figure 3.1 shows the complex CF K − (K) in the case
where K is the right-handed trefoil.
Saying that A defines a filtration means in particular that A(∂x) ≤ A(x) for any x, and
28

•o

• x(0)

1

•o

• y(−1)

j=0

•o

•

• z(−2)

•

•





−1
−2



•

−3

Ñ

−3 −2 −1

i=0

Figure 3.1: The knot Floer complex CF K − (T (2, 3)). The generators are represented by
dots in the i = 0 column with their Maslov grading in parentheses, and multiplication by U
translates an element one row down and one column to the left. The arrows represent the
nonzero terms of the differential.
we will call the part of the differential which preserves the Alexander grading the horizontal
differential, denoted ∂H . Diagramatically, ∂H simply consists of those arrows which are
horizontal. For example, in Figure 3.1, we have ∂y = U x + z, and ∂H y = U x. If we restrict
our attention to a single Alexander grading – that is, a single row – of CF K ∞ , we get a
filtered chain complex which has homology isomorphic to F. The “simplest” such filtered
complex would be one which has generators paired into acyclic summands, and a single
isolated generator of homology, which has no arrow going into or out of it. It was shown in [35,
Proposition 11.52] that one can always find such a basis for CF K ∞ (K), called a horizontally
simplified basis. In other words, we can choose a generating set G = {xi , yi , z, |1 ≤ i ≤ N },
such that
• ∂H (yi ) = U ri xi , for some ri > 0,
• ∂H (xi ) = ∂H (z) = 0.
With respect to this basis, the homology of each subquotient complex C{j = k} (each row)
29

is generated by the class U l z (where l differs from k by a constant). All other generators are
paired by horizontal arrows, and can therefore be canceled as in Chapter 2.
We refer back to Figure 3.1 to make one more observation. The basis {x, y, z} shown there
is horizontally simplified, but because we are considering the subcomplex CF K − (K), rather
than all of CF K ∞ (K), it is not only U k z which generates homology: x is also homologically
nontrivial (the horizontal arrow which would cancel it has been “cut off”). Starting with
any nontrivial knot K, if we cancel horizontal arrows in CF K − (K), we are left with some
elements which are eventually canceled for high enough powers of U , but not for lower powers
of U .
We are now ready to construct the object of interest in this section, the reduced knot
Floer complex. It will be convenient to think of CF K − (K) as being (Z, U )-filtered, by the
Alexander grading A, rather than Z ⊕ Z-filtered. So, when we refer to the filtration level of
an element, we will mean its j-coordinate in the diagram (although, for aesthetic reasons,
we will maintain the appearance of different i-coordinates). In this case, the map U is a
filtered chain map of degree 1; i.e., A(U x) = A(x) − 1.
Let K be a knot in S 3 . To simplify notation, let us define C0 := CF K − (K). After
choosing a horizontally simplified basis

{xi , yi , z|1 ≤ i ≤ N },

we will reduce the complex by “canceling” all of the horizontal arrows, as in Chapter 2.
More precisely, let h1 be the F-linear map on C0 which inverts the horizontal differential
going from y1 to x1 , and all of its U -translates, so that h1 (U r1 +n x1 ) = U n y1 for all n ≥ 0,
and h1 is zero on all other homogeneous elements.

30

We now define a filtered chain complex C1 which is freely generated over F as follows.
The generators for C1 over F are obtained from the homogeneous elements of C0 by removing
U r1 +n x1 and U n y1 for all n ≥ 0. Define maps f1 : C0 → C1 and g1 : C1 → C0 by

f1 = π ◦ (I + ∂h1 ),

g1 = (I + h1 ∂) ◦ ι,

where π and ι are the natural projection and inclusion maps. Further, we define a differential
and U map on C1 :

∂1 := π ◦ (∂ + ∂h1 ∂) ◦ ι,

U1 := π ◦ (U + ∂h1 U ) ◦ ι.

(3.3)

The filtration A1 on C1 is induced by inclusion, A1 (x) := A(ιx). Since the maps ∂, U
and h1 are all filtered, so are the maps f1 , g1 , ∂1 , and U1 defined above. Let us consider the
map U1 in more detail. Recall that on the complex C0 , U is a homogeneous map of degree
1. The map U1 , however, will not be homogeneous. There are two cases to consider. For the
generator U r1 −1 x1 (the highest remaining U -power which was not canceled by a horizontal
arrow),

U1 (U r1 −1 x1 ) = (π ◦ (U + ∂h1 U ) ◦ ι) (U r1 −1 x1 )
= π ◦ U r1 x1 + ∂h1 (U r1 x1 )

(3.4)

= π ◦ (U r1 x1 + ∂y1 ).
Since we chose a basis which is horizontally simplified, U r1 x1 is the only term in ∂y1 which

31

has filtration level equal to that of y1 . That is,

A(U r1 x1 + ∂y1 ) < A(y1 ).

It follows from (3.4) then, that

A1 U1 (U r1 −1 x1 ) < A(y1 ) = A1 (U r1 −1 x1 ) − 1,

(3.5)

so the map U1 decreases the filtration level by more than 1. On all other generators of C1 ,
however, the map U1 is equal to π ◦ U ◦ ι, and therefore still decreases the filtration level
by exactly 1. Therefore, the complex C1 is of the type mentioned in Equation (2.3); it is
(Z, U )-filtered, and U is a filtered map which decreases the filtration level by at least one.
Remark 3.1.1. By its definition, the map h1 increases the Maslov grading by 1, and as a
result, the map U1 still lowers the Maslov grading by exactly 2.
Lemma 3.1.2. C1 is a (Z, U )-filtered chain deformation retract of C0 .
Proof. We first verify the chain homotopy equivalence. For any homogeneous element x
for which π(x) = 0,

f1 ◦ g1 (π(x)) = π ◦ (I + ∂h1 )(I + h1 ∂) ◦ ι(π(x))
= π ◦ (I + ∂h1 + h1 ∂)(x)
= π ◦ (x + ∂(0) + h1 (∂x))
= π(x),

since the image of h1 projects to zero. So, f1 ◦ g1 = IC1 .
32

Next, we consider the composition g1 ◦ f1 , in three distinct cases. First, if x is any
homogeneous element for which π(x) = 0, then h1 (x) = 0, so

g1 ◦ f1 (x) = ((I + h1 ∂) ◦ ι) ◦ (π ◦ (I + ∂h1 )) (x)
= (I + h1 ∂)(x)
= (I + h1 ∂ + ∂h1 )(x).

Second, for n ≥ 0, we have

g1 ◦ f1 (U r1 +n x1 ) = ((I + h1 ∂) ◦ ι) ◦ (π ◦ (I + ∂h1 )) (U r1 +n x1 )
(3.6)
= (I + h1 ∂) ◦ ι ◦ π

◦ (U r1 +n x1

+ ∂(U n y

1 )).

Recall that multiplication by U lowers the Maslov grading by 2. Since ∂(U n y1 , U r1 +n x1 ) =
1, it follows that ∂(U n y1 , U k x1 ) = 0 for any k = r1 + n. So, the expression (U r1 +n x1 +
∂(U n y1 )) in (3.6) has no terms of the form U k x1 . Similarly, by considering Maslov gradings,
it also contains no elements of the form U k y1 . Because of this, the composition ι ◦ π in (3.6)
is the identity, so we again get

g1 ◦ f1 (U r1 +n x1 ) = (I + h1 ∂ + ∂h1 )(U r1 +n x1 ).

Finally, g1 ◦ f1 (U n y1 ) = 0 for all n ≥ 0 (since f1 (U n y1 ) = 0). Also,

(I + h1 ∂ + ∂h1 )(U n y1 ) = U n y1 + U n y1 + 0 = 0.

33

So, we have verified that in all cases,

g1 ◦ f1 = I + h1 ∂ + ∂h1 ,

which is to say, g1 ◦ f1 is chain homotopic to the identity.
This shows that C1 is a Z-filtered chain deformation retract of C0 . It remains to check
that this equivalence respects multiplication by U . In most cases, the fact that f1 commutes
with U is immediate, because, in most cases, all maps in the definition of f1 commute with
U . In fact, this is true for every homogeneous element except U r1 −1 x1 , on which h1 and U
do not commute. We verify the claim directly in this case,

f1 U (U r1 −1 x1 ) = π ◦ (U + ∂h1 U )(U r1 −1 x1 )
= π ◦ (U + ∂h1 U ) ◦ ι ◦ π ◦ (U r1 −1 x1 )
= U1 ◦ π ◦ (U r1 −1 x1 )
= U1 ◦ π ◦ (I + ∂h1 )(U r1 −1 x1 )
= U1 f1 (U r1 −1 x1 ).

Similarly, one can verify that g1 U1 is chain homotopic to U g1 via the (filtered) chain homotopy h1 U g1 .
Beginning with C0 , we have now reduced the number of horizontal arrows and obtained
a (Z, U )-filtered chain homotopy equivalent complex C1 . The differential ∂1 is nearly just
π ◦ ∂. The exception being that, if there was an arrow going from a homogeneous element
to U r1 +n x1 , the differential ∂1 adds an arrow from that element to the remaining image of
∂(U n y1 ) (see the discussion in Chapter 2 and Figure 2.1). Note though, that these additional
34

arrows must always decrease the filtration level (in fact, by more than 1). In particular,
the basis given for C1 is still horizontally simplified, with the horizontal differential being
π ◦ ∂H . This means we can iterate the above process, at each step moving from Ci to Ci+1
by canceling the horizontal arrows U n yi → U ri +n xi , and obtaining a (Z, U )-filtered chain
homotopy equivalent complex with a filtered chain map Ui . If we begin with a basis G for
C0 consisting of 2N + 1 elements, then CN will have no horizontal arrows, and will be said
to be a “reduced” version of CF K − (K). An example of this process of reduction is shown
for the (2, 7)-torus knot in Figure 3.2.
The process described above explicitly obtains a reduced complex after a choice of an
ordered, horizontally simplified basis. More generally, we have the following definition.
Definition 3.1.3. Let K be a knot in S 3 , and C be a (Z, U )-filtered chain complex. If C
is (Z, U )-filtered chain homotopy equivalent to CF K − (K), and the differential on C strictly
decreases the filtration level, then C is called the reduced CF K − (K), denoted CF K − (K).
The condition that the differential strictly decreases filtration level says precisely that
there are no horizontal arrows. Of course, this complex is only well-defined up to (Z, U )filtered chain homotopy equivalence of complexes with no horizontal arrows.
Note that, since CF K − (K) is Z-filtered by the Alexander grading, there is naturally a
spectral sequence whose (E0 , d0 ) page is (CF K − (K), ∂H ), which converges to

H∗ (CF K − (K), ∂) ∼
=

F(−2i)

(3.7)

i≥0

as a graded group, where, as in Figure 3.1, the subscript denotes the homological grading of
each generator (the filtration on this group depends on K). An alternative view of the above
construction, then, is that CF K − (K) is the E1 page of this spectral sequence. Further, the
35

A
•

•o

h1
9

h1
9
•o •

•o

h1
9

x1

•

3

y1

U1

2

•

x2

•

1

•o •

y2

•

0

h1
9
•o •

•





•




•o

•

•

x3

•o

•

•o •

y3

•o •

•o

h1
9
•o •











•o

•



•

•o

h2
9
•o •

z



•



•o

•



Ñ

!

•o
•


•o
•

•o
•

•

−1

•o

•

−2

•

•





•

•o

•o
•

−4

C1
•

•

3

U2
"

U3
"

2

•

1

U2
"

U3
"

0

•

•

−1
−2



U3
"

h3
9
•o •

•

•

•



•

•



•
Ñ

−3

−5

•

h3
9



•

C0

h3
9





Ó

•

h2
9

h2
9

h2
9

•
~

C2

−3
−4
−5

C3

Figure 3.2: Reducing a knot Floer complex. The complex C0 = CF K − (T (2, 7)), and
the process of reducing to C3 = CF K − (T (2, 7)). At each step, the map hi provides the
chain homotopy necessary to cancel the horizontal arrows from U n yi to U n+1 xi (the dots
colored red). Multiplication by U is always taken to be translation one down and one to the
left, unless otherwise shown with a dotted arrow.

36

filtered endomorphism U on E0 induces a filtered endomorphism on E1 , and that is precisely
the map U we have defined on CF K − (K). It is the additional information given by this
induced U map which will be useful for computing d-invariants in Chapter 4.

3.2

Sums of knots

As will be seen in this Chapter and Chapter 4, the reduced complex CF K − (K) retains
much of the information contained in CF K − (K), including, by definition, the homology
of its associated graded complex, which is denoted HF K − (K). It was shown by Ozsv´ath
and Szab´o in [52, Theorem 7.1] that CF K − behaves simply under connected sums of knots;
namely,
CF K − (K1 #K2 ) ∼
= CF K − (K1 ) ⊗F[U ] CF K − (K2 ).

(3.8)

However, these tensor product complexes are inconvenient to deal with by hand, even for
sums of knots with small knot Floer homology. Many of the applications of this thesis are
to sums of knots, and so it will be convenient to be able to reduce a complex before taking a
tensor product, in order to decrease the size of the product. The following theorem ensures
that this is possible.
Theorem 3.2.1. If K1 and K2 are knots in S 3 , then

CF K − (K1 ) ⊗F[U ] CF K − (K2 )

(3.9)

is a (Z, U )-filtered chain deformation retract of CF K − (K1 #K2 ).
Proof. After noting the relationship in equation (3.8), the result will follow from Lemma
2.2.2. To verify that the lemma applies, however, we must verify the following.
37

First, the complex CF K − (K2 ) is, by definition, freely generated over F[U ], and multiplication by U decreases the filtration level by 1. Second, recall that the maps Ui defined
as in equation (3.3), decrease the filtration level by at least 1. It follows that the map U on
the reduced complex CF K − (K1 ) also decreases the filtration level by at least 1. In other
words, we define a filtration on the tensor product by formula (2.4), and the result follows
from Lemma 2.2.2.
In order to use this effectively, we would like to further reduce the product complex (3.9),
which we will call C for brevity, to get CF K − (K1 #K2 ). The method of reduction described
explicitly above was for complexes which are freely generated over F[U ], but we should point
out here that a complex such as C can be handled similarly. This is because for sufficiently
negative i, the subcomplex
Ai = {x ∈ C|A(x) ≤ i}
is freely generated over F[U ], and the corresponding quotient

C/Ai = {x ∈ C|A(x) > i}

is finitely generated over F. So, after canceling the finite number of horizontal arrows with
filtration level greater than i, we can use the same method as before.
Remark 3.2.2. Suppose K1 and K2 are knots in S 3 , and we wish to reduce CF K − (K1 #K2 ).
Then we can first reduce CF K − (K1 ), tensor the reduced complex with CF K − (K2 ), and then
further reduce the product.
To give an idea of how this facilitates computation, we consider a simple example, the
sum of the right-handed and left-handed trefoils, T (2, 3)# − T (2, 3). Figure 3.3 shows
38

the knot Floer complexes of these two knots. To obtain CF K − (T (2, 3)# − T (2, 3)), we first
compute CF K − (T (2, 3)) and then tensor it with CF K − (−T (2, 3)). To provide contrast, we
also show in Figure 3.4 the tensor product complex CF K − (T (2, 3)) ⊗F[U ] CF K − (−T (2, 3)).

39

A
•

x

•o •

y



•o •

•

•

•

1

•

b

0

•o •

c

U
!

•
•

z



•o •

a

•

•







•



•o •

•



•o •

•
Ð

CF K − (K1 )

−2

•

Ð

−1

−3

Ð

CF K − (K2 )

CF K − (K1 )
A

•


•
Ø

•

•

•

x⊗a

•

x⊗b

•

1

•

0



Ø

x⊗c

•

ØÑ

•

•



Ø

•

•o •

ØÑ

•

•



Ó

•o •

2

−1
−2
−3

Ð

CF K − (K1 ) ⊗F[U ] CF K − (K2 )

CF K − (K1 #K2 )

Figure 3.3: A connected sum with a reduced complex. In this example, K1 is
the right-handed trefoil and K2 is the left-handed trefoil. We can first reduce to get
CF K − (K1 ), then tensor this complex with CF K − (K2 ). One more reduction gives the
complex CF K − (K1 #K2 ).

40

•


|

• •

•
•

•v •

 |

| 

•v •
| 

"

"

• •v

1
#

• • •
"

• •v •

• •v •
|



 |

|

2

"

• •v
"

•

"

•

"

• •
"

•

0
−1
−2
−3
−4

Figure 3.4: A tensor product without using the reduced complex. The tensor product
complex corresponding to the sum of the right- and left-handed trefoils. Even for knots with
the simplest nontrivial knot Floer complexes, computing with tensor product complexes
becomes tedious.

41

Chapter 4
Applications to L-space knots

4.1

Reduced complexes of L-space knots

In this chapter, we show how the reduced complex CF K − (K) can be used to elucidate some
properties of L-space knots. Recall that a rational homology 3-sphere Y is called an L-space
if – like a lens space – it has the “smallest possible” Heegaard Floer homology; i.e., for each
spinc structure t, HF (Y, t) ∼
= F. A knot K in S 3 is called an L-space knot if n-surgery on S 3
along K is an L-space, for some positive integer n. It was shown in [56, Theorem 1.2 and
Corollary 1.6], and restated more conveniently for our purposes in [25, Remark 6.6], that
L-space knots have knot Floer complexes of a particular form, which we describe here.
Proposition 4.1.1 (Ozsv´ath-Szab´o). If K admits a positive L-space surgery, then CF K − (K)
has a basis {x−k , · · · , xk } with the following properties:
• A(xi ) = ni , where n−k < n−k+1 < · · · < nk−1 < nk
• ni = −n−i
• If i ≡ k mod 2, then ∂(xi ) = 0
• If i ≡ k + 1 mod 2, then ∂(xi ) = xi−1 + U ni+1 −ni xi+1
Diagramatically, the knot Floer complex of an L-space knot has a “staircase” shape, as
shown, for example, in Figure 4.1. The basis described in Proposition 4.1.1 is, in particular,
42

A

•o

•

x2

•o •

x1

•



•o


Ñ

U

1


•

x0

•

•o

U

•

•
•

•

3
2



•

•

•

•o •
•o

•



•

•o

x−1
x−2

•o

•
•



•



•
Ñ

−2
)



•

−1
U

•

•

•
Ó

CF K − (T (3, 4))

0

−3
−4
−5

CF K − (T (3, 4))

Figure 4.1: Reducing an L-space knot complex. At the left is the staircase-shaped knot
Floer complex for the (3,4)-torus knot, and at the right is its reduction (with an intermediate
step shown in between). Each time we cancel the horizontal arrow from x2−2i−1 to x2−2i ,
we see that U takes the last remaining power of x2−2i to x2−2i−2 , so, in fact, every generator
which remains in CF K − (T (3, 4)) can be written as U k x2 for some k ≥ 0.
horizontally simplified, and so the method of reduction will proceed exactly as in Chapter
3. We include a proof of the corollary below, although the result should be more readily
evident by seeing the reduction in Figure 4.1.
Corollary 4.1.2. If K is an L-space knot, then CF K − (K) has exactly one generator of
Maslov grading −2i for each i ≥ 0, and no other generators. Further, if x is the generator
with Maslov grading −2i, and y is the generator with Maslov grading −2i − 2, then U x = y.
Proof. We will first show that if we use a basis as in Proposition 4.1.1, and proceed with
the reduction as described in Chapter 3, then we get a representative of CF K − (K) with the
desired properties. We will then show that, in fact, any representative must be isomorphic
to this one.
43

First, we consider the subquotient complex

CF K(K) := C{i = 0}.

This is a filtered complex which is chain homotopy equivalent to CF (S 3 ), and so its homology
is generated by a single element. To fix an absolute Maslov grading, this generator of the
homology of CF (S 3 ) is declared to have Maslov grading zero. It is clear from the explicit
differential given in Propostion 4.1.1 that, when K is an L-space knot, xk generates the
homology of CF K(K), and so M (xk ) = 0.
With this as our starting point, we now consider what happens through reduction. To
simplify notation, we will define
ri := ni − ni−1
to be the difference in the Alexander gradings of xi and xi−1 . We first cancel the horizontal
arrows which form the top steps of the staircases, those corresponding to

∂H (U m xk−1 ) = U m+rk xk

for m ≥ 0. Note that after canceling U rk xk , we have

U1 (U rk −1 xk ) = (π ◦ (U − ∂h1 U ) ◦ ι) (U rk −1 xk )
= π ◦ (U rk xk − ∂h1 (U rk xk ))
= π ◦ (U rk xk − ∂xk−1 )
= xk−2

44

That is, after canceling higher U -powers of xk , the map U takes the highest remaining power,
U rk −1 xk , to the next generator, xk−2 . The exact same argument applies to the cancellation
of the horizontal arrows
∂H (xk−3 ) = U rk−2 xk−2 ,
and we see that
U2 (U rk−2 −1 xk−2 ) = xk−4 .
We proceed in this fashion, until finally we see that

Uk (U r−k+2 −1 x−k+2 ) = x−k .

It follows that each of these generators of CF K − (K) can be written as

U i xk

for some i ≥ 0. Since multiplication by U lowers the Maslov grading by 2, the result follows.
Now let us call the reduced complex just constructed C, and suppose that C is a (Z, U )filtered chain homotopy equivalent complex which also has no horizontal arrows (i.e., a
different representative of CF K − (K)). Since it is filtered chain homotopy equivalent, each
subquotient complex C {j = k} must have homology isomorphic to that of C{j = k}, which
is either 0 or F, depending on k. Since there are no horizontal arrows, these subquotient
complexes have trivial differential, so they either have no generators or they have exactly 1
generator, with even homological grading. Therefore, C also has trivial differential, so it is
isomorphic to C as a filtered chain complex. The fact that the equivalence is (Z, U )-filtered

45

implies that U also takes generator to generator for C as it does for C, so in fact they are
isomorphic as (Z, U )-filtered complexes.
A concise way of stating Corollary 4.1.2 is that, for an L-space knot K1 ,

CF K − (K1 ) ∼
= F[U ](0) ,

(4.1)

where the subscript here means that the generator has Maslov grading zero. Necessarily,
this complex also has trivial differential. This means that a tensor product of the form

CF K − (K1 ) ⊗F[U ] CF K − (K2 )

(4.2)

will be isomorphic to CF K − (K2 ) as a chain complex, which we will make use of below. It
is important to note however, that the isomorphism (4.1) is not filtered (when K1 is not
the unknot). As a result, the tensor product (4.2) is not (Z, U )-filtered chain homotopy
equivalent to CF K − (K2 ).
Knowing that L-space knots must have this particularly simple reduced knot Floer complex, we make use of the behavior under connected sums to record the following observation.
Theorem 4.1.3. A knot in S 3 which admits an L-space surgery must be a prime knot.
The general argument, a proof by contradiction using Corollary 4.1.2, will be suggested
by considering an example, so we refer back to Figure 3.3. Consider the generator of maximal
Alexander grading in CF K − (K1 ), which we labeled x. We have A(x) = 1, and so

A(U x) = 0 = A(x) − 1.

46

But U x is canceled by a horizontal arrow, so when we move to CF K − (K1 ), we have

A(U x) = −1 < A(x) − 1

That is, multiplication by U “jumps” down by more than 1. When we take the tensor
product
CF K − (K1 ) ⊗F[U ] CF K − (K2 ),
this has the effect of “bending downward” the horizontal arrow from c to U b. That is, the
arrow from x ⊗ c to U x ⊗ b is not horizontal. Therefore, when we reduce this tensor product
complex, we cannot cancel these generators, and so they both remain. But, being connected
by an arrow, their Maslov gradings differ by exactly 1, so the complex CF K − (K1 #K2 ) does
not have the form described in Corollary 4.1.2. With this example as motivation, we provide
the details.
Proof of Theorem 4.1.3. We begin by noting that if negative surgery on a knot produces
an L-space, then positive surgery on its mirror image produces an L-space, so by definition
its mirror image is an L-space knot. Since a knot is a nontrivial connected sum if and only
if its mirror image is, it will be sufficient to show that positive surgery on a connected sum
can never produce an L-space. That is, we will show that no nontrivial connected sum is an
L-space knot.
Suppose K1 and K2 are two nontrivial knots in S 3 . Let Ci denote the complex CF K − (Ki ).
Choose a horizontally simplified basis for C1 , where y1 and x1 are generators such that
∂H (y1 ) = U r1 x1 . Likewise, choose a horizontally simplified basis for C2 , where Y1 and
X1 are generators such that ∂H (Y1 ) = U R1 X1 . Without loss of generality, assume that

47

r1 ≤ R1 . Our goal is to show by contradiction that CF K − (K1 #K2 ) is a complex which
cannot correspond to an L-space knot, by Corollary 4.1.2.
We will make use of Remark 3.2.2, and begin by reducing C1 to get a complex C1 . We
will denote the filtration on this reduced complex by F1 and the filtration on C2 by F2 .
Finally, the filtration on the tensor product C1 ⊗F[U ] C2 , defined as in equation (2.4), will
be denoted F .
After reducing C1 , we have

F1 (U r1 −1 x1 ) = F1 (x1 ) − (r1 − 1),

but the map U on U r1 −1 x1 lowers the filtration level by at least two, so

F1 (U r1 x1 ) < F1 (x1 ) − r1 .

As a consequence, since R1 ≥ r1 , and U always lowers the filtration level by at least 1,

F1 (U R1 x1 ) < F1 (x1 ) − R1 .

Note also that, on the freely generated complex C2 , U is a homogeneous map of degree 1, so

F2 (Y1 ) = F2 (X1 ) − R1 .

48

It follows that

F (U R1 x1 ⊗ X1 ) = F1 (U R1 x1 ) + F2 (X1 )
< F1 (x1 ) − R1 + F2 (X1 )
= F1 (x1 ) + F2 (Y1 )
= F (x1 ⊗ Y1 ).

Because of this, every term in

∂(x1 ⊗ Y1 ) = ∂x1 ⊗ Y1 + x1 ⊗ ∂Y1

has filtration level strictly less than that of x1 ⊗ Y1 . That is to say, ∂H (x1 ⊗ Y1 ) = 0. When
we reduce the tensor product complex, there is no horizontal differential to cancel x1 ⊗ Y1 ,
so it will project to a nonzero homogeneous element in CF K − (K1 #K2 ). Of course, x1 ⊗ X1
also projects to a nonzero homogeneous element in CF K − (K1 #K2 ). But,

M (x1 ⊗ Y1 ) = M (x1 ) + M (Y1 )
= M (x1 ) + M (U R1 X1 ) + 1
= M (x1 ) + M (X1 ) + 1 − 2R1
= M (x1 ⊗ X1 ) + 1 − 2R1 ,

so CF K − (K1 #K2 ) has two elements with Maslov gradings of opposite parity. By Corollary
4.1.2, K1 #K2 cannot be an L-space knot.

49

4.2

Correction terms and concordance

We now turn to our second application, pertaining to the Heegaard Floer correction terms,
or d-invariants. Given a rational homology three-sphere Y and spinc structure t, we obtain a
chain complex CF ∞ (Y, t) which is freely generated over F[U, U −1 ], as described in Chapter
3, and its associated subcomplex CF − (Y, t). The homology of this subcomplex, denoted
HF − (Y, t), consists of a direct summand isomorphic to F[U ], and possibly other terms
which are U -torsion. The correction term associated to (Y, t), denoted d(Y, t), is simply the
maximal Maslov grading of any nontorsion generator in HF − (Y, t).
Given a knot K in S 3 , let S13 (K) denote the integer homology sphere obtained from S 3
by doing Dehn surgery along K with slope 1. We can associate to this manifold a number,
d(S13 (K), t) (where t is the unique spinc structure on S13 (K)), which we will abbreviate as
d1 (K). This invariant was studied by Peters in [61], where it was shown to have the following
properties.
Proposition 4.2.1 (Theorem 1.5 and Proposition 2.1 of [61]). For any knot K in S 3 ,
• d1 (K) is an even integer
• d1 (K) is a concordance invariant of K
• If we denote by g4 (K) the smooth four-dimensional genus of K,

0 ≤ −d1 (K) ≤ 2g4 (K).

In addition, Peters gave an algorithm to compute d1 (K) from CF K ∞ (K), using the
fact that CF K ∞ (K) contains all the information needed to compute the Heegaard Floer

50

homology of manifolds arising from surgery on K. We briefly recount the idea here. For
details, see [61].
In [49, Lemma 7.11], the degrees of the maps in the integer surgery exact sequence

3 (K)) → HF + (S 3 ) → · · ·
· · · → HF + (S03 (K)) → HF + (SN

were computed, from which it was concluded in [61, Sec. 5] that1

3 (K), s ) −
d1/2 (S03 (K)) = d(SN
0

N −3
.
4

(4.3)

In particular, we have
3 (K), s ) −
d1 (K) = d(SN
0

N −1
,
4

(4.4)

3 (K), s ). For N sufficiently large, this can be
so the invariant d1 (K) is determined by d(SN
0

computed directly from CF K ∞ (K).
Let A+
0 denote the quotient complex

C{i ≥ 0 or j ≥ 0}

of CF K ∞ (K) (recall that C{S} denotes the elements with (i, j)-coordinates in S, and the
arrows between these elements). Ozsv´ath and Szab´o [52, Corollary 4.2] (c.f., [64]) show that,
1 For three-manifolds with H (Y ) ∼
= Z, Ozsv´ath and Szab´o define d±1/2 (Y ) to be the minimal grading of
1
+
an element in HF (Y, s0 ) which is in the image of U k for all k > 0 whose grading is additionally congruent
to ±1/2 mod 2, where s0 is the unique spinc -structure for which c1 (s0 ) = 0.

51

for any sufficiently large positive integer N ,

3 (K), s ) ∼ H (A+ ).
HF + N −1 (SN
0 = l 0
l+

(4.5)

4

That is, up to a shift in grading which depends on N , the homology of this complex is the
Heegaard Floer homology of the three-manifold obtained by surgery. Combining equations
(4.4) and (4.5), we see that the grading shifts cancel nicely, and d1 (K) is equal to the
k
minimum grading of a generator of H∗ (A+
0 ) which is in the image of U for all k > 0.

Remark 4.2.2. It should be pointed out that, by (4.3), we could get the same information
3 (K), s ), which is also a concordance invariant, as we get from
from the invariant d(SN
0

d1 (K). The choice N = 1 is a matter of convenience, because it gives a four-genus bound
without any shift.
We will find it convenient to work with the subcomplex CF K − (K) rather than the
quotient CF K + (K). From this point of view, d1 (K) is the maximum grading of a nontorsion generator of homology of the subcomplex

C{i ≤ 0 and j ≤ 0}.

Remark 4.2.3. To justify this this point of view, we first point out that Ozsv´ath and Szab´
o
define d− (Y, t) to be the maximal grading of a non-torsion generator in HF − (Y, t), and
observe in the proof of [49, Prop. 4.2] that

d− (Y, t) = d(Y, t) − 2

52

(0) • o

•

(−2) • o

•

•o

•

(−4) • o

•

•o

•

•

•

•o

•

•

•

•

(−6) • o



•o



2







−1

j=0

−2

•
−2

1

−1



~

−3





g

−3
i=0

1

2

Figure 4.2: Computing d1 from the knot Floer complex.
The complex
∞
CF K (T (2, 5)). The invariant d1 (T (2, 5)) can be seen to equal −2, either by considering the maximal grading of a generator of homology of the subcomplex C{i ≤ 0 and j ≤ 0}
(below and to the left of the dashed line); or by considering the minimal grading of a generator of homology of the quotient complex C{i ≥ 0 or j ≥ 0} (above or to the right of the
dotted line). We will find it easier to work with the subcomplex through the rest of this
thesis.
(recalling that U lowers grading by 2). But our definition of CF − differs from Ozsv´ath and
Szab´o’s by a shift by U −1 (see Remark 1.4.1). So, the maximal grading of a non-torsion
generator of our CF − (Y, t) is

d (Y, t) = d− (Y, t) + 2 = d(Y, t),

so we will think of the d-invariant this way.
Figure 4.2 shows how d1 can be computed from the knot Floer complex in the case of
the (2,5)-torus knot.

53

There is another concordance invariant which comes from the knot Floer complex, τ (K),
which was introduced by Ozsv´ath and Szab´o in [50], where they showed that it also gives a
lower bound for the smooth four-dimensional genus of K,

|τ (K)| ≤ g4 (K).

Given a knot Floer complex, this invariant is easily computed, and yet has been shown to
be a quite powerful four-genus bound. For example, its value on torus knots, shown in [50,
Corollary 1.7] to be
τ (T (p, q)) =

(p − 1)(q − 1)
,
2

(4.6)

was used to provide an alternate proof of the Milnor conjecture, which says that this is in fact
the four-genus of T (p, q) (see [42]). The invariant is defined from CF K ∞ (K) by considering
the subquotient complex
CF K(K) := C{i = 0}.
If we let ιk be the inclusion map

ιk : C{i = 0, j ≤ k} → C{i = 0},

we get an induced map on homology

(ιk )∗ : H∗ (C{i = 0, j ≤ k}) → H∗ (C{i = 0}).

This map is clearly an isomorphism for large enough k, and the zero map for sufficiently

54

negative values of k (since the complex is finitely generated). We can then define

τ (K) := min{k|(ιk )∗ is non-trivial}.

This quantity is additive under tensor products of complexes, and therefore τ defines a
homomorphism from the smooth concordance group to Z.
It follows from Proposition 4.1.1 that, for an L-space knot K,

τ (K) = A(xk ) = max{j|HF K(K, j) = 0},

(which is also the Seifert genus of K). In general, it was shown in [52] that

χ HF K(K, j) · T j = ∆K (T ),

(4.7)

j

where ∆K (T ) is the symmetrized Alexander polynomial of K. Since, for an L-space knot,
we can choose a basis for which the rank of CF K(K, j) is either 0 or 1 for each j, the rank
of each subcomplex is determined by its Euler characteristic, so, by Proposition 4.1.1, the
knot Floer complex contains the same amount of information as the Alexander polynomial.
In particular, τ (K) = deg ∆K (T ). That, however, is all the information τ can give in this
case. The statement that τ (K1 # − K2 ) = 0 for two L-space knots K1 and K2 is precisely
the statement that their Alexander polynomials have equal degree. In contrast, the next
theorem gives a sense in which the invariant d1 is more sensitive.
Theorem 4.2.4. Suppose that K1 and K2 are two knots in S 3 which admit positive L-space
surgeries. If
d1 (K1 # − K2 ) = d1 (−K1 #K2 ) = 0,
55

•oα •
1

αn

•o α
2

•o

β1

•


βm

•o

•
• o αn •


β2

•
•o

α1

βm

•

•
β
 1

•

K1

K2

Figure 4.3: Labeling staircase complexes. The staircases for K1 and K2 .
then
∆K1 (T ) = ∆K2 (T ).
In particular, the Alexander polynomial is a concordance invariant of L-space knots.
Proof. As mentioned above, in light of Proposition 4.1.1, the Alexander polynomial of Ki
gives equivalent information to the knot Floer complex of Ki , which we will represent by its
staircase shape. We can represent a staircase by listing the horizontal lengths in order from
left to right. By the symmetry of the Alexander polynomial, this list is also the list of vertical
lengths, in order from bottom to top. Suppose that K1 has staircase {α1 , α2 , . . . , αn }, and
K2 has staircase {β1 , β2 , . . . , βm }, as shown in Figure 4.3, and also that

d1 (K1 # − K2 ) = d1 (−K1 #K2 ) = 0.

The proof will proceed by showing first that the Alexander polynomials must have equal
degrees, then, one step at a time, that αi = βi for all i (and consequently, that m = n).
The complex CF K − (−K2 ) will take the shape of an “upside down staircase”, since
−K2 is the mirror image of an L-space knot (see Figure 4.4). In this case, τ (−K2 ) =
− deg ∆K2 (T ). We record here a particularly useful property of these complexes, which
56

•o

β1

•


•
β
 1

βm

•o

β2

•o

•
•o

βm

βm

•o

•


•

β1

β2

•

•
βm
•o
β1


•

−K2

K2

Figure 4.4: An upside-down staircase. The knot Floer complex of the L-space knot K2 ,
and that of its mirror image.
follows from direct inspection.
Remark 4.2.5. If K is an L-space knot, then the knot Floer complex of its mirror, CF K − (−K),
has a basis for which it satisfies the following:
• CF K − (−K) splits into a direct sum of complexes C2i , for each integer i, where C2i
consists of the homogeneous elements of Maslov gradings 2i and 2i − 1
• for i > 0, the complex C2i is acyclic
• for i ≤ 0, the complex C2i has homology isomorphic to F, generated by the sum of all
homogeneous elements of Maslov grading 2i
If K1 is an L-space knot, then, by Corollary 4.1.2, the complex CF K − (K1 ) is “almost”
the knot Floer complex of the unknot. More precisely, each is isomorphic to F[U ], supported
in grading zero, with the only difference being that, for the unknot, U is homogeneous
of degree one, while for K1 , U is a non-homogeneous map which decreases the Alexander
grading by at least one. It follows that the tensor product complex

CF K − (K1 ) ⊗F[U ] CF K − (−K2 ),
57

•
•
•
•
•
•



•o



•o

•

•o

•o

•





•o •

•

2
•


•

0

•

−2

•o •

−3



•



 Ð

•

•
•

−5
•
•


•o

•



•o

•
•

•

 Ñ

•
 Ñ



•

1

•

0

•

−1

•

−2

•

−3

•

−4
−5

•o •

−6



Ñ

 Ñ

3

•

•


•

 x

•o



 Ñ

•

5

2

•

−4

6

4

•

−1


•

•

3

1
•

•o •
Ñ

•









•o •

−7

Figure 4.5: The difference of two L-space knots. On the left, the complex of the
mirror image knot CF K − (−T (3, 4)). On the right, the complex CF K − (T (2, 7)) ⊗F[U ]
CF K − (−T (3, 4)). Note that it retains the same “staircase” form, but some horizontal arrows
get bent downward.
which we will denote by C, is “almost” an upside down staircase; the only difference being
that some of its “stairs” have been bent. Figure 4.5 shows an example of this, the complex
CF K − (T (2, 7)) ⊗F[U ] CF K − (−T (3, 4)) (recall that the reduced complex CF K − (T (2, 7)) is
shown in Figure 3.2).
Remark 4.2.6. In particular, C still has the properties in Remark 4.2.5, splitting into
summands C2i . Since the generator of homology of C0 is the sum of all homogeneous elements
with Maslov grading 0, its Alexander grading is the maximum of the Alexander gradings of
all of these elements (see Equation (3.1)). So, in this case, the d1 -invariant is zero if and
only if all of the elements with Maslov grading zero have Alexander grading less than or equal

58

to zero.
We will now see how the Alexander grading on the C0 summand is determined from the
shape of the staircases; i.e., from the αi ’s and βi ’s. Let us choose generators {x−n , . . . , xn }
for CF K − (K1 ) as in Proposition 4.1.1. Then the generators for CF K − (K1 ) are

xn−2i , U xn−2i , . . . , U αi+1 −1 xn−2i

for every 0 ≤ i ≤ n − 1, and U k x−n , for all k ≥ 0. Further, as in the proof of Corollary
4.1.2, we have that

U (U αi+1 −1 xn−2i ) = xn−2(i+1)

for all 0 ≤ i ≤ n − 1

We should also point out that if 0 ≤ k < αi+1 ,

A(U k xn−2i ) = τ (K1 ) − k −

(αj + αn+1−j ),
1≤j≤i

and
M (U k xn−2i ) = −2k −

2αj .
1≤j≤i

For CF K − (−K2 ), we choose a basis {y−m , . . . , ym }, so that

A(y−m+2k ) = τ (−K2 ) +

(βm+1−j + βj )
1≤j≤k

and
M (y−m+2k ) =

2βj .
1≤j≤k

59

(4.8)

We now consider each generator in C0 which has Maslov grading zero, and see what
restrictions we get on the αi ’s and βi ’s by assuming it has Alexander grading less than or
equal to zero. The first generator we consider is xn y−m , and we have that

A(xn y−m ) = A(xn ) + A(y−m )
= τ (K1 ) + τ (−K2 )
= τ (K1 ) − τ (K2 ).

In order for this to be less than or equal to zero, we must have τ (K1 ) ≤ τ (K2 ). On the
other hand, considering instead the knot −K1 #K2 , the same argument says we must also
have τ (K2 ) ≤ τ (K1 ), so τ (K1 ) = τ (K2 ), and the Alexander polynomials of K1 and K2 must
have equal degree.
The rest of the proof proceeds similarly. We next consider the generator U β1 xn y−m+2 .
If α1 > β1 , then

A(U β1 xn y−m+2 ) = τ (K1 ) − β1 + τ (−K2 ) + β1 + βm > 0,

so, in order to have d1 = 0, we must have α1 ≤ β1 . Again, considering −K1 #K2 , we must
also have β1 ≤ α1 , so α1 = β1 . Since α1 = β1 , U β1 xn = xn−2 , so

A(U β1 xn y−m+2 ) = A(xn−2 y−m+2 )
= τ (K1 ) − α1 − αn + τ (−K2 ) + β1 + βm
= − αn + βm .

60

This means we must also have αn ≥ βm ; and once again considering −K1 #K2 , we see that
in fact αn = βm .
We have to this point shown that the first elements of the lists representing these two
staircases agree, and also that the last elements agree. Taking this as our base case, we will
now work our way inductively toward the middle.
To that end, assume that αi = βi and αn+1−i = βm+1−i , for all 1 ≤ i ≤ k. Then
consider the generator

U β1 +β2 +···+βk+1 xn y−m+2k+2 = U βk+1 xn−2k y−m+2k+2 .

If αk+1 > βk+1 , then

A(U βk+1 xn−2k y−m+2k+2 ) = A(U βk+1 xn−2k ) + A(y−m+2k+2 )
= τ (K1 ) − βk+1 −

(αj + αn+1−j )
1≤j≤k

− τ (K2 ) +

(βj + βm+1−j )
1≤j≤k+1

= βm−k
> 0,

so it must be that αk+1 ≤ βk+1 ; considering −K1 #K2 gives αk+1 = βk+1 .
Since αk+1 = βk+1 , U βk+1 xn−2k = xn−2k−2 , so

61

A(U βk+1 xn−2k y−m+2k+2 ) = A(xn−2k−2 y−m+2k+2 )
= τ (K1 ) −

(αj + αn+1−j )
1≤j≤k+1

− τ (K2 ) +

(βj + βm+1−j )
1≤j≤k+1

= − αn−k + βm−k

This means αn−k ≥ βm−k , and as before, we see that αn−k = βm−k , which completes
the inductive step. A priori, n may not be equal to m, but this induction can be continued
for all i until either αi or βi does not exist. That is, until we exceed the minimum of n
and m. Assume, without loss of generality, that it is n. Upon reaching that point, we have
αi = βi , for 1 ≤ i ≤ n. But since the Alexander polynomials have equal degree,

βi ,

αi =
1≤i≤n

1≤i≤m

so n and m must be equal.
Figure 4.5 shows the example of the sum T (2, 7)# − T (3, 4). In Figure 4.6, it is exhibited
that
d1 (T (2, 7)# − T (3, 4)) = −2,
although
τ (T (2, 7)# − T (3, 4)) = 0.
This is an instance of a general fact which follows from Theorem 4.2.4.

62

•


•
•


•
 Ò

•

•
•
•


•o

 z





•
•

•

 Ó

 Ó

 Ó



•

1

•

0

•

−1

•

−2

•

−3

•

−4

•

−5

•o •

−6

•


Ó

•

•

•o

•o

 Ó

•

3
2

•
•

5
4

•
•

6



•o •

−7

Figure 4.6: Computing d1 of the difference of two L-space knots. The complex
CF K − (T (2, 7)) ⊗F[U ] CF K − (−T (3, 4)). Multiplication by U takes staircase to staircase,
but we suppress the dotted arrows to avoid obscuring the picture. Although τ T (2, 7)# −
T (3, 4) = 0, the summand C0 (shaded red) has its generator of homology with Alexander
grading 1 (above the dashed line), so d1 T (2, 7)# − T (3, 4) = 0 (in fact, d1 = −2).

63

Corollary 4.2.7. If K1 and K2 are two L-space knots whose Alexander polynomials are
distinct but have the same degree, then

τ (K1 # − K2 ) = τ (−K1 #K2 ) = 0,

but either
d1 (K1 # − K2 ) = 0

or

d1 (−K1 #K2 ) = 0.

In particular, d1 gives a stronger four-genus bound than τ for K1 # − K2 and its mirror.
For a particular class of L-space knots, Theorem 4.2.4 can be strengthened. Let f :
C2 → C be an irreducible polynomial such that f (0) = 0. Let Z be the curve in C2 through
the origin which is the zero set of f , and let B be a small ball containing the origin in C2 .
Then K = Z ∩ ∂B is called an algebraic knot. Algebraic knots are torus knots or, more
generally, iterated cables with cabling parameters (pk , qk ) satisfying qk > pk pk−1 qk−1 [32].
It was shown by Lˆe [32] that algebraic knots are determined up to isotopy by their Alexander
polynomial. With this result, we can provide an altenate proof of a corollary proved by Lˆe.
Corollary 4.2.8 (Lˆe). Algebraic knots are concordant if and only if they are isotopic.
Proof. It was shown by Moser [43] that torus knots are lens space knots, and more generally
it follows from work of Hedden [21] that the iterated cables specified by Lˆe are L-space knots.
By Theorem 4.2.4, if two L-space knots are concordant, they have equal Alexander polynomials. Lˆe showed that the Alexander polynomial determines the knot type for algebraic
knots.

64

4.3

Examples

Example 1. The example illustrated in Figure 4.6 can be generalized to the knots

Kp := T (2, p(p − 1) + 1) # − T (p, p + 1).

By examining the Alexander polynomials of torus knots, it can be seen that the lengths of
the staircase for T (p, p + 1) are {1, 2, · · · , p − 1}, whereas the staircase for T (2, q) has q−1
2
steps, all of length 1.
From this it can be seen (see Figure 4.7 for an example) that the generator of homology
of the complex C0 has Alexander grading
k

(p − i) − i

A = max
k

i=1

p
2

(4.9)

p − 2i

=
i=1

=

p
2

p−

p
−1 .
2

In general, showing that the Alexander grading of this generator is positive only shows that
d1 ≤ 0, but in this case we can get an explicit value with relative ease. Roughly speaking,
this is because the map U on CF K − (T (2, p(p − 1) + 1)) decreases the Alexander grading by
2 (at least on elements with high enough Maslov grading), and of course also decreases the

65

•
•

•
•

 Ð

•

 Ñ

•
•

•

2
 Ð

•

0

•
•

•
 x
•

4

 Ó

•
•
 Ò
•

6

 Ó

•

Ô

•

−2

 Ó

•

•

−4
 Ð

•

−6

•

Figure 4.7: Complexes for the knot Kp . A portion of the complex CF K − (T (2, 31))⊗F[U ]
CF K − (−T (6, 7)). The upper summand shown is C0 , and its generator of homology has
Alexander grading 6. The generators of homology for C−2 and C−4 (not shown) are also
above the dashed line. The generator of homology of the summand C−6 , the lower summand
shown, has the maximal Maslov grading of any generator below the dashed line, so d1 (K6 ) =
−6.
Maslov grading by 2. So, in fact, the grading in (4.9) is exactly −d1 . That is, for any p > 1,

τ (Kp ) = 0,

but




2


− p −2p
p even,

4




2
d1 (Kp ) = − p−1
p ≡ 1 mod 4,
2






2


− p−1
− 1 p ≡ 3 mod 4.
2

(4.10)

It should be pointed out that while d1 has more to say than τ for these knots, the knot
signature σ gives an even better topological four-genus bound (at least for p > 5).
Example 2. Even among sums of torus knots, however, we can find examples for which τ ,
σ and Rasmussen’s s invariant defined using Khovanov homology [65] are all equal to zero,

66

and |d1 | is arbitrarily large (for all knots discussed in this thesis, s = 2τ ). Define

Jp+ := T (2, 8p + 1)#T (4p, 4p + 1),

and then let
Jp := Jp+ # − T (4p + 1, 4p + 2).
A direct computation using (4.6) and, for example, [19, Theorem 5.2] shows that

τ (Jp ) = σ(Jp ) = 0 for all p > 0.

The staircases of the individual torus knot summands here have the type described in
Example 1. The sum of two L-space knots, as we have seen, is not an L-space knot. However,
its reduced complex has an acyclic subcomplex which is U -torsion, and the corresponding
quotient complex is isomorphic to F[U ]. Since U -torsion elements are not relevant to the
computation of d-invariants, this means we can treat the sum of staircases as a staircase, if
it is only the d-invariants we are interested in (see [2, Section 5] and [3, Section 2.4], where
Borodzik and Livingston discuss the gap functions of connected sums of algebraic knots, for
an alternate point of view). That is to say, this quotient complex is filtered isomorphic to
the reduced complex corresponding to some staircase, which we may call the “representative
staircase”.
If one of the summands is T (2, n), the resulting representative staircase can be obtained
relatively simply. In the case at hand, the representative staircase for Jp+ is given by
2p2 +5p

2p−2

2p−3

{1, · · · , 1, 3, 1, · · · , 1, 5, 1, · · · , 1, · · · , 4p − 5, 1, 1, 4p − 3, 1, 4p − 1}.
67

Figure 4.8: A representative staircase. On the left is the staircase for the torus knot
T (8, 9) (the horizontal lengths range from 1 to 7, in order). On the right is the representative staircase corresponding to the knot T (2, 17)#T (8, 9), which we have called J2+ . This
representative staircase contains all of the generators which are relevant for computing dinvariants. Here, every point where two line segments meet is assumed to be a generator (a
“dot”), and every line segment is assumed to be an arrow pointing down or to the left.
An example of this staircase is shown in Figure 4.8, for the case p = 2. Recall that, by
the symmetry of the Alexander polynomials of the summands of Jp+ , this is also the list of
vertical lengths, from bottom to top. With this, and also knowing the upside-down staircase
shape of CF K − (−T (4p + 1, 4p + 2)), we can compute the Alexander gradings as we did
above. This allows us to see that, for all p > 0,

d1 (Jp ) = −2p.

(4.11)

We suppress the explicit computations here, but instead show the p = 2 case in Figure 4.9.
There is, more generally, a family of knots whose knot Floer complexes are the direct sum
of a staircase and an acyclic complex. In addition to L-space knots, Petkova [62, Lemma
68

×

4
3
2
1
0
−1
−2

Figure 4.9: Complexes for the knot Jp . The portion of the complex CF K − (Jp ) which is
relevant for computing d1 (Jp ), in the case where p = 2. The line segments are interpreted as
in Figure 4.8. The uppermost summand is C0 . From right to left, its generators start with
an Alexander grading of 0, and increase by 1 until reaching an Alexander grading of 2p (the
generator marked with an ×). Multiplying by U takes staircase to staircase, and notice that
the Alexander grading of the × generator decreases by 2 each time. This is the basic idea
behind showing that d1 (Jp ) = −2p.

69

7] showed that “Floer homologically thin” knots – which include alternating and quasialternating knots, as well as a family of hyperbolic knots found in [20] – are in this family
(the staircase for a Floer homologically thin knot is the same as that of some (2, n)-torus
knot). Since d-invariants are defined in terms of non-torsion generators of homology, the
acyclic summands have no effect on d1 ; it is determined solely by the staircase summand.
Therefore, if K1 and K2 are knots in this family, then d1 (K1 # − K2 ) can be computed as
it was in the above examples for torus knots. In particular, if the staircases of K1 and K2
have different shapes, these knots are not concordant.
As an interesting further application of these ideas, one could investigate the linear
independence of a family of knots in the smooth concordance group. As an example, if we
let T (r, s)p,q denote the (p, q)-cable of T (r, s), the knots

K1 = T (2, 3)2,3 #T (2, 5) and K2 = T (2, 3)2,5 #T (2, 3)

can be shown to be linearly independent using d1 , although ∆K1 (T ) = ∆K2 (T ) (showing
that Corollary 4.2.7 does not extend to sums of L-space knots). In contrast, the knots

T (2, 3)2,13 #T (2, 15) and T (2, 3)2,15 #T (2, 13)

cannot be distinguished in the concordance group using d1 (that is, d1 cannot obstruct the
sliceness of the “Livingston-Melvin” knot [37]). So, while Theorem 4.2.4 settles the question
of when two L-space knots are concordant, it would be interesting to understand which
families of L-space knots can be shown to be independent using the d1 invariant.

70

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71

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