SOME INEQUALITIES FOR HEEGAARD FLOER CONCORDANCE

INVARIANTS OF SATELLITE KNOTS

By

Wenzhao Chen

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2019

SOME INEQUALITIES FOR HEEGAARD FLOER CONCORDANCE INVARIANTS OF

ABSTRACT

SATELLITE KNOTS

By

Wenzhao Chen

This thesis studies the Ozsv´ath-Stipsicz-Szab´o Υ-invariant and Rasmussen’s “local h-

invariants” of satellite knots. The ﬁrst main result is an inequality relating the Υ-invariant

of a knot and that of its cable knots. This result generalizes previous work on the Ozsv´a-

Szab´o τ -invariant of cable knots due to Hedden and Van Cott. The second main result

is a set of inequalities for the local h-invariants, showing their values of a satellite knot are

constrained by those of the companion knot and pattern knot. This result generalizes crossing

change inequalities and subadditivity. Both results are applied to study the smooth knot

concordance group: we use iterated cabling operations to construct inﬁnite-rank summands

consisting of topologically slice knots; for any slice pattern with winding number greater

than one, we show iterated satellite operations yield inﬁnite-rank subgroups; we also show

for a class of winding-number-one pattern which includes the Mazur pattern, the iterated

satellite operations still yield inﬁnite-rank subgroups.

To my family

iii

ACKNOWLEDGMENTS

Throughout my years in MSU, I have received tremendous help from many kind people

around me, especially those in the Geometry/Topology group. I deeply thank my advisor

Matt Hedden, who teaches me hour after hour, provides me with ongoing support, and

encourages me with words of wisdom. I thank Eﬃe Kalfagianni, who is always there to help

and always makes me feel welcome. I also thank Casim Abbas, Selman Akbulut, Gorapada

Bera, Andrew Donald, David Duncun, Kristen Hendricks, Samuel Lin, Abhishek Mallick,

Akos Nagy, Kyungbae Park, Katherine Raoux, Ben Schmidt, Tsvetanka Sendova, Faramarz

Vafaee, Eylem Yildiz; they all taught or helped me. Thank you all for making me feel

blessed.

I am also grateful to the kind people I met from other institutes. I would like to thank Min

Hoon Kim, Chuck Livingston, Kouki Sato, and Zhongtao Wu, who talked to me interesting

math and oﬀered me indispensable help.

Finally, I thank Anqi Chen, Jun Du, Rami Fakhry, Paata Ivanisvili, Zhaofeng Li, Sami

Merhi, Wenchuan Tian, Menglun Wang, and Xin Yang for being my supportive and won-

derful friends.

iv

TABLE OF CONTENTS

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

vi

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
1.1 The Upsilon invariant of cable knots
1.2 Local h invariants of satellite knots . . . . . . . . . . . . . . . . . . . . . . .
1.2.1
Subadditivity and crossing change inequalities . . . . . . . . . . . . .
1.2.2 The ν+-invariant of cable knots . . . . . . . . . . . . . . . . . . . . .
Iterated satellite operations and the smooth knot concordance group
1.2.3

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Heegaard Floer correction term and the Vi sequence . . . . . . . . . . . . . .
2.2 The Upsilon invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 A cabling inequality for the upsilon invariant

. . . . . . . . . .
3.1 Upsilon of (p, pn + 1)-cable . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Upsilon of (p, q)-cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Applications of the cabling inequality . . . . . . . . . . . . . . . . . . . . . .
(t) . . . . . . . . . . . . . . . . . . . .
3.3.1 Computation of Υ(T2,−3)2,2n+1
3.3.2 An inﬁnite-rank summand of topologically slice knots . . . . . . . . .

Chapter 4 The Vi-invariants of satellite knots

. . . . . . . . . . . . . . . . .
4.1 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Subadditivity and crossing change inequalities . . . . . . . . . . . . . . . . .
4.3 Proof of the cabling formula for ν+ . . . . . . . . . . . . . . . . . . . . . . .
4.4 The upsilon invariant and iterated satellite operations . . . . . . . . . . . . .
4.5 Proof of Theorem 1.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
. . . . . . . . . . . . . . . . . . . . . . . . . . .

Iterated satellite operations

1
3
6
7
8
10
11

12
12
18

22
22
31
33
33
36

38
38
42
43
47
47
49

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

50

v

LIST OF FIGURES

Figure 1.1: The Mazur pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.1: These ﬁgures shows CF K∞(T3,4) with the corresponding minimal Ft-
ﬁltered subcomplexes such that the inclusion map induces a nontrivial
map on the homology groups of Maslov grading zero: (a) corresponds to
0 ≤ t < 1, (b) corresponds to t = 1, and (c) corresponds to 1 < t ≤ 2.
. .

Figure 3.1: A compatible Heegaard diagram H for K. λ is a 2-framed longitude, and
according to our assumption that the 0-framed longitude can be chosen
not to hit αg, λ can be chosen to intersect αg twice. . . . . . . . . . . . .

Figure 3.2: A example of H(p, n) with n = 2 and p = 3, corresponding to the Heegaard
diagram shown in Figure 3.1. There is an obvious arc of ˜β connecting x4
0, which neither intersects δ nor δ(cid:48). By our convention, there is
and y1
an arc of ˜β connecting x4 and y2
0 satisfying the same property as well,
though it is not shown in the ﬁgure. The shaded region represents a
domain connecting {x1, a} and {x2, a}; the darkened color indicates the
. . .
multiplicity is 2, while the lighter colored region has multiplicity 1.
Figure 3.3: The thickened curve γ represents the -class between {x0, a} and {x6, a}.
Note that the arc δ and δ(cid:48) which connect based points do not intersect γ.

Figure 3.4: The thickened curve is an arc on ˜β connecting y

that does not
intersect δ nor δ(cid:48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and y

(1)
0

(1)
2

11

19

24

25

27

28

Figure 4.1: A Kirby diagram for W . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Figure 4.2: Another Kirby diagram for W obtained by applying handle-slide to the
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

diagram in Figure 4.1

Figure 4.3: The pattern Pm(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: Obtaining K− from K+ by a full twist . . . . . . . . . . . . . . . . . . .
Figure 4.5: An example showing C{max(i, j) ≤ V0} ⊂ F−1

(−∞, V0]

. . . . . . . . .

t

39

41

42

48

vi

Chapter 1

Introduction

A knot is called slice if it bounds a smoothly embedded disk in the 4-ball. Sliceness induces

an equivalence relation called concordance: two knots K0 and K1 are smoothly concordant
if K0#− K1 is slice. Equivalently, K0 and K1 are smoothly concordant if there is a smooth
proper embedding f : S1 × [0, 1] → S3 × [0, 1] such that f (S1 × {i}) = Ki, i = 0, 1. Under

the binary operation induced by connected sum, the concordance classes of knots form a
group C known as the smooth knot concordance group. This group was introduced by Fox

and Milnor in 1966 [10].

Determining the group structure of C is a natural question of central importance.

In
this direction, Levine proved the algebraic concordance group, a quotient group of C, is
isomorphic to Z∞⊕Z∞
4 [27]. Overall, the group structure of C is still quite mysterious.
For example, the question of whether C contains 4-torsion elements remains open. From the
group theoretic point of view, the goal might be to completely classify C, but such pursuit is

2 ⊕Z∞

often motivated by and beneﬁts from the quest for understanding topological properties of

knots, 3- and 4-manifolds. For example, the algebraic concordance group mentioned above

is motivated by a topological operation called ambient surgery, which is used to construct

surfaces embedded in the 4-ball. Another typical example is the study of topologically slice
knots. Note one can analogously deﬁne the topological knot concordance group Ctop using
locally ﬂat embeddings instead of smooth embeddings of surfaces. The kernel T of the

1

forgetful map C → Ctop consists of concordance classes of knots that are topologically slice.
T represents the diﬀerence between the smooth and topological categories; its non-triviality,
for instance, implies the existence of exotic R4. It is natural to ask for a classiﬁcation of
T . It is known T has a Z∞-summand [19, 21, 7, 38] and a Z∞
questions concerning T , like whether it contains a Z∞

2 -subgroup [16]. Many simple

2 -summand, remain open.

The satellite operation is a way to construct knots and is helpful to the study of knot
concordance. To deﬁne it, let P ⊂ S1 × D2 be a nontrivial knot and ﬁx an orientation of
the longitude S1 × {pt} on the boundary of the solid torus, where pt ∈ ∂D2. Now let K

be an oriented knot, and V be a regular neighborhood of K; let l be a push-oﬀ of K to the

boundary of V , oriented parallel to K, with lk(l, K) = r. Then the r-twisted satellite knot
Pr(K) is deﬁned to be the image of P under a homeomorphism f : S1 × D2 → V , where f
maps S1 ×{pt} to l respecting the orientation. P is called the pattern knot, and K is called

the companion knot. P (K) is understood as the untwisted satellite knot P0(K).

One key observation is that, given a pattern knot P , the corresponding satellite oper-
ation induces a well-deﬁned operation on C. On one hand, this opens up the possibility

of strengthening any concordance invariant: two knots K1 and K2 are not concordant if a

given concordance invariant takes on diﬀerent values on P (K1) and P (K2) for some P , even

though its values at K1 and K2 may coincide. On the other hand, the satellite operation

often oﬀers a convenient way to construct knots with speciﬁc properties . As a simple ex-

ample, if both the pattern knot P and the companion knot K are topologically slice, then

so is the satellite knot P (K).

Exploiting satellite operations requires understanding the eﬀect of such operations on

the concordance invariants at hand. A classical example is Litherland’s satellite formula for

the Levine-Tristram signature invariant, which was used to prove algebraic knots are weakly

2

independent [28].

In the last decade, a host of powerful concordance invariants were discovered from Hee-

gaard Floer homology. Instead of listing all of them out, we refer the interested readers to a

survey due to Hom [20]. It is worth pointing out, however, that a common feature to all of

such invariants is their ability to discern topological sliceness and smooth sliceness, and hence
they provide sharp tools to study topological slice knots. In fact, the results that T contains
a Z∞-summand and Z∞

2 -subgroups are both proved using Heegaard Floer invariants.

In

this thesis, we will focus on the eﬀect of satellite operations on two such concordance in-

variants, the Ozsv´ath-Stipsicz-Szab´o Υ-invariant and Rasmussen’s “local h-invariants”. The

results are given as inequalities constraining the value of the invariant of the satellite knot in

terms of its values on the pattern knot and companion knot. More detailed statement and

motivation are given in Section 1.1 and Section 1.2.

1.1 The Upsilon invariant of cable knots

Using Heegaard Floer theory, Ozsv´ath, Stipsicz and Szab´o recently constructed a concor-
dance homomorphism Υ : C → CP L([0, 2]), where CP L([0, 2]) is the group of piecewise linear
functions over [0, 2] [38]. The Υ function is a strong tool for distinguishing elements in C.
In particular, it can be applied to prove that T contains a Z∞-summand. Besides being a

concordance homomorphism, ΥK (t) also enjoys many nice properties. For example, it gives
|ΥK (t)| ≤ tg4(K) for 0 ≤ t ≤ 1. It also recovers the

lower bounds for the 4-ball genus:
earlier known invariant τ : the slope of ΥK (t) at t = 0 is −τ (K).

The eﬀect of cabling for the τ -invariant is thoroughly studied [12, 13, 15, 18, 42, 48]. We

restate two results for our purpose. The ﬁrst one is due to Hedden, obtained by carefully

3

comparing a particular representing Floer chain complex of a knot and that of its cable.

Theorem 1.1.1. ([15])Let K ⊂ S3 be a knot, and p > 0, n ∈ Z. Then

pτ (K) +

pn(p − 1)

2

≤ τ (Kp,pn+1) ≤ pτ (K) +

(pn + 2)(p − 1)

2

,

Later Van Cott used the genus bound and homomorphism property satisﬁed by τ , to-

gether with nice constructions of cobordism between cable knots to extend the above in-

equality to (p, q)-cables.

Theorem 1.1.2. ([48]) Let K ⊂ S3 be a knot, and (p, q) be a pair of relatively prime

numbers such that p > 0. Then

pτ (K) +

(p − 1)(q − 1)

2

≤ τ (Kp,q) ≤ pτ (K) +

(p − 1)(q + 1)

2

,

In view of the success in understanding the eﬀect of knot cabling on the τ -invariant, it

is natural to wonder what happens to Υ. In this thesis we show that a portion of Υ behaves

very similarly to τ . Indeed, adapting the strategies of Hedden and Van Cott on studying τ

to the context of Υ, we can prove the following result:

Theorem 1.1.3. Let K ⊂ S3 be a knot, and (p, q) be a pair of relatively prime numbers

such that p > 0. Then

ΥK (pt) − (p − 1)(q + 1)t

2

≤ ΥKp,q (t) ≤ ΥK (pt) − (p − 1)(q − 1)t

2

,

when 0 ≤ t ≤ 2
p.

Note that by diﬀerentiating the above inequality at t = 0, we recover Theorem 1.1.2.

4

One also easily sees this inequality is sharp by examining the case when K is the unknot:

when q > 0, then the upper bound is achieved, and when q < 0 the lower bound is achieved.

However, when p > 2 the behavior of ΥKp,q (t) for t ∈ [ 2

p , 2 − 2

p] is still unknown to the

author. For L-space knots, however, there is a cabling formula due to Tange [47].

Theorem 1.1.3 can often be used to determine the Υ function of cables with limited

knowledge of their complete knot Floer chain complexes. For an example, we show how our
(t) for n ≥ 8; these examples are, on the face

theorem can be used to deduce Υ(T2,−3)2,2n+1

of it, rather diﬃcult to compute, since none of them is an L-space knot. Despite this, armed

only with Theorem 1.1.3 and the knot Floer homology groups of the knots in this family, we

are able to obtain complete knowledge of Υ.

Perhaps more striking, however, is that Theorem 1.1.3 can be used to easily show that

certain subsets of the smooth concordance group freely generate inﬁnite-rank summands. To

this end, let D = W h+(T2,3) be the untwisted positive whitehead double of the trefoil knot

and let Jn = ((Dp,1)...)p,1 denote the n-fold iterated (p, 1)-cable of D for some ﬁxed p > 1

and some positive integer n. Theorem 1.1.3, together with general properties of Υ, yield the

following

Corollary 1.1.1. The family of knots Jn for n = 1, 2, 3, ... are linearly independent in C

and span an inﬁnite-rank summand consisting of topologically slice knots.

To the best of the author’s knowledge, Corollary 1.1.1 provides the ﬁrst satellite operator

on the smooth concordance group of topologically slice knots whose iterates (for a ﬁxed knot)

are known to be independent; moreover, in this case they are a summand. Note Jn has triv-

ial Alexander polynomial. The ﬁrst known example of inﬁnite-rank summand of knots with

5

trivial Alexander polynomial is generated by Dn,1 for n ∈ Z+, due to Kim and Park [24].

Their example, however, like the families of topologically slice knots studied by [38], involved

rather non-trivial calculuatons of Υ (for instance, those of [38] involved rather technical ca-

clulations with the bordered Floer invariants). The utility of Theorem 1.1.3 is highlighted

by the ease with which the above family is handled. We also refer the interested reader to [9]

for a host of other applications of Theorem 1.1.3 to the study of the knot concordance group.

1.2 Local h invariants of satellite knots

Given a knot K in the 3-sphere, there is a also family of integer-valued knot invariants
{Vi(K)|i ∈ Z} derived from Heegaard Floer homology, the so-called “local h-invariants”

introduced by Rasmussen [43]. The squence Vi(K) is vital to computing the Heegaard Floer

homology groups of Dehn surgeries along K. Moreover, the Vi’s are also important knot

concordance invariants. For instance, they provide lower bounds for the 4-ball genus [43],

and this has been employed by Hom and Wu to deﬁne the ν+-invariant, which is the best

4-genus bound derived from Heegaard Floer homology so far [22]. V0(K) may also be used

to give a lower bound for the smooth 4-dimensional crosscap number [2].

We give some bounds for the Vi-invariants of satellite knots. These inequalities are

inspired by Roberts’ results in [45], which concerns the behavior of the Ozsv´ath-Szab´o τ -

invariant under satellite operations. First we establish some notation. Given a pattern knot
P ⊂ S1× D2, we will also use P to denote P (U ) when viewed as knot in the 3-sphere, where

U is the unknot. More generally, Pr denotes the knot Pr(U ). The theorem is then stated

below.

6

Theorem 1.2.1. Let r be a positive integer, and P be a pattern knot with winding number
l ≥ 0. Then for any knot K ⊂ S3, and integers i1, i2 such that 0 ≤ i1 ≤ (cid:98) r

2(cid:99) and i2 ≥ 0, we

have

V

li1+i2+

lr(l−1)

2

(Pr(K)) ≤ Vi1

(K) + Vi2

(P ),

and for any other integer m, we also have

Vi2

(Pm(K)) ≥ V

li1+i2+

(Pm+r) − Vi1

( ¯K)

lr(l−1)

2

(Here ¯K is the mirror image of K).

We provide a few applications of Theorem 1.2.1.

1.2.1 Subadditivity and crossing change inequalities

Both the connected sum of two knots and crossing change can be viewed as special cases of

satellite operations. Restricting to these two cases, Theorem 1.2.1 recovers the subadditivity

and crossing change inequalities of the Vi’s, and hence can be viewed as a uniﬁcation of these

properties. More precisely, we give new proofs to the following results.

Propsition 1 (Proposition 6.1 of [3]). Let K and L be two knots in S3, and i1, i2 be two

non-negative integers, then

Vi1+i2

(K#L) ≤ Vi1

(K) + Vi2

(L).

In particular, ν+(K#L) ≤ ν+(K) + ν+(L).

Theorem 1.2.2 (Theorem 6.1 of [4] and Theorem 1.3 of [3]). Let K+ and K− be a pair

7

of knots such that K− is obtained from K+ by changing a positive crossing, then for any

non-negative integer i,

In particular,

Vi+1(K+) ≤ Vi(K−) ≤ Vi(K+).

ν+(K−) ≤ ν+(K+) ≤ ν+(K−) + 1.

1.2.2 The ν+-invariant of cable knots
In [49], Wu proved a cabling formula for the ν+-invariant: ν+(Kp,q) = pν+(K) + (p−1)(q−1)
when q ≥ (2ν+(K) − 1)p − 1, where Kp,q denotes the (p, q)-cable of K. In this note, we

2

give a new proof to a slightly weaker version of Wu’s formula. The reason for including a

less strong result is we expect our approach can be generalized to obtain bounds for the

ν+-invariant under satellite operations with other pattern knots (though one shall not hope

to get an equality in general). In fact, when establishing this cabling formula, we ﬁrst arrive

at an upper bound that follows easily from the following corollary of Theorem 1.2.1.

Propsition 2. Let P , K, r, l be as in Theorem 1.2.1. When r ≥ max(2ν+(K), 1),

ν+(Pr(K)) ≤ lν+(K) + ν+(P ) +

lr(l − 1)

2

.

Apart from giving us a bound in the case of cabling operations, Proposition 2 is of

independent interest due to its generality. For instance, it recovers the corresponding results

on the ν+-invariant appeared in Proposition 1 and Theorem 1.2.2. It also implies that for
any winding number zero slice pattern P , ν+(Pr(K)) = 0 when r ≥ 2ν+(K). Note this

observation resembles and partially implies a result of Hedden on Whitehead doubles, which

8

states τ (W h+

r (K)) = 0 if and only if r ≥ 2τ (K) [14].

To obtain a lower bound for the ν+-invariant of cable knots, we give a new proof to the

following inequality.

Propsition 3 (Theorem 1.3 of [46]). Let K be a knot in S3, and p, q be a pair of coprime

integers such that p > 0. Then

ν+(Kp,q) ≥ pν+(K) +

(p − 1)(q − 1)

2

.

In [46], Sato’s proof of Proposition 3 utilized Wu’s cabling formula in conjunction with a

full-twist inequality for the ν+-invariant. Such a proof may also be recovered using Theorem

1.2.1 (see Corollary 4.3.2 and Remark 4.3.1). To avoid circular reasoning, however, our

strategy is to compare the knot Floer chain complexes by a technical result due to the

author in [5], developed from Hedden’s method in [12, 15]. This approach can be used for

studying other pattern knots which admit a genus-one doubly pointed bordered Heegaard

diagram.

The above results then gives the following cabling formula.

Propsition 4 (A weaker version of Theorem 1.1 of [49]). Let K be a knot in S3, p > 0 and
q ≥ max(2ν+(K), 1)p + 1. Then

ν+(Kp,q) = pν+(K) +

(p − 1)(q − 1)

2

.

9

1.2.3 Iterated satellite operations and the smooth knot concor-

dance group

We ﬁrst give a bound for the Υ-invariant in terms of V0, which leads to an estimate of the

ﬁrst singularity in terms of the Osz´ath-Szab´o τ -invariant and V0.

Theorem 1.2.3. Given an arbitrary knot K, we have −2V0(K) ≤ ΥK (t) ≤ 2V0( ¯K). In
2V0(K)
τ (K) ].

particular if τ (K) > V0(K) > 0, then the ﬁrst singularity ξ of ΥK (t) is in [

1

g3(K) ,

Here g3(K) denotes the Seifert-genus of K.

Corollary 1.1.1 shows iterated cabling operations yield Z∞-subgroups in T . Here we show

such phenomenon does not limit to the cabling operation. In fact, Theorem 1.2.3 coupled
with Theorem 1.2.1 implies one may construct Z∞-subgroups with ease using general iterated

satellite operations, as stated in the following theorem.

Theorem 1.2.4. For any slice pattern P with winding number l ≥ 2, there exists a topologi-
cally slice knot K such that {P n(K)|n ∈ Z+} contains a subset that generate a Z∞ subgroup
in T .

For patterns of winding number one, we content ourselves to iterated satellite operations

with the Mazur pattern M (Figure 1.1). However, we point out that the following proposition

works for any pattern knots satisfying the condition of the main theorem of [44].

Propsition 5. Let D = W h+(T2,3), then there exists a subset of {M n(D)} that generates
a Z∞ subgroup in T .

Remark 1.2.1. Recall Feller, Park and Ray asked whether there exists a knot K such that
{M n(K)} span an inﬁnite rank subgroup (or summand) in C (or T ) (Question 1.2 of [9]).

Proposition 5 may be viewed as a partial answer to this question.

10

Figure 1.1: The Mazur pattern

Other than distingushing knot concordance classes, it is also worth pointing out that the

ﬁrst singularity ξ of ΥK (t) is conjectured to be related to the braid index b(K) by Feller and
Krcatovich in the form that b(K) ≥ 2

ξ (Question 5.1 of [8]). In view of this, one may ask the

following weaker question.

Question 1. Is it true that b(K) ≥ τ (K)

V0(K) when τ (K) > V0(K) > 0.

1.3 Outline

The thesis is organized as follows. In Chapter 2 we give a quick review of Heegaard Floer

homology, introducing various knot concordance invariants that we consider. In Chapter 3

we give a proof Theorem 1.1.3 and discuss the pertaining applications.

In Chapter 4 we

prove Theorem 1.2.1 and the relevant applications.

11

Chapter 2

Preliminaries

2.1 Heegaard Floer correction term and the Vi sequence

Let F = Z/2Z. Given a closed oriented 3-manifold Y together with a spinc-structure

s, with the choice certain auxiliary data, in [36] Oszv´ath and Szab´o associated to (Y, s)
a chain complex (CF∞(Y, s), ∂), whose underlying abelian group is a ﬁnitely generated
F[U, U−1]-module. CF∞(Y, s) admits a Z-ﬁltration called the algebraic ﬁltration, i.e.
each i ∈ Z, there is a subcomplex Fi of CF∞(Y, s) such that Fi ⊂ Fj whenever i ≤ j,
and CF∞(Y, s) = ∪iFi. This ﬁltration is compatible with the boundary map, i.e.
for
any i, ∂(Fi) ⊂ Fi. The algebraic ﬁltration allows construction of various chain groups:
CF−(Y, s) = F0, CF +(Y, s) = CF∞(Y, s)/F−1, and (cid:100)CF (Y, s) = F0/F−1. The Heegaard
Floer homology groups HF◦(Y, s), ◦ = ∞,−, +,(cid:98)· are deﬁned to be the homology groups of

for

the aforementioned chain complexes. The ﬁltered chain homotopy type of CF∞(Y, s) turns

out to be independent of the choice of auxiliary data, hence so are the isomorphism type of

the various homology groups.

Given a cobordism W from Y1 to Y2, and t ∈ Spinc(W ), there are induced maps on the

Heegaard Floer homology groups

f◦
W,t : CF◦(Y1, t|Y1) → CF◦(Y2, t|Y2).

12

This TQFT-like property allows one to bestow an absolute Q-valued grading ˜gr (called the
Maslov grading) on HF◦(Y, s) when Y is a rational homology sphere (Theorem 7.1 of [37]),

satisfying the following three properties:

• (cid:100)HF (S3) ∼= F is supported in grading zero

• The natural inclusion and projection maps between various versions of CF◦ preserve

the grading, the boundary map decreases it by one, and the U -action decreases it by

two.

• If W is a cobordism from Y1 to Y2, t ∈ Spinc(W ), and si = t|Yi, then for a homogeneous
element ξ ∈ CF◦(Y1, t1),

˜gr(f◦

W,t(ξ)) − ˜gr(ξ) =

c1(t)2 − 3σ(W ) − 2χ(W )

4

.

On the other hand, for any rational homology sphere Y , HF∞(Y, s) = F[U, U−1] by
Theorem 10.1 of [35]. Using this structure theorem and the absolute grading of HF◦,

Ozsv´ath and Szab´o deﬁned a numerical invariant called the correction term d [33]:

Deﬁnition 2.1.1. Let Y be a rational homology sphere and s ∈ Spinc(Y ),

d(Y, s) = min{ ˜gr(ξ)|ξ ∈ Im(HF∞(Y, s) → HF +(Y, s))}.

One notable property of the d-invariant is the rational homology cobordism invariance:
if W is a cobordism between two rational homology spheres Y1 and Y2, and H∗(W ; Q) =
H∗(S3 × I; Q), then for any t ∈ Spinc(W ), d(Y1, t|Y1) = d(Y2, t|Y2).

The rational homology cobordism invariance of the d-invariant is used to construct knot

13

concordance invariants. Manolescu and Owens deﬁned δ(K) for a knot K to be 2d(Σ(K), t0),

where Σ(K) is the double branched cover and t0 corresponds to the unique spin structure

[31]. Jabuka deﬁned further invariants in this vein using higher order branched covers [23].

Other than using the branched-cover techinque as the source of constructing 3-manifolds,
Peters studied the correction terms of ±1-surgery along knots [41]. Peters’ construction can

be taken further by using larger surgery coeﬃcients, and this is neatly manifested as the

Vi-invariants which we recall below.

The Vi’s are best deﬁned in terms of knot Floer chain complexes. Recall a K ⊂ S3 induces
another Z-ﬁltration, called the Alexander ﬁltration, on some representative of CF∞(S3)

(since there is only one spinc-structure on S3 we suppressed it from the notation). The

Alexander ﬁltration is also compatible with the boundary map. Together with the algebraic
ﬁltraion and the Maslov grading, this gives rise to a Z ⊕ Z-ﬁltered, Maslov graded chain
complex CF K∞(K). This is called the knot Floer chain complex and its bi-ﬁltered chain
homotopy type is a knot invariant [34, 43]. Recall in general, given a Z-ﬁltered chain complex
(C,F), one may deﬁne the ﬁltration degree function degF : C → Z ∪ {−∞} by

min{i | x ∈ Fi} x (cid:54)= 0

−∞

degF (x) =

x = 0

Hence the algebraic ﬁltration induces a ﬁltration degree on the knot Floer chain complex,

called algebraic degree (or algebraic grading) Alg. Similarly there is an Alexander degree (or

Alexander grading) Alex induced by the Alexander ﬁltration. Multiplication by U decreases

both the algebraic grading and Alexander grading by one.

Denote C = CF K∞(K) for convenience. Following the standard notation in the liter-

14

ature, we use indices i and j to refer to the algebraic grading and the Alexander grading
respectively. For instance, C{i < a, j < b} refers to the ﬁltered subcomplex generated

by elements with algebraic grading less than a and Alexander grading less than b, while
C{i ≥ a or j ≥ b} = C/C{i < a, j < b}.

Given an integer s, consider the natural projection map

vs : C{max(i, j − s) ≥ 0} → C{i ≥ 0}.

By Theorem 2.3 of [40], vs can be identiﬁed with the cobordism map

FW(cid:48)

m(K),t : CF +(S3

m(K), t|S3

m(K)) → CF +(S3).

Here m is a suﬃciently large integer, W(cid:48)

m(K) is the orientation reversal of the 2-handle

attachment cobordism from S3 to S3

m(K), and t is the spinc-structure satisfying

(cid:104)c1(t), [ ˆFK ](cid:105) + m = 2s,

where ˆFK is a capped-oﬀ Seifert surface for K. We often denote t|S3
Spinc(S3

m(K)). Denote by T +

d the image of

m(K) by [s] ∈ Z/mZ =

HF∞(S3

m(K), [s]) → HF +(S3

m(K), [s]).

d = F[U−1] with 1 supported in grading d = d(S3

m(K), [s]). Also note HF +(S3) =

Here T +
T +
0 . Deﬁne

Vs(K) = dim ker{(vs)∗|T +

d : T +

0 }.
d → T +

15

In other words, U−Vs(K) ∈ T +

d is mapped to 1 ∈ T +

Note the U -action decreases the grading by two, and the grading shift of FW(cid:48)
depends on m and s, so d(S3

m(K),t.
m(K),t only
m(K), [s]) may be expressed in terms of m, s, and Vs(K) when

0 under the cobordism map FW(cid:48)

m is large. More generally, we have following formula due to Ni and Wu.

Propsition 6 (Proposition 1.6 of [32]). Suppose p, q > 0 and 0 ≤ i ≤ p − 1. Then

d(S3

p/q(K), [i]) = d(L(p, q), [i]) − 2 max(V(cid:98) i

q(cid:99)(K), V(cid:98) p+q−i−1

q

(cid:99)(K)),

where L(p, q) denotes the lens space.

It follows immediately from the rational homology cobordism invariance of d and Propo-

sition 6 that the Vi’s are indeed knot concordance invariants. Moreover, Vi’s provide lower

bounds for 4-genus thanks to the following properties.

Propsition 7 (Proposition 7.6 of [43]). For any integer i and knot K, we have Vi+1(K) ≤
Vi(K) ≤ Vi+1(K) + 1.

Propsition 8 (Corollary 7.4 of [43]). Given an arbitrary knot K in S3, we have Vg4(K)(K) =
0. In particular, Vi(K) + i ≤ g4(K) when Vi(K) > 0.

Proof. Assume g4(K) > 0 for nontriviality, and pick m to be a suﬃciently large integer.

m(K), i) = T +

d ⊕HFred(S3

Write HF +(S3
The cobordism W(cid:48)

m(K), i), where HFred(S3

m(K), i) = HF +(S3

m(K), i)/T +
d .

m(K) induces the map T +

d → T +

0 = HF +(S3) which we used to deﬁne Vi.

However, we may also deﬁne Vi by turning the cobordism around and examine the induced

map

HF +(−S3) → HF +(−S3−m(K), i) → T +−d,

16

whose rank of the kernel is easily seen to be Vi by looking at the grading shift (note this

map is surjective since the cobordism is negative deﬁnite).

Note the map HF +(−S3) → HF +(−S3−m(K), i) ﬁts into the long exact sequence given

in Remark 9.20 of [35]:

··· → HF +(−S3

0(K), i) → HF +(−S3) → HF +(−S3−m(K), i) → ···

(Note we chose m big enough so that only one spinc-structure of −S3

0(K) matters).

When s = g4(K), the cobordism map HF +(−S3

0(K), i) → HF +(−S3) factors through
HF +(Σg4(K) × S1, g4(K)), which is trivial by the adjunction inequality (Theorem 7.1 of
[35]). Therefore HF +(−S3

0(K), i) → HF +(−S3) is trivial and hence

HF +(−S3) → HF +(−S3−m(K), i) → T +−d

is an isomorphism. Therefore, Vg4(K) = 0. The inequality Vi(K) + i ≤ g4(K) now follows

readily with the help Proposition 7.

In [22], Hom and Wu deﬁned the ν+-invariant of a knot.

Deﬁnition 2.1.2. Given a knot K,

ν+(K) = min{i|Vi(K) = 0}.

It follows immediately from Proposition 8 that ν+(K) ≤ g4(K).

17

2.2 The Upsilon invariant

In this section, we brieﬂy review the construction and basic properties of the upsilon invari-

ant. The original deﬁnition of ΥK (t) is based on the t-modiﬁed knot Floer chain complex

[38]. Shortly thereafter, Livingston reformulated ΥK (t) in terms of the complete knot Floer
chain complex CF K∞(K) [29]. We ﬁnd it convenient to work with Livingston’s deﬁnition,

which we recall below.

Denote CF K∞(K) by C(K) for convenience. For any t ∈ [0, 2], one can deﬁne a ﬁltration

on C(K) as follows. First, deﬁne a real-valued (grading) function on C(K) by

Ft =

t
2

Alex + (1 − t
2

)Alg,

which is a convex linear combination of Alexander and algebraic gradings. Associated to
this function, one can construct a ﬁltration given by (C(K),Ft)s = F−1
(−∞, s]. It is easy
to see that the ﬁltration induced by Ft is compatible with the diﬀerential of C(K), i.e.
Ft(∂x) ≤ Ft(x), ∀x ∈ C(K). Let

t

ν(C(K),Ft) = min{s ∈ R|H0((C(K),Ft)s) → H0(C(K)) is nontrivial}.

Here H0 stands for the homology group with Maslov grading 0.

Deﬁnition 2.2.1. ΥK (t) = −2ν(C(K),Ft).

An example is included below to help the reader digest the deﬁnition.

Example 1. We compute the upsilon invariant of the (3, 4)-torus knot T3,4. CF K∞(T3,4)
is shown in Figure 2.1. Here we only depict a generating set of CF K∞(T3,4) that contains

18

all the elements of Maslov grading zero, which are x1, x2 and x3. In general, the Ft-ﬁltered
subcomplexes (C(K),Ft)s correspond to planar regions bounded by straight lines with non-
positive slopes, when t varies from 0 to 2, the slope varies from −∞ to 0, and when s

increases the boundary is shifted upwards.

y1

x1

x2

y2

x3

y1

x1

x2

y2

x3

y1

x1

x2

y2

x3

(a)

(b)

(c)

Figure 2.1: These ﬁgures shows CF K∞(T3,4) with the corresponding minimal Ft-ﬁltered
subcomplexes such that the inclusion map induces a nontrivial map on the homology groups
of Maslov grading zero: (a) corresponds to 0 ≤ t < 1, (b) corresponds to t = 1, and (c)
corresponds to 1 < t ≤ 2.

For T3,4, when 0 ≤ t < 1, one can see from Figure 2.1(a) that the minimal Ft-ﬁltered
subcomplexes such that H0((C(T3,4),Ft)s) → H0(C(T3,4)) is nontrivial must have its bound-
ary containing x1, which has (Alg, Alex)-ﬁltration level (0, 3). Therefore ν(C(T3,4),Ft) =
2 · 3 + (1 − t
2 = −3t when 0 ≤ t < 1. Similarly one can see
(1) = −3 and t = 1 is the singularity of ΥT3,4
ΥT3,4
(t).
One can see from Figure 2.1(b) that minimal F1-ﬁltered subcomplexes contains 3-cycles that
can generate H0(C∞(T3,4)): when t < 1, the minimal subcomplexes hinge on x1, while when

(t) = 3t − 6 when 1 < t ≤ 2. ΥT3,4

(t) = −2 · 3t

t

2) · 0 = 3t

2 , so ΥT3,4

t > 1, the minimal subcomplexes hinges on x3.

We recall a few general properties of the upsilon invariant. Let CP L([0, 2]) denote the

group of piece-wise linear function over [0, 2].

19

Theorem 2.2.1 ([38]). For any knot K ⊂ S3, ΥK (t) ∈ CP L([0, 2]). Moreover, the up-
silon function Υ : C → CP L([0, 2]) is a concordance homomorphism satisfying the following

properties.

(1) (Symmetry)

(2) (4-genus bound)

for 0 ≤ t ≤ 1,

ΥK (t) = ΥK (2 − t),

|ΥK (t)| ≤ tν+(K) ≤ tg4(K)

(3) (Recovers the Ozsv´ath-Szab´o τ -invariant)

There is  > 0 such that

ΥK (t) = −τ (K)t

for t ∈ [0, ].

Other than these nice properties, it is also shown that ΥK (1) can be used to give an

lower bound to the smooth 4-dimensional crosscap number [39]. When it comes constructing
concordance homomorphisms from C to Z∞, it is helpful to consider the slope jump of the

upsilon invariant:

Υ(cid:48)
K (t0) − lim
t(cid:37)t0
where t0 ∈ (0, 2). It is clear from the deﬁnition that ∆Υ(cid:48)

K (t0) = lim
t(cid:38)t0

K (t0) is non-zero only at singular

∆Υ(cid:48)

Υ(cid:48)
K (t0),

points of the ΥK (t). More generally, we have the following property concerning the slope

jump.

Theorem 2.2.2 ([38]). For any knot K and any t0 ∈ (0, 2),

2 ∆Υ(cid:48)
t0

K (t0) is an integer.

20

Remark 2.2.1. In view of the above theorem, given any sequence {ξn} such that ξn ∈ [0, 2],
one can then construct a homomorphism C → Z∞ via [K] (cid:55)→ ( ξn

2 ∆Υ(cid:48)

K (ξn))∞
n=1.

Finally, we simply point out the idea of constructing the upsilon invariant has led to

the discovery of many other variants. This include the upsilon type invariants associated to

south-west type planar regaions [1], the secondary upsilon invariant [25], and the involutive

upsilon invariant [17].

21

Chapter 3

A cabling inequality for the upsilon

invariant

The proof of Theorem 1.1.3 will be divided into two parts:

in Section 3.1 we will prove

the inequality for the (p, pn + 1) cable of a knot by adapting Hedden’s strategy in [12, 15],

and then in Section 3.2 we will upgrade the inequality to cover the (p, q)-cable of a knot by

applying Van Cott’s argument in [48].

3.1 Upsilon of (p, pn + 1)-cable

Following [12, 13, 15], we will begin with introducing a nice Heegaard diagram which encodes

both the original knot K and its cable Kp,pn+1.

For any knot K ⊂ S3, let H = (Σ,{α1, ..., αg},{β1, ..., βg−1, µ}, w, z) be a compatible

Heegaard diagram for it. Moreover, by stabilizing we can assume µ to be the meridian of

the knot K and that it only intersects αg, and there is a 0-framed longitude of the knot K

on Σ which does not intersect αg. From now on, we will always assume that the Heegaard

diagram H for K satisﬁes all these properties.

Let H = (Σ,{α1, ..., αg},{β1, ..., βg−1, µ}, w, z) be a Heegaard diagram for the knot K as
above. By modifying µ and adding an extra base point z(cid:48), we can construct a new Heegaard
diagram with three base points H(p, n) = (Σ,{α1, ..., αg},{β1, ..., βg−1, ˜β}, w, z, z(cid:48)). More

22

precisely, ˜β is obtained by winding µ along an n-framed longitude (p− 1) times, and the new
base point z(cid:48) is placed at the tip of the winding region such that the arc δ(cid:48) connecting w and
z(cid:48) has intersection number p with ˜β. Note ˜β can be deformed to µ through an isotopy that
does not cross the base points {w, z}. See Figure 3.1 and Figure 3.2 for an example. The

power of H(p, n) lies in the fact that it speciﬁes both K and Kp,pn+1 at the same time, as

pointed out by the following lemma.

Lemma 3.1.1. (Lemma 2.2 of [12]) Let H(p, n) be a Heegaard diagram described as above.

Then

1. Ignoring z(cid:48), we get a doubly-pointed diagram H(p, n, w, z) which speciﬁes K.

2. Ignoring z, we get a doubly-pointed diagram H(p, n, w, z(cid:48)) which speciﬁes the cable knot

Kp,pn+1.

This implies CF K∞(H(p, n, w, z)) and CF K∞(H(p, n, w, z(cid:48))) are closely related. In fact
both complexes are isomorphic to CF∞(H(p, n, w)) if forgetting the Alexander ﬁltration.

Therefore, in order to get a more transparent correspondence between these two complexes,

we will compare the Alexander gradings of the intersection points with respect to the two
diﬀerent base points z and z(cid:48).

For the sake of a clearer discussion, we ﬁx some notation and terminology to deal with
the intersection points. For convenience, we assume n ≥ 0 through out the discussion and

remark that the case when n < 0 can be handled in a similar way. Note ˜β intersects αg at
2(p − 1)n + 1 points, and we label them as x0, ..., x2(p−1)n, starting at the out-most layer

from left to right, and then the second layer from left to right, and so on. On the other

(k)
hand, ˜β could also intersect other α-curves besides αg, and we label these points by y
0 ,...,
2(p−1)−1. Here k enumerates the intersections of the n-framed longitude with αi, i (cid:54)= g,

(k)

y

23

and the order of this enumeration is irrelevant. The lower index is again ordered following a

(k)
layer by layer convention, from outside to inside, but we require that y
can be connected
0
to x2n by an arc on ˜β which neither intersects δ nor δ(cid:48), the short arcs connecting the base

points. See Figure 3.2 for an example. The generators will be partitioned into p classes:
all the generators of the form {x2i, a} or {y

2i , b} will be called even intersection points

(k)

(k)
2(cid:100) i+1

n (cid:101)−1

, b} will be called (p − (cid:100) i+1

or 0-intersection points, and odd intersection points otherwise; odd generators of the form
n (cid:101))-intersection points. Here a, b are
{x2i+1, a} or {y
(g − 1)-tuple in Symg−1(Σ). Note that essentially we are classifying odd intersection points
into (p − 1) classes by the following principle: if its ˜β-component sits on the i-th layer (we
count the layers from outside to inside), then it is called a (p − i)-intersection point.

Figure 3.1: A compatible Heegaard diagram H for K. λ is a 2-framed longitude, and
according to our assumption that the 0-framed longitude can be chosen not to hit αg, λ can
be chosen to intersect αg twice.

We denote the Alexander grading by A (by A(cid:48)) when we use the base point z (base point

z(cid:48)). The comparison of Alexander ﬁltrations is summarized in the following proposition.

Propsition 9. With the choice of Heegaard diagrams as described above and let x be an

24

Figure 3.2: A example of H(p, n) with n = 2 and p = 3, corresponding to the Heegaard
diagram shown in Figure 3.1. There is an obvious arc of ˜β connecting x4 and y1
0, which
neither intersects δ nor δ(cid:48). By our convention, there is an arc of ˜β connecting x4 and y2
0
satisfying the same property as well, though it is not shown in the ﬁgure. The shaded
region represents a domain connecting {x1, a} and {x2, a}; the darkened color indicates the
multiplicity is 2, while the lighter colored region has multiplicity 1.

l-intersection point, where l ∈ {0, 1, ..., p − 1}, then

A(cid:48)(x) = pA(x) +

pn(p − 1)

2

+ l.

Proposition 9 can be viewed as a generalization of the comparison used in[12, 15], in
which only {xi, a} for i ≤ n were shown to satisfy the above equation. In studying τ , having
just a comparison for {xi, a} for i ≤ n would suﬃce: ﬁrst, Hedden observed that in the
case when |n| is suﬃciently large they account for the top Alexander graded generators of
(cid:92)CF K(Kp,pn+1) that determine τ (Kp,pn+1); second, the behavior of τ for small n can be

deduced from the large-n case by using crossing change inequality of τ .
In contrast, the
lower Alexander graded elements of CF K∞(Kp,pn+1) may play a role in Υ, even though

they do not aﬀect τ . Therefore in the current paper we have to carry out a comparison for

25

all types of generators. To accomplish this goal, we quote and extend some of the lemmas

used in [12, 15] below, after which Proposition 9 will follow easily.

Lemma 3.1.2. When 1 ≤ j ≤ (p − 1)n, we have

A({x2j−1, a}) − A({x2j, a}) = 0

A(cid:48)({x2j−1, a}) − A(cid:48)({x2j, a}) = p − (cid:100) j

n

(cid:101)

For an arbitrary k, when 0 ≤ i ≤ (p − 2), we have

A({y

(k)

2i+1, a}) − A({y

(k)

2i , a}) = 0

A(cid:48)({y

(k)

2i+1, a}) − A(cid:48)({y

(k)

2i , a}) = p − (i + 1)

(3.1)

(3.2)

(3.3)

(3.4)

Proof. Note that there is a Whitney disk φ connecting {x2j−1, a} to {x2j, a} (See Figure 3.2).
It is the product of a constant map in Symg−1(Σ) and the map represented by the disk which

connects x2j−1 and x2j, with boundary consisting of a short arc of αg and an arc of ˜β that
spirals into the winding region p − (cid:100) j
n(cid:101) times and then makes a turn out. We can see
that nw(φ) = nz(φ) = 0 and nz(cid:48)(φ) = p − (cid:100) j
n(cid:101). Therefore, A({x2j−1, a}) − A({x2j, a}) =
nz(φ) − nw(φ) = 0 and A(cid:48)({x2j−1, a}) − A(cid:48)({x2j, a}) = nz(cid:48)(φ) − nw(φ) = p − (cid:100) j
n(cid:101). We have

obtained equation (3.1.1) and (3.1.2). The proof for (3.1.3) and (3.1.4) will follow a similar

line, and hence is omitted.

26

Lemma 3.1.3. When 0 ≤ j ≤ (p − 1)n, we have

p(A({x0, a}) − A({x2j, a})) = A(cid:48)({x0, a}) − A(cid:48)({x2j, a})

(3.5)

For an arbitrary k, when 0 ≤ i ≤ (p − 2), we have

p(A({y

(k)

0 , b}) − A({y

(k)

2i , b})) = A(cid:48)({y

(k)

0 , b}) − A(cid:48)({y

(k)

2i , b})

p(A({x2n, a}) − A({y

(k)

0 , b})) = A(cid:48)({x2n, a}) − A(cid:48)({y

(k)

0 , b}).

(3.6)

(3.7)

Figure 3.3: The thickened curve γ represents the -class between {x0, a} and {x6, a}. Note
that the arc δ and δ(cid:48) which connect based points do not intersect γ.

Proof. First we prove Equation (3.1.5). Note that ({x2j, a},{x0, a}) can be represented by

a curve γ on Σ, which is obtained by ﬁrst connecting x2j to x0 along αg, and then by an arc

on ˜β which starts from x0 and winds j times counterclockwise to arrive at x2j (Figure 3.3).

27

Figure 3.4: The thickened curve is an arc on ˜β connecting y
intersect δ nor δ(cid:48).

(1)
0

and y

(1)
2

that does not

Note that [({x2j, a},{x0, a})] = 0 ∈ H1(S3, Z), hence [γ] = Σliαi + Σkiβi, with βg is
viewed as ˜β. Let c = γ − Σliαi − Σkiβi, then c bounds a domain on Σ. Note that since
δ · γ = δ(cid:48) · γ = 0, we have δ(cid:48) · c = δ(cid:48) · (−kg ˜β) = −kgp = p(δ · (−kg ˜β)) = p(δ · c), where “·”

stands for the intersection number. Equation (3.1.5) follows.

The proofs for the other two equations follow a similar line. Note the key point in the above

argument is that the -class of the two generators can be represented by a curve γ whose arc
on ˜β does not intersect the arc δ nor δ(cid:48), implying δ · γ = δ(cid:48) · γ = 0. For Equation (3.1.6),

(k)
note y
0

and y

(k)
2i can be joined by an arc on ˜β satisfying the aforementioned property (see

Figure 3.4 for an example). Recall by our convention, y
arc on ˜β which neither intersects δ nor δ(cid:48), hence Equation (3.1.7) follows.

(k)
0

can be connected to x2n by an

Let C(i) = {(g − 1) − tuples a| A({x0, a}) = i}.

28

Lemma 3.1.4. (Lemma 3.4 of [12]) Let a1 ∈ C(j1) and a2 ∈ C(j2), then

A({xi, a1}) − A({xi, a2}) = j1 − j2
A(cid:48)({xi, a1}) − A(cid:48)({xi, a2}) = p(j1 − j2).

Now we are ready to prove Proposition 9.

Proof of Proposition 9. We want to prove that if x an l-intersection point, where l ∈ {0, 1, ..., p−
1}, then A(cid:48)(x) = pA(x) + pn(p−1)
pA({x0, a}) + pn(p−1)
A(cid:48)(u) = p(A({x0, a})− A(u)), then we have A(cid:48)(u) = pA(u) + pn(p−1)

+ l. As pointed out in Lemma 2.5 of [15], A(cid:48)({x0, a}) =
. Note that for any other intersection point u, as long as A(cid:48)({x0, a})−

as well. Now the case

2

2

2

when l = 0 (even intersection points) follows easily from this obersvation, Lemma 3.1.3, and

Lemma 3.1.4. The other cases is then an easy consequence of the l = 0 case and Lemma

3.1.2.

Let C = CF∞(H(p, n, w)) be the chain complex obtained by forgetting the Alexander

2 A + (1 − t

ﬁltraion. And let (Ft) = t
2)Alg be two grading
functions on C deﬁned by using the two Alexander gradings A and A(cid:48) corresponding to z
and z(cid:48) respectively. Then the ﬁltrations corresponding to (Ft) and (Ft)(cid:48) satisfy the following

2)Alg, and (Ft)(cid:48) = t

2 A(cid:48) + (1 − t

relation.

Lemma 3.1.5. For p, n ∈ Z, p > 0, and 0 ≤ t ≤ 2

p, we have

(C,F(cid:48)
t )

s+

pn(p−1)t

4

⊂ (C,Fpt)s ⊂ (C,F(cid:48)
t )

s+

(pn+2)(p−1)t

.

4

29

Proof. Let x be a generator of C. Assume U−kx ∈ (C,Fpt)s, then

pt
2

(A(x) + k) + (1 − pt
2

)k =

pt
2

A(x) + k ≤ s.

Combine the above inequality with Proposion 9, we have

t
2
≤ t
2
pt
=
2
≤s +

(A(cid:48)(x) + k) + (1 − t
)k
2
pn(p − 1)
+ p − 1 + k) + (1 − t
2

(pA(x) +

)k

2
pn(p − 1)

t
2

A(x) + k +
(pn + 2)(p − 1)t

2

.

4

(p − 1)

+

t
2

Hence U−kx ∈ (C,F(cid:48)
t )

s+

(pn+2)(p−1)t

, and therefore

4

(C,Fpt)s ⊂ (C,F(cid:48)
t )

s+

(pn+2)(p−1)t

.

4

Similarly, if we assume U−kx /∈ (C,Fpt)s, then

pt
2

(A(x) + k) + (1 − pt
2

)k > s.

Again, in view of the above inequality and Proposion 9, we have

(A(cid:48)(x) + k) + (1 − t
)k
2
pn(p − 1)
+ k) + (1 − t
2

(pA(x) +

)k

2
pn(p − 1)

t
2
≥ t
2
pt
2

=

>s +

t
2

A(x) + k +
pn(p − 1)t

4

2

30

Hence U−kx /∈ (C,F(cid:48)
t )

s+

pn(p−1)t

4

, and therefore

(C,F(cid:48)
t )

s+

pn(p−1)t

4

⊂ (C,Fpt)s.

Proof of Theorem 1.1.3 for (p, pn + 1)-cable. Recall that

ν(C,Ft) = min{s| H0((C,Ft)s) → H0(C) is nontrivial},

and ν(C,Ft

(cid:48)

) is understood similarly. Now set s = ν(C,Fpt) in Lemma 3.1.5, we have

ν(C,Fpt) +

pn(p − 1)t

4

≤ ν(C,Ft

(cid:48)

) ≤ ν(C,Fpt) +

(pn + 2)(p − 1)t

4

.

Recall ΥK (pt) = −2ν(C,Fpt) and ΥKp,pn+1

(t) = −2ν(C,F(cid:48)

t), so by mutiplying −2 the

above inequality translates to

ΥK (pt) − (pn + 2)(p − 1)t

2

≤ ΥKp,pn+1

(t) ≤ ΥK (pt) − pn(p − 1)t

2

.

3.2 Upsilon of (p, q)-cable

Recall we denote the smooth knot concordance group by C. Let θ : C → R be a concordance
homomorphism such that |θ(K)| ≤ g4(K) and θ(Tp,q) = (p−1)(q−1)

when p, q > 0. In [48],

2

31

Van Cott proved that if we ﬁx a knot K and p > 0, and let

h(l) = θ(Kp,l) − (p − 1)l

2

,

then we have

−(p − 1) ≤ h(n) − h(r) ≤ 0,

(3.8)

when n > r such that both n and r relatively prime to p.

Remark 3.2.1. The concordance homorphism studied by Van Cott has range Z rather than
R, but by checking the argument in [48], it is straightforward to see that this choice will not

aﬀect the inequality stated above.

Now note that ﬁxing t ∈ (0, 2/p],
−ΥTp,q (t)

bounds the four-genus, and

t

−ΥK (t)

t

be

−ΥK (t)
= (p−1)(q−1)

t

2

is a concordance homorphism which lower

when q > 0 ([30]). So we can take θ to

and apply inequality (3.2.1), from which we get

0 ≤ ¯h(n, t) − ¯h(r, t) ≤ (p − 1)t,

(3.9)

where ¯h(n, t) = ΥKp,n(t) + (p−1)nt
well, and hence it holds for 0 ≤ t ≤ 2
p.

2

. It is easy to see inequality (3.2.2) is true at t = 0 as

Following essentially the argument of Corollary 3 in [48], we conclude the proof of our

main theorem as below:

Proof of Theorem 1.1.3 for (p, q)-cable. Recall 0 ≤ t ≤ 2

p. First we will show that

ΥKp,q (t) ≥ ΥK (pt) − (p − 1)(q + 1)t

2

.

32

Take r to be any integer such that q ≥ pr + 1, then by inequality (3.2.2) we have

¯h(q, t) − ¯h(pr + 1, t) ≥ 0.

In view of the deﬁnition of ¯h, the above inequality translates to

ΥKp,q (t) ≥ ΥKp,pr+1

(t) − (p − 1)(q − pr − 1)t

2

.

(3.10)

From the previous subsection, we have

ΥKp,pr+1

(t) ≥ ΥK (pt) − (pr + 2)(p − 1)t

2

.

Combining this and inequality (3.2.3), we get

ΥKp,q (t) ≥ ΥK (pt) − (p − 1)(q + 1)t

2

.

The other half of the inequality follows from an analogous argument by considering ¯h(pl +
1, t) − ¯h(q, t) ≥ 0, where l is an integer such that q ≤ pl + 1. We omit the details.

3.3 Applications of the cabling inequality

3.3.1 Computation of Υ(T2,−3)2,2n+1(t)

In this subsection, we show how one can compute Υ(T2,−3)2,2n+1
(t) by using our theorem
together with (cid:92)HF K((T2,−3)2,2n+1), for n ≥ 8. Note none of these knots is an L-space

knot. For easier illustration, we only give full procedure of the computation for the case

33

K = (T2,−3)2,17. The general case can be done in a similar way.

By Proposition 4.1 of [12], for i ≥ 0, we have

(cid:92)HF K(K, i) ∼=



(2)

(1)

(1) ⊕ F
(0) ⊕ F

(i−8)

F

F

F

F

F

0

i = 10

i = 9

(0)

i = 8

(−1)

i = 7

0 ≤ i ≤ 6

otherwise

Here the subindex stands for the Maslov grading. Note that by using the symme-
try (cid:92)HF Kd(K, i) = (cid:92)HF Kd−2i(K,−i) [34], the above equation actually tells us the whole
(cid:92)HF K(K). Now thinking CF K∞(K) as (cid:92)HF K(K)⊗F[U, U−1] when regarded as an F[U, U−1]-

module, we see the lattice points supporting generators with Maslov grading 0 are (0, 7),
(7, 0), (i, 8 − i), where −1 ≤ i ≤ 9. Here, for example, (0, 7) means the corresponding

generator has algebraic grading 0 and Alexander grading 7.

Note by Theorem 1.2 of [15], τ (K) = 7. In view of Theorem 13.1 in [29], we see that for

t ∈ [0, ], ΥK (t) = −2s, where s = t
2 and  is suﬃciently small. In other
words, when t is small, the ΥK is determined by the Ft grading of the generator at (0, 7).

2 · 7 + (1− t

2)· 0 = 7t

Now by Theorem 7.1 in [29], singularities of ΥK (t) can only occur at time t when there is a
line of slope 1 − 2
grading 0. The only t ∈ (0, 1) satisfying this property is 2
passes through the lattice points (0, 7) and (−1, 9).

t that contains at least two lattice points supporting generators of Maslov
3, giving a line of slope −2 that

34

So far, we can see that ΥK (t) is either one of the two below, depending on whether 2

3 is

a singular point or not.

ΥK (t) =

−7t,

2 − 10t,



t ∈ [0, 2
3]

t ∈ [ 2

3 , 1]

(3.11)

(3.12)

Or

ΥK (t) = −7t,

t ∈ [0, 1].

Note T2,−3 is alternating, so we can apply Theorem 1.14 in [38] to obtain ΥT2,−3
1 − |1 − t|, when t ∈ [0, 2]. Applying Theorem 1.1.3 we see that when 1

2 ≤ t ≤ 1, we have

(t) =

2 − 11t ≤ ΥK (t) ≤ 2 − 10t.

Now we see only (3.3.1) satisﬁes this constraint and hence ΥK (t) is determined. More

generally, we have

Propsition 10. For n ≥ 8,

Υ(T2,−3)2,2n+1

(t) =



−(n − 1)t,

2 − (n + 2)t,

t ∈ [0, 2
3]

t ∈ [ 2

3 , 1]

Proof. Same as the discussion above. We refer the reader to [12] for the formula of (cid:92)HF K((T2,−3)2,2n+1).

35

3.3.2 An inﬁnite-rank summand of topologically slice knots

Let D denote the untwisted positive whitehead double of the trefoil knot. Fix an integer

p > 2 and let Jn = ((Dp,1)...)p,1 denote the n-fold iterated (p, 1)-cable of D for some positive

integer n. Recall Corollary 1.1.1 states that Jn for n = 1, 2, 3, ... are linearly independent in
C and span an inﬁnite-rank summand consisting of topologically slice knots. To prove this,

we ﬁrst establish two lemmas.

Lemma 3.3.1. Let ξn be the ﬁrst singularity of ΥJn(t), then ξn ∈ [ 1
pn ,
ξi < ξj whenever i > j.

2

1+pn ]. In particular,

Proof. We ﬁrst deal with the lower bound. Recall for any knot K, ΥK (t) = −τ (K)t when
t < 1
g3(K) [29]. Note τ (D) = g3(D) = 1 by [14] and hence τ (Jn) = pn by [15]. This implies
g3(Jn) = pn since we have pn ≤ g4(Jn) ≤ g3(Jn) ≤ pn. Therefore, ξn ≥ 1
pn .
We move to establish the upper bound. Note CF K∞(D) ∼= CF K∞(T2,3)⊕A, where A is an
(t) = |1− t|− 1. In particular, ΥD(t) =
(t) ≥ pt − 2 − (p − 1)t = t − 2 when

1

acyclic chain complex [16]. Therefore ΥD(t) = ΥT2,3
t − 2 when 1 ≤ t ≤ 2. Apply Theorem 1.1.3 we get ΥJ1
p ≤ t ≤ 2
p. Inductively we have ΥJn(t) ≥ t−2 when 1
∃ > 0 such that ΥJn(
1+pn + ) =
which is a contradiction. Therefore, ξn ≤ 2

1+pn + ) = −pn(

2

2

1+pn .

pn ≤ t ≤ 2
−2pn
1+pn − pn < 2

pn . Suppose ξn > 2
1+pn − 2 < 2

1+pn , then
1+pn + − 2,

Let ∆Υ(cid:48)
K (t0) denote the slope change of ΥK (t) at t0, i.e. ∆Υ(cid:48)
Υ(cid:48)
K (t0). Recall in general

Υ(cid:48)
K (t0) −
K (t0) is an integer [38]. The following lemma shows

K (t0) = limt(cid:38)t0

2 ∆Υ(cid:48)
t0

limt(cid:37)t0
in some cases, we can determine the value of

2 ∆Υ(cid:48)
t0

K (t0).

Lemma 3.3.2. Let K be a knot in S3 such that τ (K) ≥ 0 and let ξ be the ﬁrst singularity

of ΥK (t). If 0 < ξ <

4

g3(K)+τ (K), then ξ

2∆Υ(cid:48)

K (ξ) = 1.

36

Proof. Depicting the chain complex CF K∞(K) as lattice points in the plane, by Theorem
7.1 (3) of [29], we know there is a line of slope 1 − 2
ξ containing at least two lattice points
(i, j) and (i(cid:48), j(cid:48)) supporting generators of Maslov grading 0. Since ξ is the ﬁrst singularity,
by Theorem 13.1 of [29] we know, say, (i, j) = (0, τ (K)). So we have j(cid:48)−τ (K)

= 1− 2

i(cid:48)

ξ , which

2i(cid:48)

4

i(cid:48)−j(cid:48)+τ (K)

<

implies 0 < ξ =
g3(K)+τ (K). This together with the genus bound property of
knot Floer homology |i(cid:48) − j(cid:48)| ≤ g3(K) would constrain |i(cid:48)| = 1. By Theorem 7.1 (4) of [29],
2∆Υ(cid:48)

K (ξ) = |i(cid:48)| = 1.

ξ

Proof of Corollary 1.1.1. Note all Jn have trivial Alexander polynomial and hence are topo-

logically slice [11]. The linear independence follows from Lemma 3.3.2: suppose ΣkiJni = 0
in C for some ki (cid:54)= 0 and n1 > ... > nl, since ΥΣkiJni
singularity at ξn1, which contradicts to ΥΣkiJni
by Lemma 3.3.2 and Lemma 3.3.3, ξn
phism C −→ Z∞ given by [K] (cid:55)→ ( ξn

(ξn) = 1. Now one can easily see the homomor-
K (ξn))∞

(t) it possesses ﬁrst
(t) ≡ 0. To see they span a summand, note

n=1 is an isomorphism when restricted to

2 ∆Υ(cid:48)
2 ∆Υ(cid:48)

(t) = ΣkiΥJni

Jn

the subgroup spanned by Jn and hence the conclusion follows.

Remark 3.3.1. One can actually replace D by any topologically slice knot K with τ (K) =

g(K) = 1 and even consider mixed iterated cable ((Kp1,1)...)pn,1. We chose Jn for the sake

of an easier illustration. The linear independence of certain subfamilies of mixed iterated

cables of D were also observed by Feller, Park, and Ray [9].

37

Chapter 4

The Vi-invariants of satellite knots

In this chapter we give a proof of Theorem 1.2.1, and apply it to prove the subaddtivity and

crossing change inequalities of the Vi-invariant. We also examine the the case of cabling.

4.1 Proof of Theorem 1.2.1

The strategy of the proof is to construct a negative deﬁnite cobordism between S3

r (C)#S3

n(P )

and S3

l2r+n

(Pr(C)) for some large integer n. This leads to a comparison of the correction

terms of these 3-manifolds. The statement in Theorem 1.2.1 then follows from expressing

the correction terms by Ni and Wu’s surgery formula.

Proof of Theorem 1.2.1. Let n be an integer such that n > 2i2. Let W be the four manifold
obtained by attaching two 2-handles to the split link C ∪ P , with the framing on C being r

and on P being n (see Figure 4.1). Let WC (r) and WP (n) to be the 4-manifolds obtained

by a single 2-handle attachment along C and P with framing r and n respectively, then

W = Wr(C)(cid:92)Wn(P ). On the other hand, by sliding the 2-handle attached to P across the

other 2-handle algebraically l times, we obtain a diﬀerent Kirby diagram for W as shown in

Figure 2. This latter point of view allows us to cut W along S3
(Pr(C)), decomposing
it as W1 ∪ W2, where W1 is constructed by a single 2-handle attachment along Pr(C) with
framing l2r + n, and W2 is obtained by attaching a 2-handle along C in the upper boundary

l2r+n

38

of S3

l2r+n

(Pr(C)) × [0, 1].

r

C

n

P

Figure 4.1: A Kirby diagram for W

r

r

C

l2r + n

P

Figure 4.2: Another Kirby diagram for W obtained by applying handle-slide to the diagram
in Figure 4.1

Note W2 must be positive deﬁnite. Therefore, by reversing the orientation of W2, we

obtain a negative deﬁnite cobordism W2 : S3

r (C)#S3

n(P ) → S3

(Pr(C)).

l2r+n

Recall that given a knot K ⊂ S3 and an integer p, we identify Spinc(S3

p(K)) and Z/pZ by
p(K), where t is any spinc-structure on Wp(K) such that (cid:104)c1(t), [ ˆFK ](cid:105) + p ≡

setting [i] = t|S3

2i (mod 2p). Here ˆFK is a capped-oﬀ a Seifert surface for K.

Take si1

∈ Spinc(Wr(C)) such that (cid:104)c1(si1

), [ ˆFC ](cid:105) + r = 2i1, and si2

∈ Spinc(Wn(P ))

such that (cid:104)c1(si2

), [ ˆFP ](cid:105) + n = 2i2. Let s ∈ Spinc(W2) be the restriction of si1

(cid:92)si2

to W2,

39

then s|S3

r (C)#S3

n(P ) = [i1]#[i2]. Note

(cid:104)c1(si1

(cid:92)si2

), l[ ˆFC ] + [ ˆFP ](cid:105) + l2r + n = (cid:104)c1(si1

), l[ ˆFC ](cid:105) + (cid:104)c1(si2

), [ ˆFP ](cid:105) + l2r + n

= l(2i1 − r) + (2i2 − n) + l2r + n

lr(l − 1)

).

2

= 2(li1 + i2 +

Hence s|S3

l2r+n

(Pr(C)) = [li1 + i2 + lr(l−1)

2

].

Since (W2, s) is a negative deﬁnite cobordism from (S3

r (C)#S3

n(P ), [i1]#[i2]) to (S3

(Pr(C)), [li1+

l2r+n

i2 + lr(l−1)

2

]), in view of the proof of Proposition 9.9 in [33], we have

d(S3

l2r+n

(Pr(C)), [li1 + i2 +

lr(l − 1)

2

]) − d(S3

n(P ), [i1]#[i2])

r (C)#S3
≥ c1(s)2 + 1

.

4

By additivity of the d-invariant (Theorem 4.3 of [33]),

d(S3

r (C)#S3

n(P ), [i1]#[i2]) = d(S3

r (C), [i1]) + d(S3

n(P ), [i2]).

Since i1 ≤ (cid:98) r

2(cid:99) and i2 ≤ (cid:98) n

2(cid:99), by Propostion 6, we have

d(S3

d(S3

r (C), [i1]) = d(L(r, 1), i1) − 2Vi1
r (P ), [i2]) = d(L(n, 1), i2) − 2Vi2

(C),

(P ).

(4.1)

(4.2)

Note 2(li1 + i2 + lr(l−1)

2

) ≤ lr + n + lr(l − 1) = n + l2r, so again by Proposition 6, we have

d(S3

l2r+n

(Pr(C)), [li1 + i2 +

2
d(L(l2r + n, 1), li1 + i2 +

]) =
lr(l − 1)

lr(l − 1)

2

40

) − 2V

li1+i2+

lr(l−1)

(Pr(C)).

2

(4.3)

Note d-invariants of lens spaces are computed in Proposition 4.8 of [33]. More precisely,
given p > 0 and 0 ≤ i < p + 1,

d(L(p, 1), i) =

(2i − p)2 − p

4p

.

(4.4)

Now we compute c1(s)2 as following (note c2

1 changes sign when we reverse the orientation

of the underlying manifold)

c1(s)2 = c1(si1(cid:92)si2)2 − c1((si1(cid:92)si2)|W1)2

= c1(si1)2 + c1(si2)2 − c1((si1(cid:92)si2)|W1)2
= −(2i1 − r)2

− (2i2 − n)2

r

(2li1 + 2i2 + lr(l − 1) − (n + l2r))2

+

n

n + l2r

(4.5)

.

m

C

P

Figure 4.3: The pattern Pm(C)

Substituting the terms in Equation (4.1) by the corresponding terms in Equation (4.2)

to (4.5) we obtain

V

li1+i2+

lr(l−1)

2

(Pr(C)) ≤ Vi1

(C) + Vi2

(P ).

This ﬁnishes the proof of the ﬁrst inequality in the statement of Theorem 1.2.1. Now we
move to prove the second one. To begin, note given a pattern knot P , a knot C ⊂ S3 and an

integer m, we may construct a new pattern Pm(C) by infecting P along the meridian of the

41

solid torus by C with m-twist (see Figure 4.3). Note that (Pm(C))r(−C) is concordant to

Pm+r, and then applying the inequality we proved above with pattern Pm(C) and companion
−C gives the second inequality in the statement of Theorem 1.2.1.

4.2 Subadditivity and crossing change inequalities

Proof of Proposition 1. Recall we want to prove Vi1+i2

(K#L) ≤ Vi1

(K) + Vi2

(L). Note

the connected sum operation is a special case of the satellite operation with the pattern

having geometric winding number l = 1. Also note that the number of twist will not be

relevant in this case, so we may take r to be arbitrarily large. With these at mind, applying

Theorem 1.2.1 with pattern knot being K and companion knot being L yields the desired

inequality. The corresponding statement on the ν+-invariant then follows readily from the

above inequality and the deﬁnition of ν+.

Proof of Theorem 1.2.2. We ﬁrst prove Vi+1(K+) ≤ Vi(K−). Note one can obtain K+ from
K− from via a satellite operation with the pattern knot being K− with winding number 2,

the companion being the unknot U and the number of twist being 1. Applying Theorem

1.2.1 with i1 = 0 and i2 = i yields the claimed inequality.

K+

K−

Figure 4.4: Obtaining K− from K+ by a full twist

Next we prove Vi(K−) ≤ Vi(K+). Note one can obtain K− by using K+ as a pattern

42

knot (appropriately placed in the solid torus) with winding number 0, and applying to the

unknot with a full twist (see Figure 4.4). Applying the ﬁrst inequality in Theorem 1.2.1 with

i1 = 0 and i2 = i gives the desired result.

The results on the ν+-invariant then follows easily from the deﬁnition of ν+.

4.3 Proof of the cabling formula for ν+

This section is devoted to prove Proposition 2 and Proposition 3, and hence proving the

cabling formula for ν+ as stated in Proposition 4.

First, we prove Theorem 2, which gives an upper bound on the ν+-invariant of satellite

knots.

Proof of Proposition 2. Applying Theorem 1.2.1 with i1 = ν+(K) and i2 = ν+(P ) yields

V

lν+(K)+ν+(P )+

lr(l−1)

2

(Pr(K)) = 0, and hence by deﬁnition of the ν+-invariant we have

ν+(Pr(K)) ≤ lν+(K) + ν+(P ) +

lr(l − 1)

2

.

From Proposition 2 we derive the following upper bound for the ν+-invariant of cable

knots.

Corollary 4.3.1. Let K be a knot in S3, p > 0 and q ≥ max(2ν+(K), 1)p + 1. Then

ν+(Kp,q) ≤ pν+(K) +

(p − 1)(q − 1)

2

.

Proof. Let q = rp + d, where r and d are positive integers such that 0 < d < p. Then Kp,q

43

can be viewed as the r-twisted satellite knot with companion being K and pattern being

the torus knot Tp,d. By Proposition 2, ν+(Kp,q) ≤ pν+(K) + ν+(Tp,d) + pr(p−1)
ν+(Tp,d) = (p−1)(d−1)

and hence the claim follows.

2

2

. Note

Proof of Proposition 3. We ﬁrst establish the inequality in the case q = pr + 1. Recall

from Proposition 9 that one may choose auxiliary data so that the resulting knot Floer
chain complexes CF K∞(K) and CF K∞(Kp,pr+1) for K and Kp,pr+1 satisfy the following

properties

• As F[U, U−1]-modules, CF K∞(K) and CF K∞(Kp,pr+1) have the same generating
set G, whose elements are of algebraic grading zero.

• CF K∞(K) = CF K∞(Kp,pr+1) as algebraically-ﬁltered chain complexes.
• for any x ∈ G, A(cid:48)(x) = pA(x) + pr(p−1)
for CF K∞(K) and CF K∞(Kp,pr+1) respectively, and l ∈ {0, 1, ..., p − 1}.

2 + l, where A and A(cid:48) are the Alexander grading

Let C = CF K∞(K) and Cp,pr+1 = CF K∞(Kp,pr+1). Note the third property implies to
A(cid:48)(U ix) = pA(U ix) + pr(p−1)
+ l + (p − 1)i. With this at mind, it is straightforward to see
for any s ∈ Z, C{max(i, j − s) ≥ 0} is a quotient complex of Cp,pr+1{max(i, j − s(cid:48)) ≥ 0}
when p(s + 1) + pr(p−1)

− 1 ≥ s(cid:48).

2

2

Now take s(cid:48) = ν+(Kp,pr+1). By deﬁnition of the ν+-invariant, the projection map

Cp,pr+1{max(i, j − s(cid:48)) ≥ 0} → Cp,pr+1{i ≥ 0}

induces an isomorphism on the F[U−1]-summand on homology level. Let s be the least

44

integer such that p(s + 1) + pr(p−1)

2

− 1 ≥ ν+(K), it is easy to see

s = (cid:100) ν+(Kp,pr+1) + 1 − p − pr(p − 1)/2

p

(cid:101).

(4.6)

Since the aforementioned projection map factorizes through the quotient map

Cp,pr+1{max(i, j − s(cid:48)) ≥ 0} → C{max(i, j − s) ≥ 0},

one sees the projection C{max(i, j − s) ≥ 0} → C{i ≥ 0} also induces an isomorphism
on the F[U−1]-summand on homology level. Therefore, s ≥ ν+(K). Substituting s in this

inequality by Equation 4.6 and simplifying the expression we have

ν+(Kp,pr+1) ≥ pν+(K) +

pr(p − 1)

2

.

Now we move to prove the inequality for a general q. Pick a suﬃciently large integer r
so that q ≤ pr + 1. Note Kp,q can be obtained from Kp,pr+1 by resolving (pr + 1− q)(p− 1)
crossings, and hence there is a genus (pr+1−q)(p−1)

cobordism between these two knots. By

2

the genus bound property of the ν+-invariant, we have

ν+(Kp,pr+1# − Kp,q) ≤ (pr + 1 − q)(p − 1)

2

.

Note by subadditivity, ν+(Kp,pr+1) − ν+(Kp,q) ≤ ν+(Kp,pr+1# − Kp,q). Therefore,

ν+(Kp,pr+1) − ν+(Kp,q) ≤ (pr + 1 − q)(p − 1)

2

.

45

So

ν+(Kp,q) ≥ ν+(Kp,pr+1) − (pr + 1 − q)(p − 1)

≥ pν+(K) +

= pν+(K) +

pr(p − 1)
(p − 1)(q − 1)

2

2

2

− (pr + 1 − q)(p − 1)

2

Proof of Proposition 4. This follows readily from Proposition 3 and Corollary 4.3.1.

Finally we point out Theorem 2 also partially recovers the full-twist inequality appeared

in [46]. More precisely, a knot J is said to be obtained from a full-twist operation on K

with l-linking if there is a pattern knot P of winding number l such that P (U ) = K, and

P1(U ) = J. We have

Corollary 4.3.2 (A weaker version of Theorem 1.1 in [46]). Let J and K be knots in S3

such that J is obtained from a full-twist operation on K with l-linking, then

ν+(J# − K) ≤ l(l − 1)

2

.

Proof. Apply Proposition 2 with the companion being the unknot, P = K# − K and

r = 1.

Remark 4.3.1. Using the subadditivity of the ν+-invariant, Corollary 4.3.2 implies ν+(J)−
l(l−1)
2 ≤ ν+(K). Applying this repeatedly, we have

ν+(Kp,q) ≥ ν+(Kp,pn+q) − pn(p − 1)

2

for any n ≥ 0. The way Sato proved Proposition 3 is to pick n suﬃciently large in the above

inequality so that ν+(Kp,pn+q) can be expressed using Wu’s cabling formula for ν+.

46

4.4 The upsilon invariant and iterated satellite opera-

tions

In this section we give an estimate of the ﬁrst singularity of the upsilon invariant in terms

of Vi and τ , and apply it to study iterated satellite operations.

4.5 Proof of Theorem 1.2.3

This section is devoted to prove Theorem 1.2.3, we begin with an equivalent deﬁnition of

the V0-invariant.

Propsition 11. Let C = CF K∞(K) be a knot Floer chain complex for a given knot K,

then

−2V0(K) = max{gr(x)|x ∈ Image(ι∗ : H∗(C{max(i, j) ≤ 0}) → H∗(C))}

Proof. By the deﬁnition of V0(K) we have H∗(C{max(i, j) ≥ 0}) = T +−2V0
⊕ HFred. By
examining the exact triangle induced by 0 → U · C{max(i, j) ≤ 0} → C → C{max(i, j) ≥
0} → 0, one easily sees H∗(C{max(i, j) ≤ 0}) = T −
⊕ HFred and the proposition

−2V0

follows.

Proof of Theorem 1.2.3. Let for t ∈ [0, 2], let Ft = t
deﬁned on C = CF K∞(K). Recall

2 Alex + (1− t

2)Alg be a grading function

ΥK (t) = −2 min{s ∈ R|H0(F−1

t

(−∞, s]) → H0(C) is nontrivial}.

To see ΥK (t) ≥ −2V0(K), we ﬁrst claim H0(C{max(i, j) ≤ V0}) → H0(C) is nontrivial.

47

F−1

t

(−∞, V0]

C{max(i, j) ≤ V0}

Figure 4.5: An example showing C{max(i, j) ≤ V0} ⊂ F−1

t

(−∞, V0]

Since the U -action shifts the Maslov grading down by 2, the claim follows from Proposition 11
and C{max(i, j) ≤ V0} = U−V0C{max(i, j) ≤ 0}. It is easy to see for any t, C{max(i, j) ≤
V0} ⊂ F−1
(−∞, V0(K)]) → H0(C) is nontrivial (see
Figure 5.1). The inequality ΥK (t) ≥ −2V0(K) then follows from the deﬁnition of the Υ
function. Note Υ ¯K (t) = −ΥK (t), and hence we have ΥK (t) ≤ 2V0( ¯K).

(−∞, V0(K)] and therefore H0(F−1

t

t

We move to see the bounds for the ﬁrst singularity ξ when τ (K) > V0(K) > 0. Recall

ΥK (t) = −τ (K)t for t ∈ [0, ξ].
ΥK (t) < −2V0(K) for t ∈ (
2V0(K)
τ (K) ,
established above. Hence ξ ≤ 2V0(K)

If ξ >

2V0(K)
τ (K) , then for some  > 0, we would have

2V0(K)
τ (K) + ], which is a contradiction to the bounds we

τ (K) . The lower bound

1

g3(K) comes from general properties

of the Υ function [29].

Remark 4.5.1. By the same argument, one may prove similar inequalities using Vi for i ≤ 0

48

other than just V0.

4.6

Iterated satellite operations

In this section we prove Theorem 1.2.4 and Proposition 5. First, note the following result

follows immediately Theorem 1.2.3.

V0(Kn) = ∞,
Propsition 12. Let {Kn|n ∈ Z+} be a family of knots such that limn→∞ τ (Kn)
then there exists a subset of {Kn|n ∈ Z+} which generate a Z∞ subgroup in the smooth knot
concordance group C.

Proof. Note in view of Theorem 1.2.3, we may choose a subsequence of knots Kni whose

upsilon functions have distinct ﬁrst singularities, and hence are linearly independent in

CP L([0, 2]), which implies the statement.

Proof of Theorem 1.2.4. By Proposition 1.2 of [45], there is a positive integer m = m(P ) such
that for any knot C, τ (P (C)) ≥ lτ (C) − m. Now take K to be a topologically slice knot so
that lτ (K)−m > (l− 1

2)nτ (K). For instance, one may
take K to be the (p, 1)-cable of the W h+(T2,3) for a suﬃciently large p. On the other hand, by
(P (K)) ≤ V0(K) + V0(P−1(U )), which implies V0(P (K)) ≤

2)τ (K), then we have τ (P n(K)) ≥ (l− 1

Theorem 1.2.1, we have V l(l−1)
V0(K) + V0(P−1(U )) + l(l−1)
limn→∞ τ (P n(K))

. Therefore, V0(P n(K)) ≤ V0(K) + n(V0(P−1(U )) + l(l−1)
V0(P n(K)) = ∞, and hence concluding the proof by applying Proposition 12.

). So

2

2

2

unknot pattern (see Figure 5.2), S3

Proof of Proposition 5. By Corollary 2.2 of [6], since M is a winding number one non-zero
0(D) are Z-homology corbordant, and
hence V0(M n(D)) = V0(D) = 1. By Theorem 1.4 of [26], τ (M n(D)) = n + 1 = g(M n(D)).
Therefore, limn→∞ τ (M n(K))

V0(M n(K)) = ∞, and hence the claim follows from Proposition 12.

0(M n(D)) and S3

49

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