HEEGAARD FLOER HOMOLOGY AND L-SPACE KNOTS
By
Faramarz Vafaee

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

ABSTRACT
HEEGAARD FLOER HOMOLOGY AND L-SPACE KNOTS
By
Faramarz Vafaee
Heegaard Floer theory consists of a set of invariants of three- and four-dimensional manifolds. Three-manifolds with the simplest Heegaard Floer invariants are called L-spaces, and
the name stems from the fact that lens spaces are L-spaces. The overarching goal of the
dissertation is to understand L-spaces better. More specifically, this dissertation could be
considered as a step towards finding topological characterizations of L-spaces and L-space
knots without referencing Heegaard Floer homology. We study knots in S 3 that admit positive L-space Dehn surgeries. In particular, we give new examples of knots in S 3 within
both the families of hyperbolic and satellite knots admitting L-space surgeries. It should
be pointed out that for satellite knot examples, we use Berge-Gabai knots (i.e. knots in
S 1 × D 2 with non-trivial solid torus Dehn surgeries) as the pattern. Moreover, we study the
relationship between satellite knots and L-space surgeries in the general setting, i.e. when
the pattern is an arbitrary knot in S 1 × D 2 .

To my parents, Fatemeh Vakili and Hossein Vafaee

iii

ACKNOWLEDGMENTS

First and foremost, I feel indebted to my advisor, Matthew Hedden, for all his trust,
encouragement, and invaluable supervision throughout my graduate studies. His patience,
extensive knowledge, and creative thinking have been a vast source of inspiration for me.
He was available for advice and academic help whenever I needed and gently guided me for
deeper understanding, no matter how late or inconvenient the time was. It is hard to express
in words how thankful I am for his unwavering support over the last four years.
I would like to take this opportunity to thank my dissertation committee members, Ron
Fintushel, Effie Kalfagianni, Ben Schmidt, and Bob Bell who have accommodated my timing
constraints despite their full schedules.
I would also like to acknowledge and thank my coauthors, Jen Hom and Tye Lidman.
The work that appears in Chapter 3 is joint with them.
This dissertation would not be possible without the friendly academic environment at
Michigan State University. I am grateful for helpful conversations with my colleagues: Adam
Giambrone, Andrew Donald (who made the last few months of graduate school, mathematically, very enjoyable for me), Chris Hays, Luke Williams, Tianran Chen, and Christine Lee.
A special thanks is due to David Krcatovich, my best friend in the department, with whom
I have had numerous helpful discussions. Also, Selman Akbulut was always willing to help
and so generous with his time. Moreover, I had the pleasure of having fruitful conversations
with many researchers outside of MSU from each and every one of which I had things to
learn, and the quality of my research got considerably enhanced by these interactions: Jen
Hom, Tye Lidman, Rachel Roberts, Tom Mark, Liam Watson, Allison Moore, among others.
And last but definitely not least, I want to express my deepest gratitude to my beloved
iv

parents, and also my brother Farhad. Their unending love, encouragement, and support
have been a constant source of comfort and counsel for me during the last six years.

v

TABLE OF CONTENTS

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

vii

Chapter 1 Introduction . . . . . . . . .
1.1 Hyperbolic L-space knots . . . . . .
1.2 Satellite knots and L-space surgery
1.3 Further results and discussion . . .

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3
6
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Chapter 2 Twisted torus knots and L-space surgery . . . . .
2.1 Background and preliminary lemmas . . . . . . . . . . . . .
2.1.1 (1, 1) knots and genus one Heegaard diagrams . . . .
2.1.2 Lifted Heegaard diagrams, L-space knots, and CF K −
2.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . .
2.3 Directions for future research . . . . . . . . . . . . . . . . .

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33

Chapter 3 Berge-Gabai knots and L-space satellite operations
3.1 Berge-Gabai knots . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Surgery on P (K) . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Input from Heegaard Floer theory . . . . . . . . . . . . . . . . .
3.4 Proof of Theorem 1.2.3 . . . . . . . . . . . . . . . . . . . . . . .
3.5 Left-orderability . . . . . . . . . . . . . . . . . . . . . . . . . . .

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52

BIBLIOGRAPHY

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LIST OF FIGURES

Figure 1.1

A (p, q) torus knot with r positive full twists on s adjacent strands. (Here,
p denotes the longitudinal winding.) The arc τ is a one-bridge, i.e. it
divides the knot into two arcs, where one arc is unknotted and the other
arc can be trivialized (unknotted) by sliding one or both of its endpoints
along the a priori unknotted arc. In order to make sense of adjacency of
strands, we need to have the standard presentation of a torus knot. Note
that where the twist occurs is irrelevant.
. . . . . . . . . . . . . . . .

5

Figure 1.2

Berge-Gabai knots are knots in S 1 × D 2 with non-trivial solid tori fillings.
Such knots are always the closure of the braid (σb σb−1 . . . σ1 )(σw−1 σw−2 . . . σ1 )t
where 0 ≤ b ≤ w − 2, and |t| ≥ 1. (a) An example of a braid in a solid
cylinder I × D 2 that closes to form a Berge-Gabai knot with b = 2, t = 3,
and w = 5. (The fact that the picture depicted above represents a BergeGabai knot is verified in [Gab90, Example 3.8].) Recall that we write
t = t0 + qw, where here t0 = 3 and q = 0. (b) An immersed annulus A
that can be arranged to be an embedded surface in V = S 1 × D 2 joining
P to T = ∂V by performing oriented cut and paste and adding a 2πt/w
twist. Note that the embedded surface A provides, in the exterior of P , a
homology from wℓ + tm in T to Λ in J = ∂nb(P ). . . . . . . . . . . . .
7

Figure 2.1

A (3, 4) torus knot with two positive full twists on two adjacent strands.
The one-bridge is indicated by τ . . . . . . . . . . . . . . . . . . . . . .

14

The process of obtaining a genus one Heegaard diagram for the (3, 4) torus
knot with two positive full twists on two adjacent strands. In the algorithm
discussed in Section 2.1.1, Figure 2.2(b) corresponds to Step 1, and also
Figure 2.2(c) corresponds to, simultaneously, implementing Step 2 and
Step 3. Note that the torus (in bold) corresponds to a neighborhood of
the bold curve of Figure 2.1(b). Note also that the α curve is drawn in
red and the β curve is drawn in blue. . . . . . . . . . . . . . . . . . . .

17

The process of drawing a genus one Heegaard diagram for the (4, 5) torus
knot with two positive full twists on three adjacent strands. Figure 2.3(b),
Figure 2.3(c), and Figure 2.3(d) correspond to Step 1, Step 2, and Step
3, respectively, in the algorithm discussed in Section 2.1.1. The α curve
is drawn in red. The base points must pass to the left of each other,
as otherwise we would create more crossings rather than simplify the arc
living in the torus complement. . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.2

Figure 2.3

vii

Figure 2.4

Figure 2.5

Heegaard diagrams on the torus, represented by a rectangle with opposite
sides identified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

By an isotopy, the shaded region disappears and the Heegaard diagram
will have two less intersection points. . . . . . . . . . . . . . . . . . . .

22

Figure 2.6

(a) A portion of the Heegaard diagram for K = K(p, kp ± 1; p − 1, r)
lifted to C, where r is an arbitrary integer. Note that m is the number of
intersection points in the genus one Heegaard diagram of K. It is assumed,
fixing α
˜ a connected component of π −1 (α), that we need n connected
components of π −1 (β) to obtain a complete list of all the m intersection
points between α and β downstairs. (b) A portion of the Heegaard diagram
for the (3, 4) torus knot with two positive full twists on two adjacent
strands, lifted to C. Note that the base points specified in the picture
depicted above are the only relevant base points needed to compute CF K − . 25

Figure 2.7

CF K − (K(3, 4; 2, 2))

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25

Figure 2.8

The process of drawing a genus one Heegaard diagram for K(4, 5; 2, 1).
The α curve in each step is oriented. This example indicates the pattern
when s ∈ {2, p − 2} and r = 1. In general when r = 1, to go from (c)
to (d), w first drags (s − 1) sub-arcs, all oriented in the same direction.
In the next winding it drags (s − 2) additional sub-arcs, all oriented in
the same direction but opposite to those of the first (s − 1) sub-arcs.
Dragging oppositely oriented sub-arcs does not occur in this example since
s = 2. Note that the orientation is irrelevant once the Heegaard diagram
is completed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

The base point w drags more than one sub-arc of α. The picture depicted
above is schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

A portion of the knot complex lifted to C, when the sub-arcs with different
orientations have been dragged. . . . . . . . . . . . . . . . . . . . . . .

30

A sub-diagram of a lifted Heegaard diagram, fixing one connected component of α
˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Part of a knot complex, representing the phenomenon of having two generators in one Alexander grading. . . . . . . . . . . . . . . . . . . . . .

31

A genus one Heegaard diagram for K(4, 5; 2, 1), as well as its lift to C . .

32

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

viii

Chapter 1
Introduction
In [OS04d], Oszv´ath and Szab´o introduced Heegaard Floer theory, which produces a set
of invariants of three- and four-dimensional manifolds. One example of such invariants is
HF (Y ), which associates a graded abelian group to a closed 3-manifold Y . When Y is a
rational homology three-sphere, rk HF (Y ) ≥ |H1 (Y ; Z)| [OS04c]. If equality is achieved,
then Y is called an L-space. Examples include lens spaces, and more generally, all connected
sums of manifolds with elliptic geometry [OS05b] (or equivalently, with finite fundamental
group by the Geometrization Theorem [KL08]). L-spaces are of interest for various reasons.
For instance, such manifolds do not admit co-orientable taut foliations [OS04a, Theorem 1.4].
A knot K ⊂ S 3 is called an L-space knot if it admits a positive L-space Dehn surgery.
Any knot with a positive lens space surgery is then an L-space knot. In [Ber], Berge gave a
conjecturally complete list of knots that admit lens space surgeries. Therefore it is natural to
look beyond Berge’s list for L-space knots. Examples include the (−2, 3, 2n+1) pretzel knots
(for positive integers n) [BH96, FS80, OS05b], which are known to live outside of Berge’s
collection when n ≥ 5 [Mat00]. We should first note that the pretzel knots (2, 3, 1), (2, 3, 3),
and (2, 3, 5) are isotopic to the (2, 5), (3, 4), and (3, 5) torus knots, respectively. Torus knots,
a proper subfamily of Berge knots, are well-known to admit lens space surgeries [Mos71].
The hyperbolic pretzel knot (2, 3, 7) is also known to have two lens space surgeries [FS80].
The knot (2, 3, 9) has two finite, non-cyclic surgeries [BH96]. Finally, the remaining knots,
(2, 3, 2n + 1) for n ≥ 5, are known to have Seifert fibered L-space surgeries with infinite
1

fundamental group [OS05b]. It is also proved in [LM13] that these 3-strand pretzel knots
are the only pretzel knots with L-space surgeries. Another source of L-space knots is within
the set of cable knots. By combining work of Hedden [Hed09] and Hom [Hom11a], the
(m, n) cable of a knot, K, is an L-space knot if and only if K is an L-space knot and
n/m ≥ 2g(K) − 1. Also, if K is a quasi-alternating knot with unknotting number one,
then the preimage of an unknotting arc in the branched double cover of K is a knot in an
L-space with an S 3 surgery (see [OS05a, Section 8.3]). The dual to this curve is therefore
a knot in S 3 with an L-space surgery, so either it or its mirror image is an L-space knot.
However, at present, there is no explicit parametrization of the knots that arise in this way.
Quasi-alternating knots were first introduced by Ozsv´ath and Szab´o in [OS05c]. This class
of knots appeared in the context of the Floer homology of links as a natural generalization of
alternating links. This comes from the fact that the branched double cover of an alternating
link, and more generally a quasi-alternating link, is an L-space [OS05c, Proposition 3.3]. We
remind the reader that the branched double cover Σ(K) of a knot K with unknotting number
one can be obtained by doing Dehn surgery on a knot, K, in S 3 [Mon73] (the Montesinos
trick). Therefore, for the case that K is quasi-alternating with unknotting number one, if
the surgery coefficient in the branched double cover is positive, then K will be an L-space
knot.
L-space knots are those knots that admit positive Dehn surgeries so that the resulting
three-manifolds have the same Heegaard Floer homology as lens spaces. One of the most
prominent problems in relating Heegaard Floer homology to low-dimensional topology is to
find topological characterizations of L-spaces and L-space knots. Ghiggini proved in [Ghi08]
that genus one L-space knots are fibered (see also [OS05b]). Later, Ni [Ni07] showed that all
L-space knots are fibered. It was proved by Hedden [Hed10] that knots in S 3 with positive
2

L-space surgeries are strongly quasipositive. A strongly quasipositive knot is a knot that
has a particular type of Seifert surface, called a quasipositive surface. Quasipositive surfaces
are those surfaces obtained from n parallel disks by attaching positive bands. A knot or
link is strongly quasipositive if it can be realized as the boundary of such a surface. Note
that for a fibered knot K ⊂ S 3 , being strongly quasipositive is equivalent to having the
corresponding open book decomposition associated to (F, K), where F is the fiber surface,
inducing the unique tight contact structure on S 3 [Hed10, Proposition 2.1]. In particular,
L-space knots induce the standard tight contact structure on S 3 . (We refer the reader to
[Etn06, OS04] for a review of contact geometry.) Unfortunately, it is not the case that every
strongly quasipositive fibered knot admits a non-trivial L-space Dehn surgery. An example
of such a knot is the (2, 1) cable of the right-handed trefoil. It is strongly quasipositive by
[Hed10, Corollary 1.3]. However, an exploration of its knot Floer homology groups (found
in [Hed05]) reveal that it cannot admit an L-space surgery.
In [Sch49], Schubert showed that every knot in S 3 decomposes as a connected sum of
prime knots in a unique way (up to reordering). We also recall that every knot in S 3 is
either hyperbolic, a satellite, or a torus knot. It is well-known that positive torus knots
admit positive lens space surgeries [Mos71], and therefore are L-space knots. In this work
we demonstrate new examples of L-space knots within both the families of hyperbolic and
satellite knots. (We should point out that L-space knots are prime [Krc13, Theorem 1.2].)

1.1

Hyperbolic L-space knots

The primary purpose of this section is to investigate L-space knots in the family of twisted
torus knots, K(p, q; s, r), which are defined to be, roughly speaking, the (p, q) torus knots with

3

r full twists on s adjacent strands where 0 < s < p. See Figure 1.2. Twisted torus knots
are an interesting class of knots that have been studied in several contexts. These knots
represent many different knot types. Morimoto and Yamada [MY] and Lee [Lee12] have
constructed twisted torus knots which are cables. Morimoto has also shown that infinitely
many twisted torus knots are composite [Mor94]. Guntel has shown that infinitely many
twisted torus knots are torus knots [Gun12]. Also, a complete characterization of twisted
torus knots which are isotopic to the unknot has been done by Lee in [Lee13]. Moriah
and Sedgwick have shown that certain hyperbolic twisted torus knots have minimal genus
Heegaard splittings which are unique up to isotopy [MS09]. In terms of the relationship
between twisted torus knots and their Floer homology, however, not much was previously
known.
Watson proved in [Wat09] that the knots K(3, 3k + 2; 2, 1) are L-space knots (k > 0). We
generalize this result in Corollary 2.2.3 by showing that all twisted torus knots K(3, q; 2, s),
for all q, s > 0, admit L-space surgeries (in order for a twisted (3, q) torus link to be a knot,
q must be an integer that does not divide 3).
In Chapter 2, we classify all the L-space twisted (p, q) torus knots with q = kp ± 1 (see
[Vaf14]). The question of what happens when q = kp±1 remains unanswered. Our examples
include the L-space pretzel knots as a proper subfamily since the (−2, 3, 2m + 3) pretzel knot
is isotopic to K(3, 4; 2, m) for m ≥ 1.
We now state the main result of this section. For p ≥ 2, k ≥ 1, r > 0 and 0 < s < p:
Theorem 1.1.1. The twisted torus knot, K(p, kp ± 1; s, r), is an L-space knot if and only if
either s = p − 1 or s ∈ {2, p − 2} and r = 1.

A key ingredient of the proof is the observation that all of the twisted torus knots being
4

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full twists

torus knot

s strands

on

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r

(p, q)

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τ

Figure 1.1 A (p, q) torus knot with r positive full twists on s adjacent strands. (Here, p denotes
the longitudinal winding.) The arc τ is a one-bridge, i.e. it divides the knot into two arcs, where
one arc is unknotted and the other arc can be trivialized (unknotted) by sliding one or both of
its endpoints along the a priori unknotted arc. In order to make sense of adjacency of strands,
we need to have the standard presentation of a torus knot. Note that where the twist occurs is
irrelevant.

studied in Theorem 1.1.1 are (1, 1) knots, that is, knots that can be placed in one-bridge
position with respect to a genus one Heegaard splitting of S 3 . Thus, the knot is comprised
of two properly embedded unknotted arcs, one in each of the two solid tori of the Heegaard
splitting. These arcs meet along their endpoints so that their union is equal to the knot.
From the perspective of knot Floer homology, (1,1) knots are particularly appealing. It
was first observed by Goda, Morifuji, and Matsuda [GMM05] that (1, 1) knots are exactly
those knots that can be presented by a doubly-pointed Heegaard diagram of genus one. The
chain complex for knot Floer homology is defined in terms of a doubly-pointed Heegaard
diagram. As shown by Ozsv´ath and Szab´o [OS04b], for knots admitting a genus one diagram,
knot Floer homology can be computed combinatorially and efficiently.
Recall that the tunnel number of a knot K in S 3 is the minimum number of mutually

5

disjoint arcs with endpoints on K so that the exterior of the resulting 1-complex is homotopy
equivalent to a handlebody. It is known that all of the twisted torus knots of Theorem 1.1.1,
as well as all of the Berge knots, are (1, 1) knots (see, for instance, [MSY96] for the case
of twisted torus knots). It is also known that all (1, 1) knots have tunnel number one. In
[Mot14] Motegi shows that there exist infinitely many hyperbolic L-space knots with tunnel
number two (see also [BM14, Question 24]). We propose the following question:
Question 1.1.2. Is there a non-satellite L-space knot with tunnel number greater than two?

1.2

Satellite knots and L-space surgery

As stated previously, cabling an L-space knot, when the ratio n/m of the cabling parameters
is large enough, is an L-space satellite operation. (We recall that a cabled knot is a special
case of a satellite knot; namely, the pattern is an (m, n)-torus knot.) We generalize this
result by introducing a new L-space satellite operation using Berge-Gabai knots [Gab90] as
the pattern.
Definition 1.2.1. A knot P ⊂ S 1 ×D 2 is called a Berge-Gabai knot if it admits a non-trivial
solid torus filling.1
To see that this satellite operation is a generalization of cabling, it should be noted that
any torus knot with the obvious solid torus embedding is a Berge-Gabai knot [Sei33]. Note
also that any Berge-Gabai knot P which is isotopic to a positive braid, when considered as a
knot in S 3 , admits a positive lens space surgery; for if performing appropriate surgery on P
in one of the solid tori in the genus one Heegaard splitting of S 3 returns a solid torus, then
1 Berge-Gabai knots, in the literature, are defined to be 1-bridge braids in solid tori with non-trivial solid
tori fillings. We relax that definition to include torus knots as a proper subfamily.

6

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Λ

ℓ

A

µ

m

(a)

(b)

Figure 1.2 Berge-Gabai knots are knots in S 1 × D2 with non-trivial solid tori fillings. Such knots
are always the closure of the braid (σb σb−1 . . . σ1 )(σw−1 σw−2 . . . σ1 )t where 0 ≤ b ≤ w − 2, and
|t| ≥ 1. (a) An example of a braid in a solid cylinder I × D 2 that closes to form a Berge-Gabai knot
with b = 2, t = 3, and w = 5. (The fact that the picture depicted above represents a Berge-Gabai
knot is verified in [Gab90, Example 3.8].) Recall that we write t = t0 + qw, where here t0 = 3 and
q = 0. (b) An immersed annulus A that can be arranged to be an embedded surface in V = S 1 ×D 2
joining P to T = ∂V by performing oriented cut and paste and adding a 2πt/w twist. Note that
the embedded surface A provides, in the exterior of P , a homology from wℓ + tm in T to Λ in
J = ∂nb(P ).

the corresponding surgery on the knot in S 3 will result in a lens space. For positive braids,
this surgery is positive by Lemma 3.1.1 and [Mos71, Proposition 3.2].
It is shown in [Gab89] that any Berge-Gabai knot must be either a torus knot or a 1bridge braid in S 1 × D 2 . More precisely, every Berge-Gabai knot P ⊂ V = S 1 × D 2 is
necessarily of the following form. (For a sufficient condition determining when a knot of this
form is a Berge-Gabai knot, see [Gab90, Lemma 3.2].) Let w denote the braid index of P .
In the braid group Bw let σi denote the generator of Bw that performs a positive half twist
on strands i and i + 1. Let σ = σb σb−1 . . . σ1 be a braid in Bw with 0 ≤ b ≤ w − 2 and let
t be a nonzero integer. Place σ in a solid cylinder and glue the ends by a 2πt/w twist, i.e.,

7

form the closure of the braid word (σb σb−1 . . . σ1 )(σw−1 σw−2 . . . σ1 )t . We only consider the
case where this construction produces a knot, rather than a link. This construction forms
a torus knot if b = 0 and a 1-bridge representation of P in V if 1 ≤ b ≤ w − 2. We call b
the bridge width, and t the twist number of P . Note that the twist number can be written
as t = t0 + qw for some integers t0 and q where t0 can be chosen so that 1 ≤ t0 ≤ w − 1.2
See Figure 1.2(a). Also, note that if b = 0 then the possibility of t0 = w − 1 is disallowed as
otherwise we would obtain a link with at least two components [Gab90].
Remark 1.2.2. Note that if t < 0, then the braid σ = (σb σb−1 . . . σ1 )(σw−1 σw−2 . . . σ1 )t is
isotopic to a negative braid:

σ ∼ (σb σb−1 . . . σ1 )(σw−1 σw−2 . . . σ1 )t
∼ (σw−1 σw−2 . . . σb+1 )−1 (σw−1 σw−2 . . . σ1 )t+1 .

We are now ready to state the main result. Let P (K) denote a satellite knot with pattern
P and companion K.
Theorem 1.2.3. Let P be a Berge-Gabai knot with bridge width b, twist number t, and
winding number w, and let K be a non-trivial knot in S 3 . Then the satellite P (K) is an
L-space knot if and only if K is an L-space knot and b+tw
2 ≥ 2g(K) − 1.
w

Note that when b = 0, we can take w = m and t = n, and Theorem 1.2.3 reduces to the
cabling result of [Hed09, Hom11a]. A version of the “if” direction of Theorem 1.2.3 appears
2 Our construction of Berge-Gabai knots, which enables us to define them up to isotopy of the knot in
1
S × D2 , is slightly different than that of Gabai [Gab90]. In Gabai’s original construction, he always took
q = 0 and considered knots in the solid torus up to homeomorphism of S 1 × D2 taking one knot to the
other.

8

in [Mot14, Proposition 7.2].
The outline of the proof of Theorem 1.2.3 is as follows. By applying techniques developed
in [Gab90, Gor83] to carefully explore the framing change of the solid torus surgered along
P , we prove the “if” direction of the theorem. More precisely, surgery on P (K) corresponds
to first doing surgery on P (namely removing a neighborhood of P from S 1 × D 2 and
Dehn filling along the new toroidal boundary component) and, second, attaching this to the
exterior of K. Therefore, if one chooses the filling on P such that the result is a solid torus
(using that P is a Berge-Gabai knot), then the overarching surgery on P (K) corresponds to
attaching a solid torus to the exterior of K (performing surgery on K). Moreover, note that
by positively twisting P by performing a positive Dehn twist on S 1 × D 2 (i.e., increasing
q), we can obtain an infinite family of Berge-Gabai knots. Fixing an L-space knot K, for
sufficiently large q, the satellite P (K) will admit a positive L-space surgery. Finally, the
“only if” direction is proved by methods similar to those used in [Hom11a].
In order to prove Theorem 1.2.3, we establish the following lemma, which may be of
independent interest.
Lemma 1.2.4. Let P ⊂ S 1 × D 2 be a negative braid and K ⊂ S 3 be an arbitrary knot.
Then the satellite knot P (K) is never an L-space knot.
We point out that Lemma 1.2.4 can be extended more generally to the case that P is a
homogeneous braid which is not isotopic to a positive braid [Sta78, Theorem 2]. The proof
of Lemma 1.2.4 was inspired by the arguments in [BM14].
We have the following corollary concerning the Ozsv´ath-Szab´o concordance invariant τ
and the smooth 4-ball genus.
Corollary 1.2.5. Let P ⊂ S 1 × D 2 be a Berge-Gabai knot and K ⊂ S 3 be an L-space knot.
9

If b+tw
2 ≥ 2g(K) − 1, then
w

τ (P (K)) = τ (P ) + wτ (K),
and
g4 (P (K)) = g4 (P ) + wg4 (K),
where τ (P ), respectively g4 (P ), denotes τ , respectively the 4-ball genus, of the knot obtained
from the standard embedding of S 1 × D 2 into S 3 .
Proof. If J is an L-space knot, then τ (J) = g4 (J) = g(J) by [OS05b, Corollary 1.6] and
[Ni07, Corollary 1.3]. Furthermore, by Lemma 3.2.4,

g(P (K)) = g(P ) + wg(K).

By assumption, K is an L-space knot. Since P is a Berge-Gabai knot with a positive
twist number, it follows that P is isotopic to a positive braid. Therefore, by the discussion
following Definition 1.2.1, P has a positive lens space surgery, and thus is an L-space knot.
Furthermore, by Theorem 1.2.3, we also have that P (K) is an L-space knot, and the result
follows.

1.3

Further results and discussion

Theorem 1.2.3 allows one to construct new examples of L-spaces as follows. First, begin
with any L-space knot and then satellite with a Berge-Gabai knot satisfying the conditions
in Theorem 1.2.3. Sufficiently large positive surgery will then result in an L-space. Using
this technique, we will construct L-spaces with any number of hyperbolic and Seifert fibered

10

pieces in the JSJ decomposition.
Theorem 1.3.1. Let r and s be non-negative integers such that at least one is non-zero.
Then there exist infinitely many irreducible L-spaces whose JSJ decompositions consist of
exactly r hyperbolic pieces and s Seifert fibered pieces.
As discussed, an L-space cannot admit a co-orientable taut foliation. Therefore, Theorem 1.3.1 will yield irreducible rational homology spheres without co-orientable taut foliations
whose JSJ decompositions consist of any numbers of hyperbolic and Seifert fibered pieces.
We remark that all rational homology spheres with Sol geometry are L-spaces [BGW13].
It is also natural to ask in what sense Theorem 1.2.3 generalizes; in particular, given a
satellite knot which is an L-space knot, what must hold for the pattern or the companion?
We propose the following conjecture (see also [BM14, Question 22]).
Conjecture 1.3.2. If P (K) is an L-space knot, then so are K and P .
Similarly, we conjecture that the converse holds as well, contingent on the pattern being
embedded “nicely” in the solid torus (e.g., as a strongly quasipositive braid closure) and
sufficiently “positively twisted” (akin to the condition in Theorem 1.2.3). We will not attempt
to make these notions precise in the dissertation.
As supporting evidence for Conjecture 1.3.2, we will study it from the viewpoint of leftorderability. Recall that a non-trivial group G is left-orderable if there exists a left-invariant
total order on G (see Section 3.5 for a more detailed discussion). We recall the conjecture
of Boyer, Gordon, and Watson relating Heegaard Floer homology to the left-orderability of
three-manifold groups.
Conjecture 1.3.3 (Boyer-Gordon-Watson [BGW13]). Let Y be an irreducible rational homology sphere. Then Y is an L-space if and only if π1 (Y ) is not left-orderable.
11

We point out that the computational strengths of Heegaard Floer homology and leftorderability tend to be fairly different. It is hopeful that if Conjecture 1.3.3 is true then
the strengths of each theory could be combined to derive new topological consequences. We
utilize this philosophy to establish Conjecture 1.3.2 under the assumption of Conjecture 1.3.3.
Proposition 1.3.4. Assuming Conjecture 1.3.3, if P (K) is an L-space knot, then so are P
and K.
Conjecture 1.3.2 can be also viewed from the perspective of strongly quasipositive knots.
Recall that L-space knots are strongly quasipositive [Hed10, Proposition 2.1]. Therefore it
makes sense to ask that for a strongly quasipositive fibered satellite knot P (K), what must
hold for the pattern P and companion K? We propose the following:
Question 1.3.5. For a fibered satellite knot P (K) with pattern a non-trivial knot P ⊂
S 1 × D 2 (i.e., not isotopic to the unknot when considered as a knot in S 3 ) and companion
K ⊂ S 3 , is it true that strongly quasipositiveness of any of the two implies the strongly
quasipositiveness of the third.

12

Chapter 2
Twisted torus knots and L-space
surgery
This chapter is mainly based on [Vaf14]. The focus of Chapter 2 is to prove Theorem 1.1.1.
Section 2.1 introduces the theory of (1, 1) knots and presents how to draw a genus one
Heegaard diagram for (1, 1) knots via an explicit example. Section 2.2 contains the proof
of the main result (Theorem 1.1.1), as well as the corollaries. In the final section, we state
some questions that address future research.

2.1

Background and preliminary lemmas

We start this section by showing that the knots K(p, kp ± 1; s, r) are (1, 1) knots. Next,
we explain an algorithm which produces genus one Heegaard diagrams for the twisted torus
knots with a (1, 1) decomposition. Finally, we assemble some preliminary facts needed to
prove Theorem 1.1.1.

2.1.1

(1, 1) knots and genus one Heegaard diagrams

Let p and q be relatively prime positive integers and let r and s be integers. We denote the
knot illustrated in Figure 1.2 by K(p, q; s, r). Let τ be the arc indicated in Figure 1.2. By
untying the crossings of the r full twists above the arc through edge slides along the arc, we
13

τ

τ

(b)

(a)

Figure 2.1 A (3, 4) torus knot with two positive full twists on two adjacent strands. The one-bridge
is indicated by τ .

will show that τ becomes a one-bridge for K(p, q; s, r) provided that q = kp ± 1. See Figure
2.1 for an explicit example. It has been a long standing question of whether or not any
twisted torus knot, with q that is not of the form kp ± 1, is a (1, 1) knot. In 1991, Morimoto,
Sakuma, and Yokota conjectured that the answer is negative:
Conjecture 2.1.1 ([MSY96], Conjecture 1.3). K(p, q; 2, r) admits no (1, 1) decomposition
unless either p ≡ ±1 (mod q), or q ≡ ±1 (mod p), or r = 0, ±1.

Having s = 2 does not seem to play an important role in the conjecture and, in fact, we
expect a similar conjecture to hold when the twisting is on any number of strands. Bowman,
Taylor, and Zupan have proved this conjecture when the number of twists is large [BTZ14,
Theorem 1.1].
In the rest of this subsection, we give an explicit construction of a genus one doubly-

14

pointed Heegaard diagram via a specific example, namely K = K(3, 4; 2, 2). See Figure 2.1.
This example should help clarify the strategy we use for our calculations.
We now describe a procedure to see that the arc τ (indicated in Figure 2.1) is a one-bridge,
i.e. it divides the knot K into two arcs, where one arc is a priori unknotted and the other
arc can be trivialized (unknotted) by sliding one or both endpoints of this arc along the bold
curve in Figure 2.1(b). (See [Ord06] for a detailed discussion on how to produce a genus one
Heegaard diagram for a certain family of (1, 1) knots.) The closed curve indicated in bold is
the union of the one-bridge, τ , and the a priori unknotted arc. Therefore, its neighborhood
is an unknotted torus. In Figure 2.2 we show, diagrammatically, how to use the one-bridge
and the unknotting process to obtain a Heegaard diagram for the knot K. (The red and
blue curves in Figure 2.2 (α and β curves respectively) are the boundaries of the meridional
disks corresponding to the two solid tori of the genus one Heegaard splitting of S 3 .) We do
this by trivializing the arc living in the complement of the torus. To begin, move the z base
point in the counterclockwise direction, making sure that the z base point passes to the left
of the w base point, as otherwise we would create more crossings rather than simplify the
arc. See Figure 2.2(b). Now move the w base point in the clockwise direction, passing to
the left of the z base point. See Figure 2.2(c). That completes the construction of the genus
one Heegaard diagram. See Figure 2.2(d).
This construction can be generalized to an algorithm with three steps to produce a genus
one Heegaard diagram for K(p, kp±1; s, r). Note that the number of longitudinal and meridional windings is dictated by the arc living in the torus complement:

Step 1: Wind the z base point once around the torus in the counter clockwise direction.
Note that z traverses the torus (k + r) times meridionally.
15

Step 2: Wind the w base point (s − 2) times in the clockwise direction. Note that each time
w traverses the torus (k + r) times meridionally.
Step 3: Finally, wind the w base point (p − s) times, longitudinally, to completely trivialize
the arc (in the sense that the planar projection of the arc no longer has any self-intersection).
Note that each longitudinal winding goes through k meridional moves.
Remark 2.1.2. To trivialize the part of the knot that lives outside of the torus, we isotope
the base points, z and w, on the torus which forces the α curve to be perturbed. Specifically,
in a neighborhood of the base points, the isotopy drags one (or more) sub-arc(s) of α.

Note that the Heegaard diagram in Figure 2.2(d) may be represented by a rectangle with
canonical identification implicit. See Figure 2.4(a).

2.1.2

Lifted Heegaard diagrams, L-space knots, and CF K −

For K ⊂ S 3 a knot, let CF K − (K) denote the knot Floer complex associated to K [OS04b].
Fortunately, computing CF K − (K) for a (1, 1) knot K is purely combinatorial. We refer the
interested reader to [OS04b, p.89] and [GMM05] for further details. To analyze holomorphic
disks in the torus, it is convenient to pass to the universal covering space π : C → T . Given
the base points z and w in T , π −1 (z) and π −1 (w) lift to affine lattices Z and W , respectively.
Also let {α
˜ i } and {β˜j } be the connected components of π −1 (α) and π −1 (β), respectively.
Now, given two intersection points x and y between α and β, the element φ ∈ π2 (x, y) is
a Whitney disk that has Maslov index one and admits a holomorphic representative if and
only if there is a bigon φ˜ ∈ π2 (˜
x, y˜) with Maslov index one, where x˜ and y˜ are lifts of x and
˜ ∼
y, intersection points between α
˜ i and β˜j (for some i and j). In particular, M(φ)
= M(φ).
See [OS04b] for the notation
16

z
w

z

w

(a)

(b)

w

w

z

z

2
1

34 5 6
7
8
9

(d)

(c)

Figure 2.2 The process of obtaining a genus one Heegaard diagram for the (3, 4) torus knot with
two positive full twists on two adjacent strands. In the algorithm discussed in Section 2.1.1,
Figure 2.2(b) corresponds to Step 1, and also Figure 2.2(c) corresponds to, simultaneously, implementing Step 2 and Step 3. Note that the torus (in bold) corresponds to a neighborhood of the
bold curve of Figure 2.1(b). Note also that the α curve is drawn in red and the β curve is drawn
in blue.

17

z
z

w

w

(a)

(b)

w

z

z
w

(d)

(c)

Figure 2.3 The process of drawing a genus one Heegaard diagram for the (4, 5) torus knot with
two positive full twists on three adjacent strands. Figure 2.3(b), Figure 2.3(c), and Figure 2.3(d)
correspond to Step 1, Step 2, and Step 3, respectively, in the algorithm discussed in Section 2.1.1.
The α curve is drawn in red. The base points must pass to the left of each other, as otherwise we
would create more crossings rather than simplify the arc living in the torus complement.

18

used above. Figure 2.6(b) shows a Heegaard diagram for K = K(3, 4; 2, 2) that has been
lifted to C. Also, Figure 2.7 represents CF K − (K). An L-space knot K can be thought of as
a knot with the simplest knot Floer invariants. To make sense of this fact, note that [OS04c]

(−1)m rk HF K m (K, s)T s ,

∆K (T ) =

(2.1.2.1)

m,s

where ∆K (T ) is the symmetrized Alexander polynomial of K. We observe that the total
rank of HF K(K) is bounded below by the sum of the absolute values of the coefficients of
the Alexander polynomial of K. A necessary condition for K to be an L-space knot is for
this bound to be sharp. The following lemma turns out to be useful during the course of
proving Part (c) of Theorem 2.2.1. See [OS05b, Theorem 1.2] for the complete statement.
Lemma 2.1.3. Assume that K ⊂ S 3 is a knot for which there is an integer p such that
Sp3 (K) is an L-space. Then

rk HF K(K, s) ≤ 1 ∀s ∈ Z.

In particular, all of the non-zero coefficients of ∆K (T ) are ±1.

Therefore, if the absolute value of one of the coefficients of ∆K (T ) is greater than one, then
K is not an L-space knot. We end this subsection by noting that a knot Floer complex with
a staircase-shape (as in Figure 2.7) represents an L-space knot. Such a complex has a basis

19

{x1 , x2 , ..., xm } for CF K ∞ (K) (defined in [OS04b]) such that

∂xi = xi−1 + xi+1 for i even
∂xi =

0

(2.1.2.2)

otherwise,

where the arrow from xi to xi−1 is horizontal and the arrow from xi to xi+1 is vertical.
(We refer the reader to [Hom11b, Section 6] for the concept of a knot Floer complex with a
staircase-shape.) The following corollary is a consequence of [Hom11b, Remark 6.6].
Corollary 2.1.4. For a knot K ⊂ S 3 , if CF K − (K) has a staircase-shape, then K is an
L-space knot.

2.2

Proof of the main theorem

This section is devoted to the proof of the main result of this chapter. For the sake of the
proof, it will convenient to restate Theorem 1.1.1 in the following equivalent form:
Theorem 2.2.1. For p ≥ 2, k ≥ 1, r > 0 and 0 < s < p, we have that K(p, kp ± 1; s, r):
(a) is an L-space knot if s = p − 1,
(b) is an L-space knot if r = 1 and s ∈ {2, p − 2}, and
(c) does not admit any L-space surgeries otherwise.

We prove part (a) by explicitly computing the knot Floer complex of K(p, kp ± 1; p −
1, r). Parts (b) and (c) are proved by focusing on the similarities and differences of the
corresponding complexes to those of K(p, kp ± 1; p − 1, r). The key to the proof is in
identifying whether or not the knot Floer complex associated to K(p, kp ± 1; s, r) has a
staircase-shape (Corollary 2.1.4).
20

. . .

. . .

. . .

z
1

2

. . .

z
3

4
w

5

6

7

8

9

w

(a) A Heegaard diagram for the (3, 4) torus (b) The general form of a Heegaard diagram

knot with two positive full twists on two ad- for K(p, kp ± 1; p − 1, r), where r is an arjacent strands
bitrary integer

Figure 2.4 Heegaard diagrams on the torus, represented by a rectangle with opposite sides identified
Proof of Theorem 2.2.1(a). It will help to break the proof into two steps:
Proof Step 1: We show that K(p, kp ± 1; p − 1, r) can be presented by a genus one
Heegaard diagram with the general form given in Figure 2.4(b).
Case 1: We first consider the case K(p, kp + 1; p − 1, r). The case p = 2 is trivial. The
construction of a Heegaard diagram in the case when p = 3 was given in Section 2.1. Also
Figure 2.3 shows the process for K = K(4, 5; 3, 2).
To obtain a Heegaard diagram when p ≥ 5 we can follow a similar procedure. Note that
the w base point winds around the longitude of the torus once in the case p = 3, twice in
the case p = 4, and p − 2 times in general. Moreover, in each longitudinal winding, the
w base point traverses the torus k + r times meridionally, except for the last longitudinal
winding where α traverses the torus only k times meridionally. The latter fact holds since
we are twisting p − 1 strands of the (p, kp + 1) torus knot (set s = p − 1 in Step 3 of the
algorithm given in Section 2.1). Note that as a result of s = p − 1, we always drag only one

21

z

z

w

yy
y
y
yy
y
yy
yyyy
y

w

(a)

(b)

Figure 2.5 By an isotopy, the shaded region disappears and the Heegaard diagram will have two
less intersection points.
sub-arc of α around the torus (Remark 2.1.2). Translating the resulting Heegaard diagram
obtained this way into the rectangular representation of the torus, we get the general form
of Figure 2.4(b).
Case 2: For the case q = kp − 1 we will have a similar setup, though the base points
have to pass to the right of each other, not to the left. In this case, there will always be two
intersection points of α and β that can be removed by an isotopy (see Figure 2.5(a)). To
indicate the general case, we consider K = K(3, 5; 2, 1). The resulting Heegaard diagram is
isotopic to a Heegaard diagram for K(3, 4; 2, 2) shown in Figure 2.5(b). As in Case 1, the
Heegaard diagram will have the general form of Figure 2.4(b).
Proof Step 2: In this step, the goal is to calculate the filtered chain complex CF K − (K)
for K = K(p, kp ± 1; p − 1, r). Figure 2.7 shows CF K − (K(3, 4, 2, 2)). We claim that, in
general, CF K − (K) has the same staircase-shape.
As in Section 2.1.2 we lift the diagrams, obtained in Step 1, to C . Fix a connected
component α
˜ of π −1 (α). We claim that such a component is a union of “N”-shapes (Fig-

22

ure 2.6(a)). To see this fact, we notice that the lift of a genus one Heegaard diagram can be
obtained by gluing together infinitely many copies of the rectangular form of the Heegaard
diagram in the plane (gluing from the sides of the rectangles). Figure 2.6(b) represents a
portion of such a lift for a specific example. Pick an intersection point and start moving it
along the α
˜ curve. (For example, pick the intersection point 9 on α
˜ in Figure 2.6(b) and
start moving it upward.) The direction of the motion will reverse by turning around either
of the z or w base points. (In Figure 2.6(b), the direction of the motion will change from
upward to downward, and also from downward to upward, by going from 1 to 2, and from
3 to 4, respectively.) Note that the rectangular form of the genus one Heegaard diagram
of K, as depicted in Figure 2.4(b), consists of a single β arc, together with α arcs having
endpoints on the edge(s) of the rectangle. Note also that there are only two α arcs with both
of their endpoints lying on one edge of the rectangle (namely the arcs that turn around the
base points). Therefore, by thinking of the lift of the diagram in C as coming from infinitely
many rectangles glued together along the sides and fixing a connected component of π −1 (α),
the change in the direction of the motion (equivalently, turning around either the z or w base
point) never happens twice in a single rectangle.1 Moreover, to recover all the intersection
points in the lift, only two changes of direction are needed. As a result, we get the shape of
the lifted digram as claimed.
Let us first consider the example, CF K − (K(3, 4; 2, 2)) whose Heegaard diagram is given
in Figure 2.6(b). Given a pair of intersection points x and y, the moduli space of holomorphic
representatives of Whitney disks φ ∈ π2 (x, y) with Maslov index one, modulo reparametrization, is either empty or consists of one map. In what follows, we write x → y if the moduli
space consists of one such map, and if so, we record how many times it passes over the z
1 Note that we do not distinguish between the z and w base points downstairs, and their lifts in C.

23

and w base points:
• 2 → 1, 6 → 5, 8 → 7 using one z base point,
• 3 → 9 using two z base points,
• 6 → 7, 8 → 9, 3 → 4 using one w base point, and
• 2 → 5 using two w base points.
From Figure 2.6(b), it is easy to see that we need four β˜ lines to generate the whole nine
intersection points in the lifted Heegaard diagram, i.e. fixing α
˜ , by using only four connected
components of the lift of β we can obtain a lift of all the intersection points between α and
β. Starting from β˜4 , See [OS04b] for the notation. By a similar method, we can find the
remaining Whitney disks in the list above and use them to complete the ordering of the
Alexander gradings. At this point, we can obtain the staircase-shape of Figure 2.7.
For the general case of Figure 2.6(a), it is straightforward to observe that our strategy
can be extended. Assume that {x1 , x2 , ..., xm } is the set of intersection points between α
and β curves coming from the genus one Heegaard diagram of K = K(p, kp + 1; p − 1, r)
(see Figure 2.4(b) and Figure 2.6(a)). Assume also that, fixing α
˜ a connected component of
π −1 (α), we need n connected components of π −1 (β) to recover all the m intersection points
downstairs between α and β (Figure 2.6(a)). Our strategy is first ordering the generators
based on their Alexander gradings and, second, finding all the differentials. Using the “N”shape of Figure 2.6(a) and starting from β˜n , there are three intersection points (xm , xm−1
and xm−2 ) with one disk xm−1 → xm using one w base point and one other disk xm−1 →
xm−2 using the z base point(s). Note that there exists no other non-trivial Whitney disk
with Maslov index one connecting xm−1 to another intersection point of Figure 2.6(a). Also
on β˜n−1 , there is one disk xm−3 → xm−2 using the w base point(s). Continuing this process,
24

β˜1

˜1
β

β˜2

z
x1

z
1

w

2

6

5

5

7

x3

x2

˜2
β

β˜3

.
.
.

˜n−1
β
xm−4

8

9

xm−2

xm−3

˜n
β
xm−2

7

xm−1

w

β˜4

xm

z
3

9

w

4

(b)

(a)

Figure 2.6 (a) A portion of the Heegaard diagram for K = K(p, kp ± 1; p − 1, r) lifted to C, where r
is an arbitrary integer. Note that m is the number of intersection points in the genus one Heegaard
diagram of K. It is assumed, fixing α
˜ a connected component of π −1 (α), that we need n connected
components of π −1 (β) to obtain a complete list of all the m intersection points between α and
β downstairs. (b) A portion of the Heegaard diagram for the (3, 4) torus knot with two positive
full twists on two adjacent strands, lifted to C. Note that the base points specified in the picture
depicted above are the only relevant base points needed to compute CF K − .
4
3

9
8
7
6
5

2
1
.
.
.

Figure 2.7 CF K − (K(3, 4; 2, 2))

25

we deduce that

A(xm ) > A(xm−1 ) > A(xm−2 ) > A(xm−3 ) > ... > A(x1 ).

By noting that there is no other non-trivial Whitney disk with Maslov index one, we see that
the set {x1 , x2 , ..., xm } forms a basis for CF K − (K) such that there are three intersection
points (3, 4 and 9) with one disk 4 → 3 using one w base point and one other disk 9 → 3
using two z base points. Thus, in terms of the Alexander gradings A(i) of the intersection
points, i ∈ {1, 2, ..., 9} , we have that:
˜ − nw (φ)
˜ = −1, and
• A(3) − A(4) = nz (φ)
˜ − nw (φ)
˜ = 2.
• A(3) − A(9) = nz (φ)

∂xi = xi−1 + xi+1 for i even
∂xi =

0

otherwise.

This formula for the differentials (which is the same as (2.1.2.2)), together with the existence of three intersection points on each β˜j line of Figure 2.6(a) with exactly two disks
using different base point types (i.e. z and w), gives the staircase-shape of CF K − (K) (see
the discussion about a knot Floer complex with a staircase-shape in Section 2.1.2). Now,
Corollary 2.1.4 completes the proof.

Proof of (b) and (c). Let K(p, q; s, r) be a twisted torus knot where 2 ≤ s ≤ p − 2. We
discuss the case when q = kp + 1 and leave the case q = kp − 1 to the reader. Since we
apply the same algorithm, as used in Part (a), to obtain a Heegaard diagram, we will only
26

highlight the differences in this case. Recalling the algorithm explained in Section 2.1.1, we
first wind z once in the counterclockwise direction (Step 1). Then we wind the w base point
(s − 2) times in the clockwise direction, traversing the torus (k + r) times meridionally in
each winding (Step 2). Finally, we wind the w base point (p − s) more times around the
torus longitudinally (Step 3). Note that in the latter step, w goes through only k meridional
moves in each winding.
It will be convenient to pick an arbitrary orientation for the α curve. Note that, unlike
Part (a), more than one sub-arc will be dragged since 2 ≤ s ≤ p − 2 (Remark 2.1.2). With
the α curve oriented, either these sub-arcs will have all the same orientation or there will
be at least one pair of sub-arcs with opposite orientations. The case for only two sub-arcs
can be seen in Figure 2.9. Figure 2.8 shows the process of constructing a Heegaard diagram
for K(4, 5, 2, 1), which indicates the pattern, particularly in the case when s ∈ {2, p − 2}.
Claim. Unless s ∈ {2, p − 2} and r = 1, the trivializing process will drag oppositely oriented
sub-arcs.
Proof. Suppose r = 1. The first longitudinal traversal of Step 3 drags no additional subarcs. The second traversal of Step 3, however, drags (s − 1) sub-arcs, all oriented in the
same direction. The next winding drags (s − 2) additional sub-arcs, all oriented in the same
direction but opposite to those of the first (s − 1) sub-arcs. This opposite orientation will
clearly not occur if s = 2. Suppose s = p − 2. Then in Step 3 the w base point is wound
longitudinally around the torus p − (p − 2) = 2 times (twice). Hence, only sub-arcs with the
same direction will be dragged. If r ≥ 2 the full twists of Step 1 create future oppositely
oriented sub-arcs in Step 3, i.e. the w base point will be dragging sub-arcs with opposite
orientations, starting the second longitudinal traverse of Step 3. More specifically, if the

27

z
z

w

w

(b)

(a)

w

w

z

z

(c)

(d)

Figure 2.8 The process of drawing a genus one Heegaard diagram for K(4, 5; 2, 1). The α curve in
each step is oriented. This example indicates the pattern when s ∈ {2, p − 2} and r = 1. In general
when r = 1, to go from (c) to (d), w first drags (s − 1) sub-arcs, all oriented in the same direction.
In the next winding it drags (s − 2) additional sub-arcs, all oriented in the same direction but
opposite to those of the first (s − 1) sub-arcs. Dragging oppositely oriented sub-arcs does not occur
in this example since s = 2. Note that the orientation is irrelevant once the Heegaard diagram is
completed.

28

w

w

Figure 2.9 The base point w drags more than one sub-arc of α. The picture depicted above is
schematic.

number of full twists is greater than one, each additional twist will create two oppositely
oriented sub-arcs and the w base point will drag both of these sub-arcs after the first (s − 1)
longitudinal windings.
Since the hypotheses of Part (b) imply that the sub-arcs have the same orientation, a
similar argument to Part (a), once we lift the diagram to C, shows that the ordering of the
Alexander gradings of the intersection points will follow the same manner as in the case s =
p−1. More precisely, if we think of the lift of the Heegaard diagram as coming from infinitely
many rectangles glued together, by picking an intersection point and moving it along a fixed
connected component α
˜ of π −1 (α), we see that the picked point, during its motion, never
turns around the z (or w) base point twice in a single rectangle. Therefore, although the lifted
diagrams are not looking the same as Part (a), we claim that the corresponding complexes
have the staircase-shape. In particular, for the case s = 2 (respectively s = p − 2), we
need four (respectively 2p − 4) changes of direction2 to recover all the intersection points of
downstairs. For the specific example of K(4, 5; 2, 1) depicted in Figure 2.13(b):

A(6) > A(5) > A(9) > A(8) > A(7) > A(4) > A(1) > A(11) > A(10) > A(3) > A(2).

Exploring the Whitney disks in the lifted diagram will give a staircase-shape for the asso2 We remind the reader that by changing direction we mean turning around one of the base points (z or
w).

29

ciated complex. To see this in the general case, note that the set of all intersection points
between α and β curves forms a basis for CF K − . Moreover, for every intersection point xi ,
either the differential vanishes, or there exist two Whitney disks with Maslov index one connecting xi to another two distinct intersection points, using z and w base points alternatively.
(This shows that the differentials are of the form of (2.1.2.2).) That is, for each intersection
point xi , either there is no arrow joining it to another intersection point, or there are two
arrows joining xi to two distinct intersection points such that one arrow is horizontal and
the other is vertical. This gives us the staircase-shape of the knot Floer complex. Finally,
Corollary 2.1.4 completes the proof of Part (b).

To prove Part (c), note that if the arcs dragged by w have different orientations, then, after
lifting the diagram to C, the following phenomenon occurs:

z

˜i
β
1

2

3

4

Figure 2.10 A portion of the knot complex lifted to C, when the sub-arcs with different orientations
have been dragged.

Claim: The associated complex does not represent an L-space knot.

Proof. As in the proof of Part (a), we can order the Alexander gradings of the intersection
points from the Whitney disks in the lifted Heegaard diagram. Let β˜1 , ..., β˜k denote the lifts
of β needed to find all of the Whitney disks. Work from β˜k to β˜1 and stop at the first β˜i
that exhibits the phenomenon in Figure above. Then part of the diagram is as Figure 2.11.

30

z

˜i
β
1

2

3

4

˜i+1
β

.
.
.

˜
β
k

w
Figure 2.11 A sub-diagram of a lifted Heegaard diagram, fixing one connected component of α˜

We analyze this by looking at the Whitney disks:
• 4 → 1, 3 → 2 using one z base point, and
• 1 → 2, 4 → 3 using one w base point.
As a result, the part of CF K − involving the intersection points, {1, 2, 3, 4}, on β˜i will look
like
3

4

2

1

Figure 2.12 Part of a knot complex, representing the phenomenon of having two generators in one
Alexander grading.

Note that the boundary map decreases the Maslov grading by one, and the U-action decreases
the grading by two. Combining these facts with the existence of the disks 1 → 2 and 4 → 3,
we find that the intersection points 2 and 4 both have the same Maslov gradings as well
as the same Alexander gradings. (We are assuming that there are no trivial Whitney disks
connecting two intersection points; if there is a bigon that does not pass over any of the base

31

z
˜1
β

2

3

w 10

z

˜2
β

11

w 8

z
2

3

4

5

6
w

7

8

9

10

w 7

z

˜3
β

1

1 4

9

11
z

˜4
β

5

w

6

(b)

(a)

Figure 2.13 A genus one Heegaard diagram for K(4, 5; 2, 1), as well as its lift to C
points, we can isotop it away.) Thus,

rk HF K(K, s) ≥ rk HF K m (K, s) = rk CF K m (K, s) ≥ 2,

where s is the Alexander grading of the intersection points 2 and 4. Now, Lemma 2.1.3
completes the proof of the claim and Part (c).
The Heegaard diagrammatic observation in Figure 2.5 can be generalized. The author
suspects that the following corollary could have been proved differently, using braid words
for instance:
Corollary 2.2.2. The twisted torus knot, K(p, kp + 1; p − 1, r), is isotopic to K(p, (k + 1)p −
1; p − 1, r − 1).
Proof. We start from the genus one Heegaard diagram of K1 = K(p, (k +1)p+1; p−1, r −1),
obtained from implementing the algorithm explained in Section 2.1.2. The proof is done by
first doing an isotopy to get rid of the two extra generators in the genus one Heegaard

32

diagram of K1 3 and, second, tracking back the drag of the w and z base points in the torus.
More precisely, after removing the extra generators, if we track back the w base point, we see
that it passes, during its p−2 longitudinal windings, to the right of z. Now, by tracking back
the z base point once around the torus, we see that it also passes to the right of w. These
facts can be verified in the example depicted in Figure 2.5(b). (Thus, while implementing
the algorithm to obtain the diagram in the first place, the base points must have passed
by the left of each other). During this process, except for the first winding of w that goes
through k meridional moves, the rest of windings traverse the torus k + r times meridionally.
Therefore, by noting that only one sub-arc of α has been dragged by the base points, we
get that the diagram obtained after doing the isotopy is a genus one Heegaard diagram for
K2 = K(p, kp + 1; p − 1, r).
When p = 3 in Theorem 1.1.1, we obtain a generalization of [Wat09, Theorem 1.2]:
Corollary 2.2.3. All twisted (3, q) torus knots admit L-space surgeries.

2.3

Directions for future research

Closely related to the main result of this chapter, one can ask the question of which operations
on knots produce L-space knots. Satellite operations are the first in line. As pointed out
in Chapter 1, the (p, q) cabling is an L-space satellite operation [Hom11a]. More generally,
Hom, Lidman and the author introduced an L-space satellite operation, using Berge-Gabai
knots as the pattern [HLV14]. By definition, a knot P ⊂ S 1 × D 2 is called a Berge-Gabai
knot if it admits a non-trivial solid torus surgery. We also suspect that one can obtain
3 Note that the phenomenon (of having two removable intersection points) in Figure 2.5, once we implement
the algorithm explained in Section 2.1.2, will always occur in the genus one Heegaard diagram of K(p, kp −
1; p − 1, r).

33

more L-space satellite operations, choosing the patterns from the the list of L-space knots
of Theorem 1.1.1. Although classifying such operations does not seem to be an easy task to
do, there is an obstruction to obtaining L-space satellite knots (Lemma 2.1.3) which can be
appealing. Let P (K) be a satellite knot with pattern P ⊂ V = S 1 × D 2 and companion K.
We recall the behavior of the Alexander polynomial of a satellite knot:

∆P (K) (T ) = ∆P (T )∆K (T w )

where w is the geometric intersection number of the pattern P with a fixed meridional disk
of V (see for instance [Lic97]). So one can attack the following question by first examining
the obstruction of Lemma 2.1.3, using algebraic methods.

Question 1: Is there a classification of L-space satellite operations?

Another interesting direction one can pursue, encouraged by the computations done in
this chapter, is to calculate the Alexander polynomials ∆K (T ) of twisted (p, q) torus knots
with q = kp ± 1 or more generally with q an arbitrary non-zero integer. In [Mor06], Morton
gives a closed formula for ∆K (T ) where K = K(p, q; 2, r) and p > q > 0.
Finally, notice that the (2, 2n + 1) torus knots have the particular property that they
admit lens space surgeries and also have branched double covers that are lens spaces. It
seems reasonable to ask what class of knots have this property. The following question was
first brought to the author’s attention by Allison Moore:

Question 2: What class of knots have surgeries and branched double covers that are lens

34

spaces (or L-spaces)?

35

Chapter 3
Berge-Gabai knots and L-space
satellite operations
This chapter is mainly based on [HLV14]. We start by providing background on 1-bridge
braids in solid tori and Dehn surgery on satellite knots. See [Ber91, Gab90, Gor83] for further
details. We then present the rank formula, found in [OS11], for the Heegaard Floer homology
of the surgered manifold obtained by doing Dehn surgery on a knot in S 3 . Theorem 1.2.3
is proved in Section 3.4. We end this chapter by clarifying the discussions in Section 1.3
and proving Theorem 1.3.1 and Proposition 1.3.4. Throughout the rest of the chapter, we
assume that P is a Berge-Gabai knot in V = S 1 × D 2 (i.e., P admits a non-trivial solid torus
surgery) unless otherwise stated. We also consider the standard embedding of S 1 × D 2 into
S 3 such that S 1 × {∗} bounds an embedded disk in S 3 . When it is clear from context, we
will not distinguish between the Berge-Gabai knot P ⊂ V and P ⊂ S 3 .

3.1

Berge-Gabai knots

The primary goal of this subsection is to highlight the Dehn surgeries on P ⊂ V that will
return a solid torus. In what follows, we provide a setup similar to that of [Gab90].
An arbitrary knot P in V is called a 1-bridge braid if P can be isotoped to be a braid
in V that lies in S 1 × ∂D 2 except for one arc that is properly embedded in V , and P is not
36

a torus knot. Gabai [Gab89] showed that any knot in a solid torus with a non-trivial solid
torus surgery must be either a torus knot or a 1-bridge braid in S 1 × D 2 , and Berge [Ber91]
classified all 1-bridge braids in S 1 × D 2 with non-trivial solid tori fillings. We denote the
braid index of P by w.
We will consider V , the exterior of P ⊂ V . Let T = ∂V and J = ∂nb(P ). We equip
T with the homological generators (m, ℓ) where ℓ is the longitude S 1 × {∗} of T and m is
{∗} × ∂D 2 ; therefore, ℓ becomes null-homologous after standardly embedding V in S 3 and
removing nb(P ). We equip J with homological generators (µ, Λ) as follows. The generator
µ is the meridian of P . Note that m is homologous to wµ in V . To define Λ, consider
the immersed annulus A connecting J to T with b arcs of self-intersection in Figure 1.2(b).
By doing oriented cut and paste to the arcs of self-intersection we can arrange A to be an
embedded surface in V joining J to T . Define Λ to be A ∩ J. Orient m, ℓ, µ, and Λ as in
Figure 1.2(b). Note that A ∩ T = wℓ + tm, and so wℓ + tm is homologous to Λ in V .
Let λ be the simple closed curve on J that is homologous to Λ − wtµ ∈ H1 (J; Z). Thus,
we have the following equalities in H1 (V ; Z):

[λ] = [Λ − wtµ]
= [wℓ + tm − wtµ]
= [wℓ],

where the last equality follows from the fact that m is homologous to wµ. In particular, λ
becomes null-homologous after standardly embedding V in S 3 and removing nb(P ). Now
the equation [λ] = [Λ − wtµ] can be used to switch from (µ, Λ)- to (µ, λ)-coordinates, where
(µ, λ) are the usual meridian-longitude coordinates on P when V is standardly embedded in
37

S 3.
We recall that a 1-bridge braid in S 1 × D 2 with winding number w, bridge width b, and
twist number t can be represented via the braid word σ = (σb σb−1 ...σ1 )(σw−1 σw−2 ...σ1 )t
where |t| ≥ 1, and 1 ≤ b ≤ w − 2. The following lemma is a consequence of [Gab90, Lemma
3.2]:
Lemma 3.1.1. Let P be a 1-bridge braid in V and s a positive integer. If filling V along a
curve α = dµ + sΛ in J yields S 1 × D 2 , then s = 1, d ∈ {b, b + 1}, and gcd(w, d) = 1.
In (µ, λ)-coordinates these possible exceptional surgeries are α = (tw + d)µ + λ where
d ∈ {b, b + 1}.1

Note that when P is an (m, n)-torus knot in V , there are infinitely many surgeries on P
that will return a solid torus, including mn + 1 = tw + b + 1; this follows, for instance, from
the proof of [Mos71, Proposition 3.2].
Let (P ; n1 /n2 ) denote the result of filling V along the curve n1 µ + n2 λ. Lemma 3.1.1
shows that if P is a Berge-Gabai knot, then (P ; pd) will be homeomorphic to S 1 × D 2 for
at least one of the coefficients pd = tw + d, d ∈ {b, b + 1}.
Note that adding a positive full-twist to all of the w strands of P results in a new knot
′

P where t changes into t + w. Correspondingly, there exists a homeomorphism of the solid
′

torus (doing a positive meridional twist), which takes P to P . Iterating this process q times,
we get the following:
Proposition 3.1.2. Let P be a Berge-Gabai knot in S 1 ×D 2 , standardly embedded in S 3 , so
′

that (P ; p) is homeomorphic to a solid torus. Let P be the knot obtained from P by adding
1 We have stated Lemma 3.1.1 so that the orientation of (µ, λ) agrees with the standard convention that
µ · λ = 1. In Gabai’s paper [Gab90], µ is oriented opposite to that of Figure 1.2(b).

38

q positive Dehn twists. Then

′
(P ; p + qw 2 ) ∼
= S 1 × D2 .

Hence if we have a Berge-Gabai knot P with twist number t, adding q full twists to all
w strands of P will produce a Berge-Gabai knot with twist number t + qw.

3.2

Surgery on P (K)

Let P (K) be a satellite knot with pattern P ⊂ V and companion K. Let f : V → nb(K) be
a homeomorphism that determines the zero framing of K, i.e., [f (S 1 × {∗})] = 0 ∈ H1 (X; Z)
where X = S 3 − nb(K). Thus P (K) = f (P ).
Recall that m, ℓ ∈ H1 (T ; Z) are the natural meridian and longitude coordinates of T =
∂V , oriented such that m · ℓ = 1. Recall also that V = V − nb(P ). Note that H1 (V ) =
Z ℓ ⊕ Z µ where µ is the class of the meridian of nb(P ). When P is viewed as a knot in
S 3 , let λ ⊂ ∂nb(P ) be the unique curve on ∂nb(P ) which is null-homologous in S 3 − nb(P )
(i.e., the zero framing of P ). That is, if f is as above, then f (λ) is the zero framing of P (K).
Thus, Sp3 /p (P (K)) ∼
= X ∪f (P ; p1 /p2 ), where the notation means ∂X and ∂(P ; p1 /p2 ) are
1 2
identified via the restriction of f to ∂(P ; p1 /p2 ) = ∂V . With the above notation:
Lemma 3.2.1 ([Gor83, Lemma 3.3]). For relatively prime integers p1 , p2 , and P ⊂ V with
winding number w:
(a) H1 ((P ; p1 /p2 ); Z) ∼
= Z ⊕ Zgcd(w,p1 ) .

39

(b) If w = 0, the kernel of H1 (∂(P ; p1 /p2 ); Z) → H1 ((P ; p1 /p2 ); Z) is the cyclic group
generated by
p1
p2 w 2
m+
ℓ.
gcd(w, p1)
gcd(w, p1 )
Note that Lemma 3.2.1 is valid regardless of whether or not P is a Berge-Gabai knot.
However, when P is a Berge-Gabai knot, we can use Lemma 3.2.1 to relate surgeries on K
and P (K) in the following sense.
Corollary 3.2.2. Let P be a Berge-Gabai knot in V with winding number w so that (P ; p) ∼
=
S 1 × D 2 . Then
3
Sp3 (P (K)) ∼
= Sp/w
2 (K).

Proof. The result essentially follows from the fact that

Sp3 (P (K)) ∼
= X ∪f (P ; p).

By assumption, (P ; p) is homeomorphic to a solid torus. Therefore, in order to find the
corresponding surgery coefficient on K, one needs to determine the slope of the meridian of
∂(P ; p) under the canonical identification with ∂V , and where it is sent under f .
Note that the slope of the meridian of (P ; p) is precisely the generator of

ker H1 (∂(P ; p); Z) → H1 ((P ; p); Z) .

Using the identification of ∂V and ∂(P ; p), we have that the slope of the meridian, in (m, ℓ)coordinates, is given by (p, w 2 ) by Lemma 3.2.1. Since f sends m (respectively ℓ) to the
40

meridian (respectively longitude) of K, the result follows.
Combining Lemma 3.1.1 with Corollary 3.2.2, we deduce the following:
Proposition 3.2.3. Let P be a Berge-Gabai knot with bridge width b = 0, winding number w,
and twist number t, and let K be an arbitrary knot in S 3 . Then for at least one d ∈ {b, b+1},

3
Sd+tw
(P (K)) ∼
= S 3d+tw (K).
w2

Note that gcd(d + tw, w 2) = 1 (see Lemma 3.1.1). We end this subsection by stating the
following lemma, which will be useful in the proof of Theorem 1.2.3. Let ∆K (T ) denote the
symmetrized Alexander polynomial of K. Recall the behavior of the Alexander polynomial
for satellites (see for instance [Lic97]):

∆P (K) (T ) = ∆P (T )∆K (T w ).

(3.2.0.1)

Lemma 3.2.4. Let P (K) be a fibered satellite knot where P has winding number w. Then

g(P (K)) = g(P ) + wg(K).

Furthermore, if P is a Berge-Gabai knot as above with t > 0, then

g(P ) =

(t − 1)(w − 1) + b
.
2

(3.2.0.2)

Proof. Since P (K) is a fibered knot, we deduce that deg ∆P (K) (T ) = g(P (K)). It also
41

follows that K and P are both fibered [HMS08]. Combining these two facts with (3.2.0.1),
we see that g(P (K)) = g(P ) + wg(K).
In order to calculate g(P ), notice that P is a positive braid if t > 0. Hence, the Seifert
surface R obtained from Seifert’s algorithm is a minimal genus Seifert surface for P [Sta78].
Then
χ(R) = 1 − 2g(P ) ⇒ w − b − t(w − 1) = 1 − 2g(P ).

3.3

Input from Heegaard Floer theory

In this subsection we mainly use the notation of [Hom11a]. Recall that an L-space Y
is a rational homology sphere with the simplest possible Heegaard Floer homology, i.e.,
rkHF (Y ) = |H1 (Y ; Z)|. We say that a knot K in S 3 is an L-space knot if it admits a
positive L-space surgery.
We let τ (K) denote the integer-valued concordance invariant from [OS03]. Let P denote
the set of all knots K for which g(K) = τ (K). (Recall from [Hed10] that for fibered knots,
g(K) = τ (K) is equivalent to being strongly quasipositive.) If K is an L-space knot, then
K ∈ P. This follows from [OS05b, Corollary 1.6] and the fact that L-space knots are fibered
[Ni07, Corollary 1.3].
Let
rkH∗ (AK
i )−1 ,

sK =
i∈Z

∞
where AK
i is the subquotient complex of CF K (K) defined in [OS08]. It is proved in

[Hom11a] that rkH∗ (AK
i ) is always odd, and so sK is always a non-negative even integer.

42

For a pair of relatively prime non-zero integers m and n, n > 0, let

m/n

tK

= 2 max(0, n(2ν(K) − 1) − m).

(3.3.0.3)

Observe that
m/n

tK

=0

if and only if

m/n ≥ 2ν(K) − 1.

(3.3.0.4)

The term ν(K) is another integer-valued invariant of K, defined in [OS11, Definition 9.1],
which is bounded below by τ (K) and above by g(K). In particular, if K ∈ P, then ν(K) =
g(K).
Let m and n be as above, and suppose that ν(K) ≥ ν(K) where K denotes the mirror
of K. (This condition is automatically satisfied for K ∈ P.) If ν(K) > 0 or m > 0, then

m/n

3
rkHF (Sm/n
(K)) = m + nsK + tK

(3.3.0.5)

by [OS11, Proposition 9.6].
By (3.3.0.5), when m > 0 we have that

m/n

3 (K) is an L-space if and only if t
Sm/n
K

= 0 and sK = 0.

(3.3.0.6)

3
By [OS04b, Theorem 4.4], the group H∗ (AK
i ) is isomorphic to HF (SN (K), [i]) for N ≫ 0

and |i| ≤ N/2. Thus, we have that

K is an L-space knot if and only if sK = 0.

(3.3.0.7)

3 (K) is an L-space if and only if m/n ≥
In fact, if K is a non-trivial L-space knot, Sm/n

43

2g(K) − 1. This follows from (3.3.0.4), (3.3.0.5), and the fact that for K a non-trivial Lspace knot, ν(K) = g(K) > 0. (The original argument for the forward direction is given in
[KMOS07].)

3.4

Proof of Theorem 1.2.3

This subsection is devoted to the proof of Theorem 1.2.3. We begin with the proof of
Lemma 1.2.4. We do not review the concept of a quasipositive Seifert surface but instead
refer the reader to [Hed10, Rud98].
Proof of Lemma 1.2.4. Suppose for contradiction that P (K) is an L-space knot. Recall that
L-space knots are fibered [Ni07, OS05b]. It is also a well-known fact that a minimal genus
Seifert surface for a negative braid can be expressed as a plumbing of negative Hopf bands
[Sta78, Theorem 2]. (See also [AO01, Theorem 1] for an explicit construction in the case of
torus knots.) Since P (K) is fibered, this implies that K is fibered and P is fibered in the
solid torus [HMS08], so the fiber for P (K) is constructed by patching the fiber for P in the
solid torus to w copies of the fiber for K. As a result, when P is a negative braid, the fiber
surface for P (K) contains (at least) as many negative Hopf bands as the one for P .
By the above description of the fiber surface, we can deplumb a negative Hopf band.
This means we can decompose the fiber surface for P (K) as a Murasugi sum, where one of
the summands is not a quasipositive surface. By [Rud98], if a Seifert surface is a Murasugi
sum, it is quasipositive if and only if all of the summands are quasipositive. Thus, the fiber
surface for P (K) is not a quasipositive surface. However, since P (K) is an L-space knot, it
is strongly quasipositive [Hed10], which gives a contradiction.
We prove Theorem 1.2.3 only for the cases where b = 0 (consequently 1 ≤ t0 ≤ w − 2)
44

and refer the reader to [Hed09, Hom11a] for the case b = 0.
Proof of Theorem 1.2.3. (⇐) The proof of this direction follows from Proposition 3.2.3,
which tells us that
3
Sd+tw
(P (K)) ∼
= S 3d+tw (K).
w2

3
Since K is a non-trivial L-space knot and b+tw
2 ≥ 2g(K) − 1 > 0, it follows that S d+tw (K)
w

w2

is an L-space. Here we are using that d ≥ b. Therefore, P (K) is an L-space knot.
(⇒) For the case that t < 0 (see Remark 2.1.2), we apply Lemma 1.2.4 to see that P (K)
cannot be an L-space knot. Therefore, we can assume that t > 0 and P (K) is an L-space
knot. For simplicity of notation, we set m = d + t0 w + qw 2 where d ∈ {b, b + 1} is such that
(P ; m) ∼
= S 1 × D 2 . Again from Proposition 3.2.3 we have

3 (P (K))) = rkHF (S 3
rkHF (Sm

m/w2

(K)).

(3.4.0.8)

Since P (K) is an L-space knot, it follows that g(P (K)) = τ (P (K)), and we see that

tm
P (K) = 2 max(0, 2g(P (K)) − 1 − m).

(3.4.0.9)

We first suppose that ν(K) ≥ ν(K). Since m > 0, we may combine (3.3.0.5), (3.3.0.7),
and (3.4.0.8) to obtain
m/w2

2
m + tm
P (K) = m + w sK + tK

,

or equivalently
m/w2

2
tm
P (K) = w sK + tK

45

.

(3.4.0.10)

Note that by Lemma 3.2.4, (3.2.0.2), and (3.4.0.9), we have that

tm
P (K) = max(0, 4wg(K) − 2w − 2t0 − 2qw + 2b − 2d).

(3.4.0.11)

Claim. The equality in (3.4.0.10) does not hold unless both sides are identically zero.
Proof of the Claim. If tm
= 0 then we have two cases:
P (K)
m/w2

Case 1. Suppose tK

= 0. Using (3.4.0.11), we see (3.4.0.10) is equivalent to

4wg(K) − 2w − 2t0 − 2qw + 2b − 2d = w 2 sK .

It follows that w divides 2t0 + 2d − 2b. Since d − b ∈ {0, 1} and 1 ≤ t0 ≤ w − 2, we
conclude that w = 2t0 + 2d − 2b. Since

4wg(K) − 2w − w − 2qw = w 2 sK ,

then
4g(K) − 3 − 2q = wsK .
The right side is an even number and the left side is odd which is a contradiction.
m/w2

Case 2. Suppose tK

= 0. By expanding both sides of (3.4.0.10) and again using (3.4.0.11),

we see that

4wg(K) − 2w − 2t0 − 2qw + 2b − 2d = w 2 sK + 4w 2 ν(K) − 2w 2 − 2d − 2t0 w − 2qw 2.

46

By rearranging terms, we get

4wg(K) − 2w + 2(b − t0 ) − 2qw + 2t0 w = w 2 (4ν(K) − 2 − 2q + sK ).

Therefore w divides 2(b − t0 ). Since b and t0 are both bounded above by w − 2, we
have either 2(b − t0 ) = ±w or b = t0 .
Recall that we described P as a braid closure in Chapter 1. Viewing this braid as
a mapping class of the disk with w punctures, it is straightforward to verify that if
b = t0 , the (t0 + 1)th puncture is fixed. Therefore, in this case P has at least two
components, which contradicts P being a knot. Thus, we must have 2(b − t0 ) = ±w.
Substituting and dividing by w gives:

4g(K) − 2 ± 1 − 2q + 2t0 = w(4ν(K) − 2 − 2q + sK ).

As in Case 1, comparing the parities of each side gives a contradiction.
Having proved the claim, all the terms in (3.4.0.10) are identically zero. Since sK = 0,
(3.3.0.7) gives that K is an L-space knot. Also, tm
= 0 together with (3.4.0.11) implies
P (K)
t0 + qw + d − b
≥ 2g(K) − 1.
w

(3.4.0.12)

Since 1 ≤ t0 ≤ w − 2 and (d − b) ∈ {0, 1}, we have that 0 ≤ t0 + d − b < w. Note that
2g(K) − 1 is an integer, so we deduce that (3.4.0.12) holds if and only if

q ≥ 2g(K) − 1,

47

which implies that
b + t0 w + qw 2
≥ 2g(K) − 1,
w2
as desired.
Now suppose that ν(K) < ν(K). We claim that in this case, P (K) is not an L-space
knot, which is a contradiction. Recall from [OS11, Equation (34)] that ν(K) is equal to
either τ (K) or τ (K) + 1, and from [OS03, Lemma 3.3] that τ (K) = −τ (K). Thus, when
ν(K) < ν(K), it follows that ν(K) > 0. By [OS04c, Proposition 2.5], the total rank of
HF (Y ), for a closed three-manifold Y , is independent of the orientation of Y , i.e.,

rkHF (Y ) = rkHF (−Y ).

(3.4.0.13)

By combining (3.4.0.13), Proposition 3.2.3, and the fact that

3 (K) ∼ −S 3
Sm/n
=
−m/n (K),

(3.4.0.14)

we deduce that
3 (P (K))) = rkHF (S 3
rkHF (Sm

−m/w2

(3.4.0.15)

(K)).

By combining (3.3.0.5), (3.3.0.7), and (3.4.0.15), since P (K) is an L-space knot, we have
−m/w2
.
K

2
m + tm
P (K) = −m + w sK + t

−m/w2
K

Using (3.3.0.3) and the fact that ν(K) > 0, we observe that t

Claim. The equality in (3.4.0.16) never holds.
48

(3.4.0.16)

= 0.

Proof of the Claim. We prove the claim by considering the following two cases:
Case 1. Suppose tm
= 0. Using (3.4.0.11), by expanding both sides of (3.4.0.16) we get that
P (K)

d + t0 w + qw 2 + 4wg(K) − 2w − 2t0 − 2qw + 2b − 2d
= −d − t0 w − qw 2 + w 2 sK + 4w 2ν(K) − 2w 2 + 2d + 2t0 w + 2qw 2.

A similar reasoning as in Case 1 of the previous part of the proof shows that this
equality gives a contradiction.
Case 2. Suppose tm
= 0. Using (3.4.0.11), we see that (3.4.0.16) is equivalent to
P (K)

d + t0 w + qw 2 = −d − t0 w − qw 2 + w 2 sK + 4w 2 ν(K) − 2w 2 + 2d + 2t0 w + 2qw 2 .

This equation reduces to 2w 2 = w 2 sK + 4w 2 ν(K). However, this equation has no
solutions, since ν(K) > 0 and sK ≥ 0.
Having proved the claim, it follows that if ν(K) < ν(K), then P (K) could not have been an
L-space knot. This completes the proof.

3.5

Left-orderability

Recall that a non-trivial group G is left-orderable if there exists a left-invariant total order
on G. Examples of left-orderable groups include Z and Homeo+ (R), while any group with
torsion (e.g., a finite group) is not left-orderable. It is natural to ask which three-manifold
groups can be left-ordered. Such groups are well-suited for this study due to the following
theorem.
49

Theorem 3.5.1 (Boyer-Rolfsen-Wiest [BRW05]). Let Y be a compact, connected, irreducible, P 2 -irreducible three-manifold. If there exists a non-trivial homomorphism f : π1 (Y ) →
G where G is left-orderable, then π1 (Y ) is left-orderable. In particular, if there exists a nonzero degree map from Y to Y ′ , where π1 (Y ′ ) is left-orderable, then π1 (Y ) is left-orderable.
Rather than define P 2 -irreducible, we simply point out that if Y is orientable, then
irreducibility implies P 2 -irreducibility. For compact, orientable, irreducible three-manifolds
with b1 > 0, it then follows that their fundamental groups are always left-orderable. However,
there are more interesting phenomena for rational homology spheres; for example +3/2surgery on the left-handed trefoil has left-orderable fundamental group, while −3/2-surgery
has torsion-free, non-left-orderable fundamental group (this can be deduced for instance
from [BRW05, Theorem 1.3]). Surprisingly, the left-orderability of the fundamental groups
of three-manifolds is conjecturally characterized by Heegaard Floer homology. The following
conjecture was made in [BGW13]:
Conjecture 1.3.3 (Boyer-Gordon-Watson). Let Y be an irreducible rational homology sphere.
Then Y is an L-space if and only if π1 (Y ) is not left-orderable.
There exists a large amount of support for this conjecture, as it is known to be true
for manifolds with Seifert or Sol geometry, branched double covers of non-split alternating
links, graph manifold integer homology spheres, and many other families of examples (see
for instance [BB13, BGW13, Pet09]). We also remark that irreducibility is necessary, as
Σ(2, 3, 7)#Σ(2, 3, 5) has non-left-orderable fundamental group, but is not an L-space.
In the proof of Proposition 1.3.4 below, we remind the reader that we will be assuming
Conjecture 1.3.3.
Proof of Proposition 1.3.4. Suppose that P (K) is an L-space knot. Then for all α ∈ Q
50

with α ≥ 2g(P (K)) − 1, we have Sα3 (P (K)) is an L-space. For all but finitely many such
α, we have that Sα3 (P (K)) is irreducible as well. Thus, by Conjecture 1.3.3, we have that
π1 (Sα3 (P (K))) is not left-orderable for α ≫ 2g(P (K)) − 1.
We first study the pattern P . By [CW11, Proposition 13], for such α, π1 (Sα3 (P )) is not
left-orderable. Furthermore, for all but finitely many α, we have that Sα3 (P ) is irreducible.
Therefore, we appeal to Conjecture 1.3.3 to conclude that P is an L-space knot.
We modify the argument of [CW11, Proposition 13] to study the companion K. Recall
that w represents the winding number of P in the solid torus V . We also consider the basis
(m, ℓ) for H1 (∂V ; Z) as given in Section 3.1. We choose n ∈ Z such that gcd(w, n) = 1
and n ≫ 2g(P (K)) − 1. As discussed, we have Sn3 (P (K)) is irreducible and π1 (Sn3 (P (K))
is not left-orderable. We consider the manifold (P ; n). We have that the kernel of i∗ :
H1 (∂(P ; n); Z) → H1 ((P ; n); Z) is generated by nm+w 2 ℓ by Lemma 3.2.1. Since gcd(w, n) =
1 by assumption, we have that the element nm + w 2 ℓ is represented by a simple closed
curve on ∂(P ; n) which bounds in (P ; n). It then follows that there exists a degree one map
φ : (P ; n) → S 1 ×D 2 , which restricts to a homeomorphism on the boundary (see for instance
[Ron95, Lemma 2.2]). Since nm + w 2 ℓ bounds in (P ; n), we must have that φ(nm + w 2 ℓ) is
isotopic to {∗} × D 2 .
By extending φ to be the identity on the exterior of K, one obtains a degree one map
from Sn3 (P (K)) to S 3

n/w2

(K). Since Sn3 (P (K)) is irreducible and π1 (Sn3 (P (K))) is not left-

orderable, we have that π1 (S 3

n/w2

(K)) is not left-orderable by Theorem 3.5.1. Since w is

fixed, by choosing sufficiently large n with gcd(w, n) = 1, we can arrange that S 3

n/w2

irreducible as well. Again, by Conjecture 1.3.3, K is an L-space knot.

51

(K) is

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