TOPICS IN CLASSICAL AND QUANTUM KNOT INVARIANTS

By

Rob McConkey

A DISSERTATION

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

Mathematics—Doctor of Philosophy

2024

ABSTRACT

We continue the work done by Kalfagianni and Lee, where they gave two sided linear bounds for

the crosscap number of alternating links in terms of certain coefficients of the Jones polynomial.

In particular we find two-sided bounds for the crosscap number of the Conway sums of strongly

alternating tangles in terms of certain coefficients of the Jones polynomial. Then we find families

of links for which the crosscap number and these coefficients of the Jones polynomial grow

independently. These families of links enable us to state the bounds for the crosscap number will

not generalize to all links.

We also study the relationship of the span of the colored Jones polynomial to the crossing

number of a family of knots cablings. In particular we use the degree of the colored Jones knot

polynomials to show that the crossing number of a ( 𝑝, 𝑞)-cable of an adequate knot with crossing

number 𝑐 is larger than 𝑞2 𝑐. As an application we determine the crossing number of 2-cables of

adequate knots.

Copyright by
ROB MCCONKEY
2024

ACKNOWLEDGEMENTS

I owe my success as a graduate student here at Michigan State to many people both in and outside

of the department. First and foremost none of this would have been possible without my advisor,

Effie Kalfagianni. Her advice and mentorship helped me grow as a mathematician and as a member

of the larger mathematical community.

I would also like to thank the members of my committee, Matt Hedden, Bob Bell, and Ben

Schmidt. Their advice both academically and professionally helped me navigate my time as a

graduate student. My committee members along with Teena Gerhardt, Andy Krause, and Tsveta

Sendova provided my with numerous opportunities which I have used to grow as an advisor and

educator in my own right. I would also like to thank Andy and Tsveta for their guidance in becoming

the educator I am today. They also helped me navigate the job market this last year, ultimately

landing a teaching focused position which has been my goal since starting my doctorate.

I would also like to thank Kevin Woods for inspiring my pursuit of mathematics and the many

conversations about graduate school and careers.

My peers here at Michigan State have also been a fundamental part of my journey. Having

others who have shared in my struggles as a graduate student was extremely important for my

perseverance throughout my time here. I also made many friends here in the department which

made coming to campus and traveling to conferences much more enjoyable. Outside of “work”

hours the time spent playing board games, watching Bachelor, or at happy hours allowed me the

breaks that were much needed.

I want to also thank my family and friends outside of the department for providing an escape

from the world of mathematics. They have also provided me with much needed support and

patience at times when I have struggled. They have also kept me humbled with their interesting

math questions.

Finally, I want to acknowledge my cats Saide and Hugs and late cat Jamalama. While they will

never understand what I do their companionship has kept me going throughout the last six years.

Part of this research was supported in the form of graduate Research Assistantships by NSF

iv

grants DMS-2004155 and DMS-2304033 and funding from the NSF/RTG grant DMS-2135960.

v

LIST OF ABBREVIATIONS .

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.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

TABLE OF CONTENTS

CHAPTER 1

.
1.1 Main Results and Layout of the Dissertation . . . . . . . . . . . . . . . . . . .

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

1
2

CHAPTER 2

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

2.1 Alternating and Adequate Links
2.2 Classic Knot Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
2.4 Colored Jones Polynomial

. 10
. . . . . . . . . . . . . . . . . . . . . . . . . 10
. 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Jones Polynomial

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CHAPTER 3

CROSSCAP NUMBER OF CONWAY SUMS . . . . . . . . . . . . . . 23
3.1 Tangles and the Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. 25
3.2 The Lower Bound .
3.3 Crosscap Number, Twist Number, and the Jones Polynomial . . . . . . . . . . . 31
3.4 Generalizing to Larger Conway Sums of Tangles . . . . . . . . . . . . . . . . . 40
3.5 Families where 𝑇𝐿 and the Crosscap Number are Independent . . . . . . . . . . 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4

CROSSING NUMBER OF CABLE KNOTS . . . . . . . . . . . . . . . 51
4.1
Jones Diameter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
. 54
Lower bounds and admissible knots . . . . . . . . . . . . . . . . . . . . . .
4.2
4.3 Non-adequacy results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Composite non-adequate knots . . . . . . . . . . . . . . . . . . . . . . . . . . 59

. .

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BIBLIOGRAPHY . .

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vi

LIST OF ABBREVIATIONS

A knot or link

A link

A copy of the circle (one-sphere)

The 3-sphere.

The Jones polynomial for a link 𝐿.

The first and last coefficients of the Jones polynomial.

The second and second to last coefficients of the Jones polynomial for 𝐿.

|𝛽𝐿 | + |𝛽′
𝐿 |

The crosscap number for a surface 𝑆.

The crosscap number for a link 𝐿.

A tangle.

𝐾

𝐿

𝑆1

𝑆3

𝑉 (𝐿)

𝛼𝐿, 𝛼′
𝐿

𝛽𝐿, 𝛽′
𝐿

𝑇𝐿

𝐶 (𝑆)

𝐶 (𝐿)

𝑇𝑖

𝐾𝑖𝑁 , 𝐾𝑖𝐷 The link resulting from closing the tangle 𝑇𝑖 via the numerator and denominator

closures respectively.

The number of link components in 𝐿.

The crossing number for a knot 𝐾.

The ( 𝑝, 𝑞) cabling of 𝐾.

The neighborhood of a knot 𝐾.

The writhe of a knot 𝐾.

𝑘 𝐿

𝑐(𝐾)

𝐾𝑝,𝑞

𝑁 (𝐾)

wr(𝐾)

𝐾#𝐾 ∗

The connected sum of knots 𝐾 and 𝐾 ∗.

𝐷 (𝐿)

G𝐴 (𝐿)

G𝐵 (𝐿)

G′

𝐴 (𝐿)

G′

𝐵 (𝐿)

A diagram for the link 𝐿.

The all A state graph for a link 𝐿.

The all B state graph for a link 𝐿.

The reduced all A state graph for a link 𝐿.

The reduced all B state graph for a link 𝐿.

vii

𝑒 𝐴, 𝑒𝐵

The number of edges in the graphs G𝐴 (𝐿) and G𝐵 (𝐿) respectively.

𝐴, 𝑒′
𝑒′
𝐵

Γ

𝑆(𝑠)

S(𝐹)

The number of edges in the graphs G′

𝐴 (𝐿) and G′

𝐵 (𝐿) respectively.

The 4-valent graph which results from turning crossings of a knot to vertices.

The Turaev surface for a Kauffman state 𝑠.

A linear skein for a surface 𝐹.

SL2((𝐶) The special linear group of degree 2 over the complex numbers.

𝑇 𝐿𝑛

Σ

The Temperley-Lieb Algebra on 𝑛 generators.

A Conway sphere.

𝑡𝑤(𝐷 (𝐿)) The twist number for a link diagram.

ℓin(𝐷 (𝐿)) The edges lost as we pass from 𝑒 𝐴 + 𝑒𝐵 to 𝑒′

𝐵.
𝐴 + 𝑒′

ℓext(𝐷 (𝐿)) Edges we lose from identification when we take the Conway sum.

𝑏 𝐴, 𝑏𝐵

𝑊−(𝐾)

𝑏(𝐺)

𝑔𝑇 (𝐷)

The bridges in a state graph.

The negative Whitehead double of the link 𝐿 with the blackboard framing.

The first Betti number of the graph 𝐺.

The Turaev genus for the knot diagram 𝐷.

𝑣 𝐴 (𝐷)

The number of vertices in G𝐴 (𝐿).

𝑣 𝐵 (𝐷)

The number of vertices in G𝐵 (𝐿).

𝐽𝐾 (𝑛)

The 𝑛th colored Jones polynomial of 𝐾.

𝑑− [𝐽𝐾 (𝑛)] The minimum degree of the colored Jones polynomial of 𝐾.

𝑑+ [𝐽𝐾 (𝑛)] The maximum degree of the colored Jones polynomial of 𝐾.

𝑑𝐽𝐾

The Jones diameter of 𝐾.

viii

CHAPTER 1

INTRODUCTION

A knot K is a smooth (or piece-wise linear) embedding of 𝑆1 into 𝑆3 or R3. A link is the embedding

of 𝑘 disjoint copies of 𝑆1 into 𝑆3 or R3. In the case of link we call the individual images of the

copies of 𝑆1 the link components and say there are 𝑘 link components, further we use either 𝐾 or 𝐿

as the standard notation for a link. We consider two links 𝐿1 and 𝐿2 to be equivalent if there exists

an isotopy which takes us from 𝐿1 to 𝐿2. Telling if two knots are equivalent is a classical question

in knot theory and one that turns out to be quite difficult.

When studying links we consider link diagrams. A link diagram is a projection of a link

𝐿 onto the plane, R2, or sometimes the 2-sphere, such that we retain over/under information at

the crossings. The link isotopy of link diagrams is done through a series of moves called the

Reidemeister moves [32]. To highlight the difficulty of the question of whether two knot diagrams

are equivalent, we note that it is a challenge to even recognize the unknot. The unknot is the trivial

knot, in other words its simplest knot diagram is a copy of 𝑆1. For the Jones polynomial, which

we introduce in chapter two, it remains an open question whether the polynomial identifies the

unknot [2].

Figure 1.1 Haken’s Gordian diagram of the unknot [12], which is difficult to recognize using
isotopy.

Given that it is difficult to recognize links via isotopy we instead use a tool known as knot

invariants. Knot invariants are other mathematical objects which we associate to links in such

1

away that if two links are equivalent then their invariant must be equivalent. In particular if we

define 𝑓 (𝐿) to be a knot invariant for a link 𝐿, then given links 𝐿1 and 𝐿2 if 𝑓 (𝐿1) ≠ 𝑓 (𝐿2) then

𝐿1 ≠ 𝐿2.

So, the study of knot theory lives largely in the realm of the study of knot invariants.

In

particular we study what knot invariants can tell us about knots and links, but also what knots and

links can tell us about the category of mathematical objects different invariants come from. For

this dissertation we will be using a family of link invariants known as quantum invariants to better

understand the geometric invariants of knots and links. As we will see in the following chapters,

geometric invariants of links are often challenging to compute for a link, while quantum invariants

tend to be more algorithmic.

Another convenient property of certain quantum invariants (e.g. Jones polyonial or colored

Jones polynomial) is that they behave nicely for a family of knots known as alternating knots and

more generally adequate knots [32, 43, 25, 11]. Further, with these families of links we have been

able to show that we can recover them from purely combinatorial approaches. For instance, the

Jones polynomial can be recovered from the Bolobas-Riordan-Tutte polynomial for graphs [11].

The remainder of this introduction will be to give a layout of the dissertation, details of each

chapter, and precise statements of the main results to be shown throughout.

1.1 Main Results and Layout of the Dissertation

This dissertation has been structured such that a knowledgeable reader can read the chapters in

any order. But for those without the necessary background in knot theory, chapter 2 will provide a

primer for concepts essential to the understanding of the results and their proofs.

1.1.1 Outline of Chapter 2

Chapter 2 will be used to provide the necessary background for the rest of the dissertation.

The first section of this chapter will be introducing two important families of links, alternating and

more generally adequate links. Further we recall graphs associated to link diagrams known as a

Kauffman state graphs. Finally we end the section by introducing the Turaev surface and recall the

information it holds about how close a link is to alternating. This section is important throughout

2

our results as we regularly consider adequate links and reference the constructions stated here.

The next section of the background will be introducing classic link invariants, in other words

those which stem from the geometry of our link. Our results in this dissertation have shared goal

of being able to say more about these invariants. Then the last two sections of this chapter will

give information about two quantum invariants, the Jones polynomial and its generalization the

colored Jones polynomial. In the Jones polynomial section we will also show how state graphs

can be used to calculate it, along with an interesting property between the Jones polynomial and

crossing number for alternating links. In this chapter we will also give a construction of the colored

Jones polynomial but one can still understand the results with the knowledge that the colored Jones

polynomial is a sequence of Laurent polynomials.

1.1.2 Outline of Chapter 3

In this chapter we will present our results relating the crosscap number of knots and the

coefficients of the Jones polynomial. We will start this chapter by defining the family of links we

will be considering which are formed by tangle addition. Afterwards, we work towards the main

result for the case where we have two tangles and then generalize to more than two tangles. Finally,

we prove that there exist infinite families of knots for which the results do not generalize.

To start, let the Jones polynomial (see definition 2.3.1 for reference) for a link 𝐿 be,

𝑉 (𝐿) = 𝛼𝐿𝑡𝑛 + 𝛽𝐿𝑡𝑛−1 + · · · + 𝛽′

𝐿𝑡𝑚+1 + 𝛼′

𝐿𝑡𝑚,

𝑛, 𝑚 ∈ Z,

where 𝛼𝐿, 𝛼′

𝐿 ≠ 0. Then define 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 |. For chapter 3 this quantity will be the component
of the Jones polynomial we will use throughout our results. The other important invariant is the

crosscap number. The crosscap number is similar to the genus of a link but for spanning surfaces

which are non-orientable. We will talk more about this in chapter 2 but a formula for the crosscap

number for a surface 𝑆 is as follows,

𝐶 (𝑆) = 2 − 𝜒(𝑆) − 𝑘,

where 𝑘 is the number of boundary components for 𝑆 and 𝜒(𝑆) the Euler characteristic of 𝑆. Then

the crosscap number 𝐶 (𝐿) for a link 𝐿 is the minimum of 𝐶 (𝑆) for all non-orientable spanning

3

surfaces for 𝐿.

Quantum knot invariants have been used to better understand classical knot invariants and the

geometry of knots and links for quite some time. In the 80’s Kauffman [28] and Murasugi [37]

showed that the span of the Jones polynomial, as seen in Theorem 2.3.8, realizes the crossing

number for alternating links. More recent results from Futer, Kalfagianni, and Purcell showed

how the coefficients of the colored Jones polynomial store information about the geometry of

incompressible surfaces in the link complement and their strong relations to geometric structures

and in particular hyperbolic geometry [17, 13, 16]. For example, specific coefficients of the colored

Jones polynomial can coarsely define the volume of large families of hyperbolic links [15, 14]. In

particular Dasbach and Lin [9] did this for all hyperbolic alternating links. Further, the Volume

Conjecture [36] predicts that the asymptotics of the colored Jones polynomial can be used to

calculate the volume of all hyperbolic links. Recent work by Kalfagianni and Lee [24] which this

chapter references, continues this pattern of using quantum invariants to calculate the crosscap

number of links.

For alternating links Kalfagianni and Lee showed that there exists linear bounds for the crosscap

number with respect to certain coefficients of the Jones polynomial (see Theorem 1.1 in [24] for

reference). Before their work the best known lower bound for all alternating links was 𝐶 (𝐾) ≥ 1.

On the other side Clark [7] observed that for any knot the crosscap number is bounded by

𝐶 (𝐾) ≤ 2𝑔(𝐾) + 1,

where 𝑔(𝐾) is the orientable genus of the knot. Kalfagianni and Lee’s results gave an exact

calculation of the crosscap number for 283 prime alternating knots on Knotinfo and improvements

of 1472 prime alternating knots. Kindred [30] has also made progress in calculating the crosscap

number, calculating it for alternating knots with less than 13 crossings. We hope our results in

this chapter will pave the way for similar calculations for non-alternating knots which are less

understood.

Motivation for such calculations also comes from the work that has been done concerning the

orientable genus of links and the Alexander polynomial of a link and further the Heegard Floer

4

homology of a link. Crowell [39] and Murasugi [38] have independently shown that the orientable

genus of an alternating link is half the degree span of the Alexander polynomial of the knot. The

Heegard Floer homology is now known to detect the genus of links [34]. There is also an algorithm

using normal surface theory to calculate the orientable knot genus [4, 22]. The hope with the

crosscap number is to find parallel results to Heegard Floer and orientable genus using quantum

invariants. As of this dissertation the crosscap number outside of alternating knots and links is still

poorly understood except for a few families of links [45].

We generalize the results of [24] to a family of links which are the Conway sum of strongly

alternating tangles. We were able to do this using the alternating property of each individual tangle

along with Lemma 3.3.4, plus a series of lemmas we prove throughout Chapter 3. These results

give us bounds on the crosscap number of a link 𝐶 (𝐿) for a family of links which are not necessarily

alternating. The tangles we will be considering will be non-splittable which means that there does

not exist a copy of 𝑆2 ⊂ 𝑆3 such that the components of 𝐿 will be on opposite sides of 𝑆2 with

𝑆2 ∩ 𝐿 = ∅. The tangle diagrams will also be twist reduced which means that any copy of 𝑆1 which

intersects 𝑇 in four points such that two are adjacent to one crossing two to another will bound a

twist region, where a twist region is a maximal collection of bigons end to end. We will state the

main result of this chapter here, refer to Definition 3.1.2 for strongly alternating, Definition 2.2.4

for crosscap number, Definition 3.1.3 and Figure 3.2 for a Conway sum, .

Theorem 1.1.1. Let 𝑇1 and 𝑇2 be non-splittable, twist reduced, strongly alternating tangles whose

Conway sum is a link 𝐿. Let 𝐶 (𝐿) be the crosscap number of 𝐿 and 𝑘 𝐿 be the number of components

of 𝐿. Then, we have

where 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 |.

(cid:25)

(cid:24)𝑇𝐿
6

− 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑘 𝐿 + 8,

A key ingredient in the proof of Theorem 1.1.1, is Theorem 1.1.2 which we state below.

Theorem 1.1.2 gives us bounds for 𝐶 (𝐿) in terms of the crosscap numbers of the closures of the

tangles which sum to 𝐿.

Theorem 1.1.2. Let 𝑇1 and 𝑇2 be non-splittable, twist reduced, strongly alternating tangles, and

5

let 𝐿 be the link formed by the Conway sum of 𝑇1 and 𝑇2. Let 𝐾𝑖𝑁 and 𝐾𝑖𝐷 be the links formed by

the numerator closure and denominator closure of 𝑇𝑖 respectively, 𝑖 ∈ {1, 2}. We have

𝑚 − 2 ≤ 𝐶 (𝐿) ≤ 𝑚 + 2

where 𝑚 = min{𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ), 𝐶 (𝐾1𝐷) + 𝐶 (𝐾2𝐷)}.

Having Theorem 1.1.2 at hand we use a result of [24] and the additivity of twist numbers for

strongly alternating tangles to find bounds for 𝐶 (𝐿) in terms of the twist number of 𝐿. Then using

these bounds and a generalization of Theorem 1.6 from [19] gives us Theorem 1.1.1.

After proving Theorem 1.1.1 for the Conway sum of two tangles we will generalize this to larger

Conway sums. The proof for the case of larger Conway sums follows in the same vein as the proof

for two tangles. Ultimately, we will end up with the following result:

Theorem 1.1.3. Let 𝑇1, 𝑇2, ..., 𝑇𝑙 be non-splittable, twist reduced, strongly alternating tangles and

let 𝐿 be the link which results from taking the Conway Sum. Then let 𝐶 (𝐿) be the crosscap number

and 𝑘 𝐿 the number of link components in 𝐿, then

(cid:25)

(cid:24)𝑇𝐿 − 2𝑙
6

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑙 + 6 + 𝑘 𝐿.

After generalizing the results of [24] we consider the question of the extent to which Theo-

rem 1.1.3 generalize to all knots. Then, we will introduce two infinite families of knots for which

there do not exist linear bounds for 𝐶 (𝐿) in terms of 𝑇𝐿 for all links 𝐿 in the given family. The

first family we will look at is a family of torus knots. A torus knot is a ( 𝑝, 𝑞)-curve on the torus

such that 𝑝 and 𝑞 are relatively prime. Notice that when we project this curve into the plane we

will have a knot diagram. For the family of torus knots we consider we will find that 𝑇𝐿 will be at

most two but 𝐶 (𝐿) can be made arbitrarily large. This family will show that there does not exist a

uniform upper bound for all links.

On the other hand we will consider a certain family of Whitehead doubles. A Whitehead

double, 𝑊 (𝐿) of a link is a satellite knot. See figure 3.11 for reference. For the Whitehead double

of a knot it will always be true that 𝐶 (𝐿) = 2 which results directly from a result in [7]. But we will

6

show that we can make 𝑇𝐿 arbitrarily large for the family we look at which shows that a universal

linear lower bound does not exist. To state this precisely we have the following result:

Theorem 1.1.4. We have the following;

(a) There exists a family of links for which 𝑇𝐿 ≤ 2, but 𝐶 (𝐿) is arbitrarily large.

(b) There exists a family of links for which 𝐶 (𝐿) ≤ 3, but 𝑇𝐿 is arbitrarily large.

We point out that the families of link considered to prove Theorem 1.1.4 are both non-hyperbolic

families. So, we are left with the following question.

Question 1.1.5. Do there exist linear bounds for 𝐶 (𝐿) in terms of 𝑇𝐿 exist for all hyperbolic links?

1.1.3 Outline of Chapter 4

In this chapter we will be using the colored Jones polynomial to get bounds for, and to calculate

the crossing number for an infinite family of cable knots. Let 𝐾 be a knot, then the crossing

number for a knot, 𝑐(𝐾) is the minimal number of crossings for all knot diagrams representing 𝐾.

The crossing number for knots has historically been intractable, especially for satellite knots. In

particular we have the following question:

Question 1.1.6. If we let 𝐶 be the companion knot for a satellite knot 𝐾, is 𝑐(𝐾) ≥ 𝑐(𝐶)? [29]

Intuitively this seems to be an obvious conjecture as the diagram resulting from taking a

cabling of 𝐶 will have 𝑐(𝐾) equal to a multiple of 𝑐(𝐶) plus a constant depending on the pattern

knot. Despite this the intractable nature of the crossing number has prevented a proof as of this

dissertation.

Our work in this chapter will start with a stronger lower bound for cables of adequate knots.

Then, using a result of [25] we give an exact calculation of the crossing number for an infinite

family of knot cablings.

To state our results, for a knot 𝐾 and co-prime integers 𝑝, 𝑞 we define 𝐾𝑝,𝑞 to be the ( 𝑝, 𝑞)

cabling of 𝐾. In other words 𝐾𝑝,𝑞 is the curve on 𝜕𝑁 (𝐾) such that it wraps 𝑝 times around the

meridian and 𝑞 times around the canonical longitude. See Figure 4.1 for reference. The writhe of

any adequate diagram 𝐷 = 𝐷 (𝐾) for a knot 𝐾 is an invariant of 𝐾. We will denote it by wr(𝐾).

7

Theorem 1.1.7. For any adequate knot 𝐾 with crossing number 𝑐(𝐾), and any coprime integers

𝑝, 𝑞, we have

𝑐(𝐾𝑝,𝑞) ≥ 𝑞2 · 𝑐(𝐾) + 1.

Theorem 1.1.7, combined with the results of [25], has significant applications in determining

crossing numbers of prime satellite knots. In particular we have the following:

Corollary 1.1.8. Let 𝐾 be an adequate knot with crossing number 𝑐(𝐾) and writhe number wr(𝐾).

If 𝑝 = 2 wr(𝐾) ± 1, then 𝐾𝑝,2 is non-adequate and 𝑐(𝐾𝑝,2) = 4𝑐(𝐾) + 1.

The proof of Corollary 1.1.8 shows that, when 𝑝 = 𝑞 wr(𝐾) ± 1, we apply the ( 𝑝, 2)-cabling

operation to an adequate diagram 𝐷 = 𝐷 (𝐾) of 𝐾 the resulting diagram is a minimum crossing

diagram of the knot 𝑐(𝐾±1,2). It should be compared with other results in the literature asserting

that the crossing numbers of some important classes of knots are realized by a “special type" of

knot diagrams. These classes include alternating and more generally adequate knots, torus knots

and Montesinos knots [28, 37, 46] and untwisted Whitehead doubles of adequate knots of zero

writhe number [25].

Theorem 1.1.7 allows to compute the crossing number of (±1, 2)-cables of adequate knots

that are equivalent to their mirror images (a.k.a. amphicheiral) since such knots are known have

wr(𝐾) = 0. One way to obtain an amphicheiral adequate knot is to take the connected sum of an

adequate knot with its mirror image (the mirror image of a knot is 𝐾 ∗ = 𝑟 (𝐾), where 𝑟 : 𝑆3 → 𝑆3

is an orientation reversing diffeomorphism.

Corollary 1.1.9. For a knot 𝐾 let 𝐾 ∗ denote the mirror image of 𝐾 and, for every 𝑚 > 0, let

𝐾 𝑚 := #𝑚 (𝐾#𝐾 ∗) denote the connected sum of 𝑚-copies of 𝐾#𝐾 ∗. Suppose that 𝐾 is adequate

with crossing number 𝑐(𝐾). Then, the (1, 2)-cable 𝐶1,2(𝐾𝑚) is non-adequate, and we have

𝑐(𝐾 𝑚

1,2) = 4𝑚𝑐(𝐾) + 1.
Note that out of the 2977 prime knots with up to 12 crossings, 1851 are listed as adequate on

Knotinfo [34] and thus Corollary 1.1.9 applies to them. The method of the proof of Theorem 1.1.7

is similar to that used in [25] to determine the crossing number of untwisted Whitehead doubles of

8

zero-writhe adequate knots. These Whitehead doubles and the cables 𝑐(𝐾±1,2) of Theorem 1.1.7

are the only infinite non-adequate, non-composite satellite knots for which the crossing numbers

have been determined.

Another result we show in this section relates to the open conjecture on the crossing number

of connected sums [29]. 𝐾 = 𝐾1#𝐾2 is defined to be a connected sum of knots 𝐾1 and 𝐾2 if there

exists a sphere such that 𝑆 ∩ 𝐾 is two points 𝑥, 𝑦 and 𝑆 bounds two 3-balls 𝐵1 and 𝐵2, such that if

𝛼 is an embedded arc on 𝑆 connecting 𝑥 and 𝑦

(𝐵𝑖 ∩ 𝐾) ∪ 𝛼 = 𝐾𝑖.

Similar to satellite knots, the crossing number of connected sums is also not well understood. There

remains an open conjecture that states 𝑐(𝐾1#𝐾2) = 𝑐(𝐾1) + 𝑐(𝐾2), where # is the connected sum of

knots. Similar to satellite knots this remains open as it is unclear if we can show that the resulting

diagram for 𝐾1#𝐾2 is a minimal crossing diagram for 𝐾1#𝐾2.

Theorem 1.1.10. Suppose that 𝐾 is an adequate knot and let 𝐾1 = 𝐾𝑝,2, where 𝑝 = 𝑞 wr(𝐾) ± 1.

Then for any adequate knot 𝐾2, the connected sum 𝐾1#𝐾2 is non-adequate and we have

𝑐(𝐾1#𝐾2) = 𝑐(𝐾1) + 𝑐(𝐾2).

9

CHAPTER 2

BACKGROUND

In this chapter we will cover the necessary background information needed throughout this docu-

ment.

2.1 Alternating and Adequate Links

In this section we will define two families of knots which will come up regularly later in this

paper. The first family of knots we will talk about is alternating knots.

Definition 2.1.1. A knot diagram 𝐷 is called alternating if tracing along the diagram the crossings

will alternate over/under. Then we call a knot 𝐾 alternating if it has a knot diagram which is

alternating.

Alternating knots generalize to alternating links by considering each of the link components

independently enters crossings in an over/under pattern. Alternating knots are well understood,

and therefore many tools exist to help us further study alternating knots and links.

A generalization of alternating knots and links which we will talk about here is a family known

as adequate knots. All alternating knots and links are also adequate, which is why this is a natural

family for us to consider next. As we will see throughout this document, adequate links have many

useful properties with respect to both the geometry of a knot or link, and also with respect to the

quantum invariants of a knot or link. It will also be an important hypothesis of many of our lemmas

and theorems.

Definition 2.1.2. Let 𝐷 (𝐿) be the diagram of a link 𝐿, we define the A resolution and B resolution

of a crossing as shown in Figure 2.1. Then a Kauffman state is a choice of resolutions for each

crossing in 𝐷 (𝐿).

Definition 2.1.3. Next we define an all A (resp. all B) state graph G𝐴 := G𝐴 (𝐷 (𝐿)) (resp.

G𝐵 (𝐷 (𝐿))) by adding edges where we performed resolutions and then contracting the simple closed

curves to vertices. Let 𝑒 𝐴 (resp. 𝑒𝐵) be the number of edges in G𝐴 (𝐷 (𝐿)) (resp. G𝐵 (𝐷 (𝐿))).

Further, if we identify all parallel edges (edges whose union bounds a disk) we find the reduced

state graph G′

𝐴 := G′

𝐴 (𝐷 (𝐿)) (resp. G′

𝐵 (𝐷 (𝐿))). Let 𝑒′

𝐴 (resp. 𝑒′

𝐵) be the number of edges in

10

Figure 2.1 Here we see an A and B resolution for a crossing.

G′

𝐴 (𝐷 (𝐿)) (resp. G′

𝐵 (𝐷 (𝐿))). See Figure 2.2 for an example of this process.

Figure 2.2 Left: the trefoil knot, Center: state circles with edges where the crossings were, Right:
(reduced) state graph.

Now we are ready to define what it means for a link to be adequate.

Definition 2.1.4. We call a link diagram A-adequate (resp. B-adequate) if the A state (resp. B

state) graph of the diagram has no one edge loops. A link diagram is called adequate if it is both

A-adequate and B-adequate, and a link is adequate if it has a diagram which is adequate.

Another definition for adequacy has to do with the count of state circles after resolving the

crossings.

Definition 2.1.5. We call a link diagram A-adequate (resp. B-adequate) if the all resolution A state

(resp. B state) has more state circles than any state 𝐴′ where all but one crossing is resolved with

an A-resolution (resp. B resolution) with the one remaining crossing resolved via a B resolution

(resp. A resolution). A link diagram is called adequate if it is both A-adequate and B-adequate,

and a link is adequate if it has a diagram which is adequate.

It is not hard to see that these two definitions are equivalent. This arises from the fact that any

one edge loop will represent a crossing which when flipped will result in a new state circle. The

one edge loop will represent where we cut our state circle to get two state circles instead of one.

11

2.1.1 Turaev Surfaces

We can further develop the idea of a state graph to find a surface called the Turaev surface of a

Kauffman state for a link diagram. We follow the construction of [47] and [8]. Let 𝑠 be a choice of

a state for a link diagram and ˆ𝑠 be the dual state. Where the dual state is defined to be the opposite

choice of resolution at each of the crossings.

To build this surface we start by allowing Γ ⊂ 𝑆2 to be the 4-valent graph arising from the

projection of the link diagram onto 𝑆2. In particular Γ will be the graph with vertices where our

link diagram has crossings. We thicken 𝑆2 to be 𝑆2 × [−1, 1], with Γ ⊂ 𝑆2 × {0}. Outside of a

neighborhood of the vertices of Γ our surface will intersect Γ × [−1, 1] inside this thickening. Then

at each vertex of Γ we will add a saddle positioned so that at 𝑆2 × {1} we will have the state circles

of 𝑠 and at 𝑆2 × {−1} will be the state circles of ˆ𝑠, see Figure 2.3 for reference. Then we cap the

state circles of both 𝑠 and ˆ𝑠 with disks to create a closed surface 𝑆(𝑠). We call the resulting surface

𝑆(𝑠) the Turaev surface of 𝑠.

Figure 2.3 A saddle at one of the verices of Γ, to create the surface 𝑆(𝑠) we cap of the state circles
shown by 𝑠 and ˆ𝑠. We also see that away from the saddle the surface will include Γ × [−1, 1]. [11]

Using the Turaev surface we are able to realize interesting properties of the link diagram. Before

investigating these properties we introduce the following lemma from [11] for 𝐺 (𝑠).

Lemma 2.1.6. 𝑆(𝑠) is an unknotted surface which means that 𝑆3\𝑆(𝑠) is the disjoint union of two

handlebodies.

Now that we better understand the Turaev surface for a link we can define a link invariant known

as the Turaev genus of a link.

12

Definition 2.1.7. Let 𝑠∗ be the all A or all B state of a link diagram. If 𝑔(𝑆(𝑠∗)) is the genus of the

surface 𝑆(𝑠∗), the Turaev genus 𝑔𝑇 (𝐿) of a link 𝐿 is the minimum for 𝑔(𝑆(𝑠∗)) over all diagrams

for 𝐿.

The Turaev genus in a sense tells us how far our link is from alternating, in particular it gives

the necessary genus for a surface on which are link diagram can be made alternating. This is shown

through the following lemma and corollary [11].

Lemma 2.1.8. When 𝑠 is the all 𝐴-state or the all B-state, the link has an alternating projection to

𝑆(𝑠).

Corollary 2.1.9. A link has Turaev genus equal to zero if and only if the link is alternating.

Corollary 2.1.9 follows directly from Lemma 2.1.8 since a link possessing an alternating

projection on the sphere is the definition of alternating. We will use Turaev genus in our proofs

later in this document, and give an explicit formula for the Turaev genus.

2.2 Classic Knot Invariants

In this section we will define the classic knot invariants which are relevant to this work. We

will also give some historical results here which are not used in our results later in the paper. The

first such invariant we will talk about is the crossing number of a knot.

Definition 2.2.1. The crossing number of a knot is the minimum number of crossings needed for

a knot across all knot diagrams of that knot.

Given a knot 𝐾 we will use 𝑐(𝐾) to represent the crossing number. If we let 𝑈 be the unknot

then 𝑐(𝑈) = 0 and the unknot is the only knot with crossing number less than 3. The trefoil is

the first nontrivial knot and has crossing number equal to three. Historically, given an arbitrary

knot diagram the crossing number is quite intractable. As shown in Figure 1.1 we can make rather

complex diagrams for the unknot using only isotopy.

Next we will define multiple knot invariants which arise from the spanning surface of a knot or

link.

Definition 2.2.2. A spanning surface for a link 𝐿 is a surface 𝑆 with boundary such that 𝜕 (𝑆) = 𝐿.

Notice in Figure 2.4 that both of these spanning surfaces are non-orientable. Spanning surfaces

13

Figure 2.4 Left: A spanning surface for the unknot, Right: A spanning surface for the trefoil.

can also be orientable, for instance we can take the complement surface of the shading, for each

of the knots in Figure 2.4. Both orientable and non-orientable surfaces give rise to link invariants.

For orientable links we can find an invariant called the knot genus. Remember that for a surface 𝑆

the genus is

𝑔(𝑆) =

2 − 𝜒(𝑆) − 𝑘
2

,

where 𝜒(𝑆) is the Euler characteristic of 𝑆 and 𝑘 the number of boundary components. Another

definition is that the genus is the maximal number of non-intersecting closed curves we can cut 𝑆

along without separating the surface. Now we are ready to define the genus of a knot.

Definition 2.2.3. The knot genus of a link 𝑔(𝐿) is the minimum genus across all orientable surfaces

𝑆 which are spanning surfaces for the link.

We can define a similar invariant using the non-orientable spanning surfaces of a knot. To do

so we must first define the crosscap number of a non-orientable surface. The crosscap number of

a non-orientable surface 𝑆 is

𝐶 (𝑆) = 2 − 𝜒(𝑆) − 𝑘

where 𝜒(𝑆) and 𝑘 are defined as above. We can also view the crosscap number of a surface as

the number of projected planes we need to construct our surface. Now we can define the crosscap

number of a link.

Definition 2.2.4. The crosscap number of a link is the minimum value for 𝐶 (𝑆) across all non-

orientable surfaces which are bounded by the link.

14

2.3

Jones Polynomial

In this section we will define another knot and link invariant called the Jones polynomial.

The Jones polynomial is a Laurent polynomial over Z and we will begin to define this invariant

by showing the construction of the Kauffman bracket polynomial. We start with a knot or link

diagram 𝐷 and then define the Kauffman bracket polynomial to be ⟨𝐷⟩. Then we get the following

relationships between ⟨𝐷⟩ and the Kauffman bracket of other diagrams. For rule 3 it relates the

polynomial for a diagram with the two diagrams where a chosen crossing is resolved in both ways.

Then rule 2 allows us to remove an unlinked trivial link component. Finally at the end we will use

rule 1 to end up with a polynomial.

1. (cid:10)

(cid:11) = 1

2. (cid:10)𝐿 ∪ (cid:11) = (−𝐴2 − 𝐴−2)⟨𝐿⟩

3. (cid:10)

(cid:11) = 𝐴 (cid:10)

(cid:11) + 𝐴−1 (cid:10)

(cid:11) .

It turns out that the Kauffman bracket polynomial is not a knot invariant under Reidemeister I moves

as this will shift the polynomial by 𝐴±3 per twist. It will hold under the other two Reidemeister

moves. We will account for this when we define the Jones polynomial using the Kauffman bracket

polynomial.

Definition 2.3.1. Let 𝐷 (𝐾) be a knot diagram of the knot 𝐾, and let ⟨𝐷 (𝐾)⟩ be the Kauffman

bracket polynomial for 𝐷 (𝐾). Then we define 𝑉 (𝐾), the Jones polynomial of 𝐾 to be

𝑉 (𝐾) = ((−𝐴)−3𝑤(𝐷 (𝐾)) ⟨𝐷 (𝐾)⟩)𝑡1/2=𝐴−2.

Another definition for the Jones polynomial exists using Kauffman state graphs. To do this

we will start by indexing the crossings for a given link diagram 𝐷, 1, 2, 3, ..., 𝑛. Then we will

define a function 𝑠 : {1, 2, ..., 𝑛} → {−1, 1} where 𝑠 maps to −1 if we resolve the crossing with a

B-resolution and 1 if we resolve with an A-resolution. Notice that a choice of function 𝑠 will give

us a Kauffman state. Further, define a value |𝑠𝐷 | to be the number of state circles that arise in the

Kauffman state defined by 𝑠. Then we can state the Kauffman bracket polynomial as follows:

15

Definition 2.3.2. If 𝐷 is a link diagram with 𝑛 crossings, the Kauffman bracket polynomial of 𝐷

can be computed as below,

⟨𝐷⟩ =

∑︁

(cid:16)

𝐴(cid:205)𝑛

𝑖=1 𝑠(𝑖) (−𝐴−2 − 𝐴2) |𝑠𝐷 |−1(cid:17)

.

𝑠

Where the first summation is over all 2𝑛 possible functions 𝑠.

From here we get to the Jones polynomial by the same formula as in Definition 2.3.1. This

relationship between the Jones polynomial and Kauffman states gives rise to many important results

relating the crossing number of an adequate link to the Jones polynomial. For these results we

will state them without proof but an interested reader can find them in Lickorish [32]. The proofs

largely rely on Definition 2.3.2.

As mentioned when we defined the crossing number of a link, the crossing number is a

historically intractable invariant to compute. However, the finding of Kauffman states and the

Kauffman bracket polynomial resulted in a variety of useful discoveries concerning the crossing

number for alternating links. We will start by letting 𝑠 𝐴 and 𝑠𝐵 be the all A-state and all B-state

respectively. Now we can formalize the alternative definition of adequacy we gave above.

Definition 2.3.3. We call a link diagram 𝐷 A-adequate if |𝑠 𝐴𝐷| > |𝑠𝐷 | for all states 𝑠 where
𝑛
∑︁

𝑠(𝑖) = 𝑛 − 2. Similarly we call a link diagram B-adequate if |𝑠𝐵𝐷 | > |𝑠𝐷 | for all states 𝑠 where

𝑖=1
𝑛
∑︁

𝑖=1

𝑠(𝑖) = 𝑛 + 2, then if 𝐷 is both A-adequate and B-adequate, 𝐷 is adequate.

Then if we let 𝑀 ⟨𝐷⟩ be the maximum degree of the Kauffman bracket polynomial for 𝐷 and

𝑚⟨𝐷⟩ be the minimum degree we get the following lemma.

Lemma 2.3.4. Let 𝐷 be a link diagram with 𝑛 crossings. Then

1. 𝑀 ⟨𝐷⟩ ≤ 𝑛 + 2|𝑠 𝐴𝐷| − 2, with equality if 𝐷 is A-adequate, and

2. 𝑚⟨𝐷⟩ ≥ −𝑛 − 2|𝑠𝐵𝐷| + 2, with equality if 𝐷 is B-adequate.

Combining the two inequalities in the previous lemma we get the following corollary,

Corollary 2.3.5. If 𝐷 is an adequate diagram, then

𝑀 ⟨𝐷⟩ − 𝑚⟨𝐷⟩ = 2𝑛 + 2|𝑠 𝐴𝐷 | + 2|𝑠𝐵𝐷| − 4.

16

Now we define a link diagram to be connected if there does not exist a simple closed curve such

that the curve separates part of the link from the rest. Where a link diagram is considered split if it

is possible to draw a simple closed curve 𝛾 such that 𝛾 ∩ 𝐷 = ∅ and 𝛾 separates 𝐷 into two pieces.

For connected diagrams, we can relate |𝑠 𝐴𝐷| + |𝑠𝐵𝐷| and the number of crossings a diagram has

in the following way,

Lemma 2.3.6. Let 𝐷 be a connected 𝑛-crossing diagram.

1. If 𝐷 is alternating, then |𝑠 𝐴𝐷| + |𝑠𝐵𝐷| = 𝑛 + 2.

2. If 𝐷 is non-alternating and strongly prime, then |𝑠 𝐴𝐷| + |𝑠𝐵𝐷| < 𝑛 + 2.

Where strongly prime is defined as follows:

Definition 2.3.7. We call a knot 𝐾 prime if for any 𝐾1 and 𝐾2 such that 𝐾 = 𝐾1#𝐾2 one of 𝐾1 or

𝐾2 must be the unknot, then we call the knot strongly prime if the diagram of the unknot contains

no crossings.

Then this can be used to prove the following conjecture by Tait [2], first shown by Kauffman,

Thistlewaite, and Murasagi. First we define 𝐵(𝑉 (𝐿)) to be the breadth of the Jones polynomial for

a given link 𝐿. As a reminder, the breadth of a Laurent polynomial is the difference of the highest

and lowest degrees of the polynomial.

Theorem 2.3.8. Let 𝐷 be a connected, 𝑛-crossing diagram of an oriented link 𝐿 with Jones

polynomial 𝑉 (𝐿). Then,

1. 𝐵(𝑉 (𝐿)) ≤ 𝑛;

2. if 𝐷 is alternating and reduced, then 𝐵(𝑉 (𝐿)) = 𝑛;

3. if 𝐷 is non-alternating and a prime diagram, then 𝐵(𝑉 (𝐿)) < 𝑛.

Where a reduced knot is a knot without a nugatory crossing or a crossing which can be removed

by a half twist.

It immediately follows that for an alternating link any diagram which realizes

the crossing number must be reduced and alternating. Then we see that for alternating links the

crossing number will be equal to 𝐵(𝑉 (𝐿)). In [32] we can also see that an adequate diagram for a

17

link also realizes the crossing number. As we will see later in the paper there are also results that

relate the crossing number of adequate links to the colored Jones polynomial.

2.4 Colored Jones Polynomial

A generalization of the Jones polynomial which we will use during this dissertation is the

colored Jones polynomial. The colored Jones polynomial is a sequence of Laurent polynomials 𝐽𝑛

in Z[𝑡, 𝑡−1] indexed by 𝑛 ∈ N. The Jones polynomial will be the case where 𝑛 = 2. Here we will

define the colored Jones polynomial via the Kauffman bracket and the Jones-Wenzel idempotent

of the Temperley-Lieb Algebra, a detailed description of which can be found in [32]. The colored

Jones polynomial may also be defined via the representation theory of quantum SL2(C) which we

will not show here but details may be found in [41], [40], and [48].

2.4.1 The Temperley-Lieb Algebra

We begin by defining a linear skein S(𝐹) over a surface 𝐹. S(𝐹) is a vector space of formal

linear sums over C of link diagrams in 𝐹 quotiented by

1. (cid:10)𝐷 ∪ (cid:11) = (−𝐴2 − 𝐴−2)⟨𝐷⟩

2. (cid:10)

(cid:11) = 𝐴 (cid:10)

(cid:11) + 𝐴−1 (cid:10)

(cid:11)

where 𝐴 is an invertible variable.

Let 𝑆(𝐷, 2𝑛) be the linear skein of the square representation of the disk, with 𝑛 marked points

along one side of the square and then 𝑛 marked points along the opposite edge of the square. Given

𝑑1, 𝑑2 ∈ 𝑆(𝐷, 2𝑛) we define a product 𝑑1𝑑2 as the juxtaposition of the two disks along the right

side of 𝑑1 and the left side of 𝑑2 where we position the marked sides of the square to fall on the left

and right.

Definition 2.4.1. Extending this product by linearity we arrive at a bilinear map

𝑆(𝐷, 2𝑛) × 𝑆(𝐷, 2𝑛) → 𝑆(𝐷, 2𝑛)

which makes 𝑆(𝐷, 2𝑛) into an algebra which we call the Temperley-Lieb Algebra, 𝑇 𝐿𝑛, over C.

𝑇 𝐿𝑛 has generators 1, 𝑒1, ..., 𝑒𝑛−1 shown below.

18

Figure 2.5 Generators of the Temperley-Lieb Algebra.

Note that in Figure 2.5 that on the left the 𝑛 represents the 𝑛 parallel strands in the generator 1,

and on the right the 𝑖 − 1 and 𝑛 − (𝑖 + 1) represent the number of parallel strands below and above

the ‘u’ shaped strands.

There is a map from 𝑇 𝐿𝑛 to S(R2) by projecting an element 𝑑 ∈ 𝑇 𝐿𝑛 into R2 and connecting

the marked points on the left side of 𝑑 to the marked points on the right of 𝑑 by 𝑛 parallel strands.

Notice that in the linear skein where 𝐹 = R2, the image is reduced to an empty diagram with a

coefficient a Laurent polynomial in 𝐴.

2.4.2 The Jones-Wenzl Idempotent

We now define the Jones-Wenzl idempotent, which is an element of the Temperley-Lieb algebra.

Definition 2.4.2. The Jones-Wenzl idempotent 𝑓𝑛 is the unique element in 𝑇 𝐿𝑛 which satisfies

(i.) 𝑓𝑛𝑒𝑖 = 0 = 𝑒𝑖 𝑓𝑛 for 1 ≤ 𝑖 ≤ 𝑛 − 1,

(ii.) 𝑓𝑛 − 1 belongs to the algebra generated by {𝑒1, ..., 𝑒𝑛−1},

(iii.) 𝑓𝑛 𝑓𝑛 = 𝑓𝑛, and

Further Δ𝑛 represents the polynomial that arises when we connect the 𝑛 marked points on each side

of 𝑓𝑛 with parallel strands in the image of S(R2), which has the following form:

Δ𝑛 =

(−1)𝑛 ( 𝐴2(𝑛+1) − 𝐴−2(𝑛+1))
( 𝐴2 − 𝐴−2)

.

We represent 𝑓𝑛 by an empty box with 𝑛 incoming parallel strands and 𝑛 outgoing parallel

strands as shown in Figure 2.6. Similarly, the closure Δ𝑛 is represented by an empty box with 𝑛

parallel incoming strands and 𝑛 parallel outgoing strands, but in this case the 𝑛 strands are closed

as shown in Figure 2.6.

19

Figure 2.6 Left: The Jones Wenzl Idempotent, Right: its closure in S(R2).

The existence of 𝑓𝑛 is proven in [32]. There is also a recursion formula for 𝑓𝑛 which is

important when working with idempotent so we will show it here diagrammatically. Notice that

in Figure 2.7 the number above any line represents that many parallel strands, so for instance the

leftmost diagram will have 𝑛 + 1 parallel incoming and outgoing strands from the box. Also, the

Δ𝑛−1
Δ𝑛

will be a rational function in 𝐴.

Figure 2.7 Recursion formula for the Jones Wenzl Idempotent.

Similar to the Kauffman bracket polynomial if we add a type-1 Reidemester move to the

𝑛-strands of 𝑓𝑛 we will have equality if we multiply the skein element by a scalar in 𝐴 [32].

Figure 2.8 Effect of adding a twist to 𝑓𝑛.

20

2.4.3 A formula for the colored Jones polynomial

Before we give the formula for the colored Jones polynomial we define a family of polynomials

defined by a recurrence relations called the Chebyshev’s polynomials of type 2 [31].

Definition 2.4.3. A Chebyshev’s polynomial of type 2 is a polynomial for an unoriented link diagram

𝐷 defined by the following recurrence relation. Let S0(𝐷) = 1 and S1(𝐷) = 𝐷. Then for 𝑛 ≥ 2,

𝑆𝑛+1(𝐷) = 𝐷𝑆𝑛 (𝐷) − 𝑆𝑛−1(𝐷).

Now we can state a formula for the colored Jones polynomial. To start, if 𝐷 is an unoriented

link diagram of a link 𝐿 with 𝑙 link components, then 𝐷 defines a multilinear map

⟨, ..., ⟩𝐷 : S(𝑆1 × 𝐼) × · · · × S(𝑆1 × 𝐼) → S(R2).

Where this is the Cartesian product of 𝑙 copies of the linear skeins of the annulus 𝑆1 × 𝐼 into S(R2).

We define this product by starting with an annulus corresponding to the blackboard framing of

each component of the link, where the boundaries of the annulus run parallel to 𝐷. Figure 2.9

shows where the boundary of the annulus runs for a diagram 𝐷 of the trefoil knot. For this setup

Figure 2.9 𝐷2 where 𝐷 is the standard trefoil diagram.

we identify 𝑆1 × 𝐼 as the annulus, giving us a map from skein elements 𝑆1 × 𝐼 into the annulus

corresponding to a link component of 𝐷.

Definition 2.4.4. The 𝑛 + 1 − 𝑘th-unreduced colored Jones polynomial 𝐽𝑛+1

𝐾 (𝑞) is the polynomial

obtained by the substitution 𝑞1/2 = 𝐴−1/2 into the polynomial

(cid:16)

(−1)𝑛 𝐴𝑛2+2𝑛(cid:17) 𝑤(𝐷)

⟨Δ𝑛, ..., Δ𝑛⟩ ∈ S(𝑅2).

21

By induction and the recursion formula of 𝑓𝑛 shown in Figure 2.7, we have that

⟨, ..., Δ𝑛, ..., ⟩ = ⟨, ..., 𝛼Δ𝑛 − Δ𝑛−1, ..., ⟩

where 𝛼 is the generator of S(𝑆1×𝐼) consisting of a parallel curve encircling the annulus. Then if we

extend by bilinearity we find a second formula for the colored Jones polynomial when substituting

in 𝑡1/2 = 𝐴−1/2 as below.

𝐽𝐾 (𝑛 + 1, 𝑡) =

(cid:18) (cid:16)

(−1)𝑛 𝐴𝑛2+2𝑛(cid:17) 𝑤(𝐷)

(cid:19)

⟨S𝑛 (𝐷)⟩

.

𝑡1/2=𝐴−1/2

In chapter 4 we will be using the colored Jones polynomial to find bounds for the crossing

number of a family of knot cablings and then calculate the crossing number for an infinite sub

family of satellite knots. In particular, we will be using the span of the colored Jones polynomial,

which is the difference between the maximum and minimum degrees of a polynomial, to calculate

information about the crossing number for satellite knots. Garoufalidis [20] showed that the

extremal degrees of the colored Jones polynomial are quasi-quadratic polynomials. As we will see

in Chapter 4 this will be relevant for the proofs of multiple theorems and a key component of the

statements of Proposition 4.1.3 and Lemma 4.1.4.

It has also been shown for certain families of knots that these extremal degrees of the colored

Jones polynomial tell us the boundary slopes of knots. Garoufalidis [20] showed this for alternating

knots, knots up to nine crossings, torus knots, and a family of 3-string pretzel knots. Futer,

Kalfagianni, and Purcell [18] showed this for adequate knots, which is the family of knots we will

use as the pattern knots for our satellite knots in Chapter 4. We mention this here, but we will not

be talking about the slopes conjecture later in the dissertation, just using the extremal degrees.

22

CHAPTER 3

CROSSCAP NUMBER OF CONWAY SUMS

In this chapter we will explore the relationship between the crosscap number for links and certain

coefficients of the Jones polynomial. The work and figures in this chapter come from a preprint by

McConkey [35].

3.1 Tangles and the Upper Bound

We start with a couple of definitions.

Definition 3.1.1. A tangle is a graph in the plane contained within a box that intersects the box

at the four corners with one-valent vertices, with all other vertices, contained inside the box, four-

valent, and given over/under crossing data. We label the four, 1-valent vertices NW, NE, SE, SW,

positioned according to Figure 3.1.

Definition 3.1.2. The closure of a tangle is the link which results when we connect the NW and NE

points along the box and SW and SE points along the box as seen in the center panel of Figure 3.1,

this is called the numerator closure. If we close as in the right hand panel of Figure 3.1 we call it

the denominator closure. A tangle is strongly alternating if both closures are prime, reduced and

alternating.

Figure 3.1 Left: A tangle inside a box with directional strands labelled, Center: numerator
closure, Right: denominator closure.

Definition 3.1.3. A Conway sphere is a 2-sphere which intersects a knot or link transversely in four

points. A Conway sum is a sum of tangles as shown in Figure 3.2. For our purposes, a Conway

sphere Σ will be positioned such that it intersects a Conway sum at the four one valent vertices for

one of the tangles in the sum. Notice if we let 𝑆 be a spanning surface for our Conway sum, then

23

𝑆 ∩ Σ will contain two arcs and a possibly empty collection of simple closed curves.

Figure 3.2 An example of a Conway sum of 𝑙 tangles.

Figure 3.3 Here we see a tangle contained within a Conway sphere. The blue dots represent the
intersection of 𝑇𝑖 with Σ. Then the dotted lines are the intersections of 𝑆 with Σ.

We are now ready to begin proving Theorem 1.1.2. We separate it into the upper and lower

bounds, beginning with the upper bound:

Lemma 3.1.4. Let 𝑇1 and 𝑇2 be a pair of non-splittable, strongly alternating tangles. Let 𝐿 be the

link formed by the Conway sum of 𝑇1 and 𝑇2. Let 𝐾𝑖𝑁 and 𝐾𝑖𝐷 be the numerator and denominator

closures, respectively, of 𝑇1 and 𝑇2. If 𝐶 (𝐿) is the crosscap number of 𝐿, then

𝐶 (𝐿) ≤ min{𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ) + 2, 𝐶 (𝐾1𝐷) + 𝐶 (𝐾2𝐷) + 2}.

Proof. Start with a pair of strongly alternating tangles, 𝑇1 and 𝑇2. Let 𝐾1𝑁 and 𝐾2𝑁 be the links

acquired by the numerator closures of the tangles. Let 𝑆1 and 𝑆2 be non-orientable spanning

surfaces which realize the crosscap numbers of 𝐾1𝑁 and 𝐾2𝑁 respectively. We find a spanning

surface 𝑆 of 𝐿 by attaching 𝑆1 and 𝑆2 with a pair of bands bounded by the strands along which the

Conway sum was taken. Notice that as we do not cut 𝑆1 and 𝑆2, 𝑆 will also be non-orientable.

Now we study the relationship between 𝐶 (𝑆) and the sum of 𝐶 (𝑆1) and 𝐶 (𝑆2). We remind the

reader that 𝐶 (𝑆) = 2 − 𝜒(𝑆) − 𝑘. The difference between 𝑆 and the disjoint union of 𝑆1 and 𝑆2 is

24

the two connecting bands used to construct 𝑆. So, 𝜒(𝑆) = 𝜒(𝑆1) + 𝜒(𝑆2) − 2.

Next we compare the number of link components in 𝐿 with the total in 𝐾1𝑁 and 𝐾2𝑁 . The

gluing of the East strands of 𝐾1𝑁 to the West strands of 𝐾2𝑁 will reduce the number of components

by 1 as we are connecting two disjoint links. The other attachment can increase or decrease the

number of components by 1, or keep it the same. Then 𝑘 𝐿 = 𝑘 𝐾1𝑁 + 𝑘 𝐾2𝑁 − 𝜖 where 𝜖 = 0, 1, or 2.

Now we substitute for 𝜒(𝑆) and 𝑘 𝐿 to find:

𝐶 (𝑆) = 2 − 𝜒(𝑆1) − 𝜒(𝑆2) + 2 − 𝑘 𝐾1𝑁 − 𝑘 𝐾2𝑁 + 𝜖 = 𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ) + 𝜖 .

Hence:

𝐶 (𝐿) ≤ 𝐶 (𝑆) = 𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ) + 2

as 𝜖 = 2, will give the weakest upper bound. By the same argument with the denominator closures

of 𝑇1 and 𝑇2, we find that

giving us the claim.

𝐶 (𝐿) ≤ 𝐶 (𝐾1𝐷) + 𝐶 (𝐾2𝐷) + 2,

□

Here we remark that Lemma 3.1.4 will hold even if we take two general tangles. Notice in the

proof that we do not use the fact that 𝑇1 or 𝑇2 are non-splittable or strongly alternating. As this will

not hold true for the other statements we included these hypotheses for uniformity.

3.2 The Lower Bound

3.2.1 Technical Lemmas

Before we show the lower bound, we will need some more background, as well as some technical

results. Lemma 3.2.1 below was discussed in the proof of Lemma 3.1.4.

Lemma 3.2.1. Let 𝐿 be the Conway sum of the tangles 𝑇1 and 𝑇2, and let 𝐾1 and 𝐾2 be closures

of 𝑇1 and 𝑇2. If 𝑘 𝐿 is the number of link components for 𝐿, and 𝑘1 and 𝑘2 the number of link

components for 𝐾1 and 𝐾2, respectively, then 𝑘 𝐿 = 𝑘1 + 𝑘2 − 𝜖 for 𝜖 = 0, 1, 2.

25

Definition 3.2.2. Let 𝐿 be a link in 𝑆3 and let 𝑁 (𝐿) be a neighborhood of 𝐿. A spanning surface 𝑆

of 𝐿 in 𝑆3 is defined to be meridianally boundary compressible if there exists a disk 𝐷 embedded

in 𝑆3\𝑁 (𝐿), such that 𝜕𝐷 = 𝛼 ∪ 𝛽 where 𝛼 = 𝐷 ∩ 𝜕𝑁 (𝐿) and 𝛽 = 𝐷 ∩ 𝑆. Notice both 𝛼 and 𝛽 are

arcs, 𝛽 does not cut off a disk of 𝑆, 𝜕𝐷 ∩ 𝜕𝑆 cuts 𝜕𝑆 into two arcs 𝜙1, 𝜙2 and 𝛼 ∪ 𝜙𝑖 is a meridian

of the link for one of 𝑖 = 1, 2 as shown in Figure 3.4. A spanning surface is said to be meridianally

boundary incompressible if no such disk exists.

Figure 3.4 This figure shows a case where 𝐿 is meridianally boundary compressible. As 𝜙1 ∪ 𝛼
create a meridian of 𝑁 (𝐿).

Definition 3.2.3. Given an alternating projection of a link 𝐿 on 𝑆2, we modify it so that in a

neighborhood of each crossing, we have a ball whose equator lies on 𝑆2 such that the over strand

runs over the ball and the under goes underneath, see Figure 3.5 for reference. We call every such

ball a Menasco ball, and we call such an embedding of 𝐿 relative to 𝑆2 a Menasco projection 𝑃

with 𝑛 crossings.

Definition 3.2.4. We say that a surface 𝑆 intersects a Menasco ball 𝐵𝑖 in a crossing band if 𝑆 ∩ 𝐵𝑖

consists of a disc bounded by the over and under strands on 𝜕𝐵𝑖 along with opposite arcs along the

equator of 𝐵𝑖. We refer the reader to figures 28-30 in [3] for reference.

Let 𝐹 = 𝑆2/(cid:208)𝑖 𝐵𝑖, where the 𝐵𝑖 are the Menasco balls for 𝐿. Given an incompressible (not

necessarily meridianally) surface 𝑆 spanning 𝐿, we can isotope 𝑆 so that

26

Figure 3.5 Here we put a crossing into Menasco form, with the over strand running across the top
of the ball and the lower strand running along the bottom.

i. 𝑆 ∩ 𝐹 is a collection of simple closed curves and arcs with endpoints on 𝐿 or the equator of a

Menasco ball.

ii. 𝑆 is disjoint wherever possible from the interior of the 𝐵𝑖, including along 𝑁 (𝐿). The only

exception will be at crossing bands.

We say such a surface isotoped in this way is in Menasco form.

We will also need Lemma 5.1 from [3] by Adams and Kindred which we state here as

Lemma 3.2.5. Lemma 3.2.5 is required to prove Lemma 3.2.6, which is proven as Corollary

5.2 in [3]. Lemma 3.2.6 is essential to our proof of the lower bound, as it guarantees the con-

nectedness of any spanning surface of the numerator or denominator closures of tangles that we

consider.

Lemma 3.2.5. An incompressible and meridianally boundary incompressible surface 𝑆 spanning

an alternating link 𝐿 can be isotoped relative to a given nontrivial Menasco projection 𝑃 to obtain

a crossing band.

Lemma 3.2.6. Any spanning surface for a non-splittable, alternating link is connected.

Proof. This proof uses induction on the number of crossings a non-splittable, alternating link

contains. As the unknot contains only one link component, any spanning surface for the unknot

must be connected. Now consider a non-splittable alternating link 𝐿 which has 𝑛 crossings, and

let 𝑆 be a spanning surface for 𝐿 then 𝑆 is either incompressible and meridianally boundary

incompressible or a finite sequence of compressions take 𝑆 to an incompressible and meridianally

boundary incompressible surface 𝑆′. Choose a reduced alternating diagram of 𝐿 and put 𝑆′ into

27

Menasco form relative to 𝐿. By Lemma 3.2.5, we can isotope 𝑆′ such that 𝑆′ contains a crossing

band in at least one of the Menasco balls, 𝑀. Further, when we cut open the link along 𝑀, we

find a spanning surface 𝑆′′ for a non-splittable alternating link with fewer crossings 𝐿′. Part of the

equator of 𝑀 replaces the crossing strands and guarantees that 𝑆′′ is a spanning surface of 𝐿′. Then,

by induction, as 𝑆′′ is a spanning surface for a link of 𝑛 − 1 crossings, it is connected. Regluing in

the crossing band does not disconnect our surface, showing that 𝑆′ and 𝑆 are connected.

□

Lemma 3.2.7. Let Σ be a Conway sphere which intersects a Conway sum 𝐿 of two strongly

alternating tangles 𝑇1 and 𝑇2 in 𝑆3 such that Σ separates 𝑇1 and 𝑇2. If we let 𝑆 be a spanning

surface of 𝐿 and 𝑆 ∩ Σ contains a simple closed curve 𝛾 such that 𝛾 does not separate the two arcs

in 𝑆 ∩ Σ on Σ, then there exists an isotopy on 𝑆 which will eliminate 𝛾.

Proof. Assume there is only one closed curve 𝛾 contained in 𝑆 ∩ Σ. First we consider the case

where cutting 𝑆 along 𝛾 and gluing in disks along the resulting boundary components results in

two disconnected closed components. Notice that this implies that 𝑆 has a disconnected closed

component, contradicting that 𝑆 is a crosscap realizing surface.

Next assume that cutting along 𝛾 and gluing in disks does not result in a closed surface

component. Then cutting 𝑆 along Σ and gluing disks along the copies of 𝛾 will result in spanning

surfaces for a closure of 𝑇1 and a closure of 𝑇2, both of which are connected by Lemma 3.2.6.

Reversing this procedure everywhere but 𝛾 results in a new connected surface 𝑆′ which also spans

𝐿. But as 𝑆′ has two additional disks, 𝜒(𝑆′) = 𝜒(𝑆) + 2, showing that 𝐶 (𝑆′) < 𝐶 (𝑆) contradicting

that 𝑆 is a crosscap realizing spanning surface.

The only remaining possibility is if cutting 𝑆 along 𝛾 and gluing disks to the two resulting

boundaries, results in a single closed surface component, 𝑈. Then 𝑈 will separate 𝑆3 into two

disjoint spaces. As 𝛾 does not separate the two arcs on Σ, one side of 𝑈 must not contain any part

of 𝑆. But then we can move 𝑈 to the opposite side of Σ and re-glue it to 𝑆 along 𝛾 to find an isotopy

of 𝑆 for which 𝛾 is no longer in Σ ∩ 𝑆.

In the case that we have multiple such closed curves along Σ we do the same as above starting

28

with the innermost closed curve. The innermost closed curve in this case is the one that bounds an

empty disk on Σ. Hence showing the claim.

□

By Lemma 3.2.7, we can choose 𝑆 such that the only simple closed curves in Σ ∩ 𝑆 are those

which bound two disks each containing an arc. Next we show that 𝑆 can be chosen so that 𝑆 ∩ Σ

contains at most one such simple closed curve.

Lemma 3.2.8. There exists a spanning surface 𝑆 for 𝐿, where 𝐿 is the Conway sum of two strongly

alternating tangles, such that 𝐶 (𝑆) = 𝐶 (𝐿) and Σ ∩ 𝑆 contains at most one closed curve.

Proof. By Lemma 3.2.7 we can assume that if Σ ∩ 𝑆 contains closed curves 𝛾1, ..., 𝛾𝑛, they each

split Σ such that the two arcs lie on opposite disks. Assume we have 𝑛 > 1 such closed curves in

Σ ∩ 𝑆, and let 𝛾1 and 𝛾2 be such that 𝛾1 bounds a disk on Σ such that no other 𝛾𝑖 are in the disk and

𝛾2 bounds a disk where the only closed curve in it is 𝛾1.

We now find a spanning surface 𝑆′ such that 𝐶 (𝑆′) = 𝐶 (𝐿) and 𝑆′ ∩ Σ contains 𝑛 − 2 closed

curves. We start with 𝑆 and cut along 𝛾1 and 𝛾2 and then glue in annuli whose boundaries are a

copy of 𝛾1 and a copy of 𝛾2. Then as the Euler characteristic of an annulus is 0, this cutting and

gluing operation will result in 𝜒(𝑆) = 𝜒(𝑆′).

It remains to show that 𝑆′ will be a connected surface. As in the proof of Lemma 3.2.7 we cut

𝑆 along Σ and glue in disks along each 𝛾𝑖 except for 𝛾1 and 𝛾2 which we glue a pair of annuli.

This results in spanning surfaces 𝑆1 and 𝑆2 for closures of 𝑇1 and 𝑇2 which by Lemma 3.2.6 are

connected. If we reverse this procedure everywhere except 𝛾1 and 𝛾2 the result will be 𝑆′ and as we

only remove disks before regluing, 𝑆′ will be connected. Hence, we have found a spanning surface

𝑆′ for 𝐿 such that 𝐶 (𝑆′) = 𝐶 (𝐿) and 𝑆′ ∩ Σ contains two fewer closed curves. Hence, repeating for

all such pairs of closed curves in 𝑆 ∩ Σ we will find the claim.

□

Lemma 3.2.9. We can choose a surface 𝑆 which spans a link 𝐿 such that 𝐶 (𝑆) = 𝐶 (𝐿) and cutting

along Σ will not give us a closed surface component.

Proof. This was shown in the proof of Lemma 3.2.7.

□

29

3.2.2 Proof of the Lower Bound of Theorem 1.1.2

In this subsection we will prove the lower bound of Theorem 1.1.2, which will be restated as

Lemma 3.2.10. We start by discussing what happens when we cut our link 𝐿 along Σ. Assume that

𝑆 is a non-orientable spanning surface for 𝐿 with 𝐶 (𝑆) = 𝐶 (𝐿). The two arcs on Σ will define how

we close 𝑇1 and 𝑇2 after cutting. We let 𝐾1 and 𝐾2 be these closures. To see that 𝐾1 and 𝐾2 are

the numerator or denominator closures consider a crossing which as a vertex in the tangle graph is

adjacent to a 1-valent vertex. If the exterior regions for a tangle are the faces bounded by the box in

the graph then one of the two exterior regions adjacent to the crossing must be included in 𝑆. This

means the boundary of this region will result in the numerator or denominator closures.

We are now ready to prove the lower bound of Theorem 1.1.2.

Lemma 3.2.10. Let 𝑇1 and 𝑇2 be non-splittable, strongly alternating tangles and 𝐿 the link resulting

from the Conway sum of 𝑇1 and 𝑇2. Also, let 𝑆 be a spanning surface of 𝐿 such that 𝐶 (𝐿) = 𝐶 (𝑆).

Then:

𝐶 (𝐾1) + 𝐶 (𝐾2) − 2 ≤ 𝐶 (𝐿).

Proof. Let 𝑆 be a non-orientable spanning surface for 𝐿 such that 𝐶 (𝐿) = 𝐶 (𝑆). By Lemma 3.2.8

𝑆 can be chosen such that the intersection of 𝑆 with Σ contains at most one closed curve 𝛾 and that

both disks 𝛾 bounds contain arcs. We cut 𝑆 along Σ and if 𝛾 exists we glue a disk to each copy to

get spanning surfaces 𝑆1 and 𝑆2 for 𝐾1 and 𝐾2 respectively. By Lemma 3.2.9 we know that 𝑆1 and

𝑆2 will not have closed components and by Lemma 3.2.6 𝑆1 and 𝑆2 must be connected as 𝐾1 and

𝐾2 are alternating.

If 𝑘1 and 𝑘2 are the number of link components for 𝐾1 and 𝐾2 respectively, then by Lemma 3.2.1

𝑘1 + 𝑘2 − 𝜖 = 𝑘 𝐿 where 𝜖 = 0, 1, 2.

Next we consider how the Euler characteristics of 𝑆 and the sum of the Euler characteristics of

𝑆1 and 𝑆2 will be related. We know that Σ ∩ 𝑆 contains two arcs and at most one closed curve by

Lemma 3.2.8. Then cutting the two arcs along Σ will increase the Euler characteristic by 2. Assume

there are 𝑛 closed curves, gluing disks along the two copies after cutting will further increase the

Euler characteristic by 2𝑛. The final consideration we have to make is whether 𝑆1 and 𝑆2 are

30

non-orientable, let 𝑡 = 0, 1, 2 be the number of 𝑆𝑖 which are orientable. Notice we will have to add

𝑡 half twist bands to make sure all the 𝑆𝑖 are non-orientable decreasing the Euler characteristic by

𝑡. Now we see that 𝜒(𝑆) = 𝜒(𝑆1) + 𝜒(𝑆2) − 2 − 2𝑛 + 𝑡. Then:

𝐶 (𝑆1) + 𝐶 (𝑆2) = 4 − 𝜒(𝑆) − 2 − 2𝑛 + 𝑡 − 𝑘 𝐿 − 𝜖 = 𝐶 (𝐿) − 2𝑛 + 𝑡 − 𝜖 .

Simplifying we find

𝐶 (𝐾1) + 𝐶 (𝐾2) + 2𝑛 − 𝑡 + 𝜖 ≤ 𝐶 (𝐿).

This will be the weakest when 𝑛 = 0, 𝑡 = 2, and 𝜖 = 0. Hence we find 𝐶 (𝐾1)+𝐶 (𝐾2)−2 ≤ 𝐶 (𝐿). □

We restate Theorem 1.1.2:

Theorem 1.1.2. Let 𝑇1 and 𝑇2 be non-splittable, twist reduced, strongly alternating tangles. Let 𝐿

be the link formed by the Conway sum of 𝑇1 and 𝑇2. Let 𝐾𝑖𝑁 be the link formed by the numerator

closure of 𝑇𝑖, 𝑖 ∈ {1, 2}, similarly 𝐾𝑖𝐷 will be the link formed by the denominator closure. If we let

𝑚 = min{𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ), 𝐶 (𝐾1𝐷) + 𝐶 (𝐾2𝐷)} then,

𝐶 (𝐾1) + 𝐶 (𝐾2) − 2 ≤ 𝐶 (𝐿) ≤ 𝑚 + 2.

Theorem 1.1.2 follows directly from Lemma 3.1.4 and Lemma 3.2.10. A similar result exists

to Theorem 1.1.2 for the cross cap number of connected sums. In particular, Clark [7] showed with

a strategy similar to our own, that if 𝐾1 and 𝐾2 are knots, then

𝐶 (𝐾1) + 𝐶 (𝐾2) − 1 ≤ 𝐶 (𝐾1#𝐾2) ≤ 𝐶 (𝐾1) + 𝐶 (𝐾2).

3.3 Crosscap Number, Twist Number, and the Jones Polynomial

3.3.1 Twist Number Bounds

Now we have a relationship between the crosscap numbers of the Conway sum of two tangles

and the closures of the tangles which compose it. Unfortunately, the bounds depend upon the

tangles and which closures we take. But we can use Theorem 1.1.2 to find bounds for 𝐶 (𝐿) entirely

dependent upon 𝐿. Before proceeding with the statements, we will need a definition.

31

Definition 3.3.1. The twist number of a link diagram or a tangle diagram is the number of twist

regions a link diagram contains, where a twist region is a maximal collection of bigon regions

contained end to end. We call a link diagram twist-reduced if any simple closed curve which meets

the link diagram transversely at four points, with two points adjacent to one crossing and the other

two another crossing, bounds a possibly empty collection of bigons arranged end to end between

the two crossings.

We take a brief pause to mention that we can take the Conway sum of more than two tangles.

In particular, for tangles 𝑇1, 𝑇2, ..., 𝑇𝑛, we can glue the eastern strands of 𝑇𝑖 to the western strands

of 𝑇𝑖+1. Then we glue the eastern strands of 𝑇𝑛 to the western strands of 𝑇1. See Figure 3.2 for an

example.

Lemma 3.3.2. Let 𝑇1, 𝑇2, ..., 𝑇𝑛 be strongly alternating tangle diagrams whose Conway sum is a
link diagram 𝐷 (𝐿). Then 𝑡𝑤(𝐷 (𝐿)) = (cid:205)𝑛

𝑖=1 𝑡𝑤(𝑇𝑖), where 𝑡𝑤(𝐷 (𝐿)) is the twist number for 𝐷 (𝐿)

and 𝑡𝑤(𝑇𝑖) the twist number for the tangle diagram 𝑇𝑖.

Proof. First notice that taking the sum of tangles will not result in new twist regions. This is

because, when taking a Conway sum, crossings that shared a twist region will still share a twist

region and we introduce no new crossings. Therefore, 𝑡𝑤(𝐷 (𝐿)) ≤ (cid:205)𝑛

𝑖=1 𝑡𝑤(𝑇𝑖).

Now assume that 𝑡𝑤(𝐷 (𝐿)) < (cid:205)𝑛

𝑖=1 𝑡𝑤(𝑇𝑖). Then for some 𝑖, a twist region in 𝑇𝑖 and a twist
region in 𝑇𝑖+1 become one region in 𝐿. This implies there exists a simple closed curve 𝛾 which

transversely intersects 𝐷 (𝐿) twice in 𝑇𝑖 and twice in 𝑇𝑖+1. If we think back to 𝑇𝑖 lying in a unit

square, then 𝛾 must intersect the north and south edges or the east and west edges of the square. In

the first case, this shows that the denominator closure is not prime, and the second, the numerator

closure is not prime. But as 𝑇𝑖 is strongly alternating, this would be a contradiction, therefore

𝑡𝑤(𝐷 (𝐿)) = (cid:205)𝑛

𝑖=1 𝑡𝑤(𝑇𝑖).

□

Next we consider the relationship between the twist number of a tangle diagram and the twist

numbers of diagrams of its closures.

Lemma 3.3.3. Let 𝑇 be a strongly alternating tangle diagram, and let 𝐷 (𝐾) be the link diagram

32

which comes from the numerator or denominator closure. Then:

𝑡𝑤(𝑇) − 2 ≤ 𝑡𝑤(𝐷 (𝐾)) ≤ 𝑡𝑤(𝑇).

Proof. The upperbound is true as we are not adding crossings when closing a tangle, and hence

cannot create new twist regions.

The lower bound stems from the fact that, when we choose a closure for 𝑇 we create two new

potential bigons. If either region is a bigon, then it joins two twist regions. If both regions are

bigons, the twist number is reduced by 2. In Figure 3.6 we see an example of a tangle where the

lower bound is sharp for both closures.

□

Figure 3.6 Left: A strongly alternating tangle with 5 twist regions. Right: The numerator closure
with 3 twist regions. The denominator also results in 3 twist regions, showing the sharpness of the
lower bound.

We will also need Theorem 3.8 from [24] which we state here. This Theorem allows us to

relate the cross cap numbers of the closures of strongly alternating tangles to twist numbers of their

diagrams.

Theorem 3.3.4. Let 𝐿 ⊂ 𝑆3 be a link of 𝑘 𝐿 components with a prime, twist-reduced, alternating

diagram 𝐷 (𝐿). Suppose that 𝐷 (𝐿) has 𝑡𝑤(𝐷 (𝐿)) ≥ 2 twist regions. Let 𝐶 (𝐿) denote the crosscap

number of 𝐿. We have

(cid:24)𝑡𝑤(𝐷 (𝐿))
3

(cid:25)

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 𝑡𝑤(𝐷 (𝐿)) + 2 − 𝑘 𝐿

Furthermore, both bounds are sharp.

33

Now that we have that the twist number is additive for strongly alternating tangles by

Lemma 3.3.2 and can relate 𝐶 (𝐾𝑖) and 𝑡𝑤(𝑇𝑖) by Lemma 3.3.3 and Theorem 3.3.4, we are ready to

state Theorem 3.3.5

Theorem 3.3.5. Let 𝑇1 and 𝑇2 be diagrams of non-splittable, strongly alternating, twist-reduced

tangles whose Conway sum is a link diagram 𝐷 (𝐿). Let 𝐶 (𝐿) be the crosscap number of 𝐿,

𝑡𝑤(𝐷 (𝐿)) be the twist number of 𝐷 (𝐿) and 𝑘 𝐿 be the number of link components in 𝐿. Then

(cid:24)𝑡𝑤(𝐷 (𝐿))
3

(cid:25)

− 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 𝑡𝑤(𝐷 (𝐿)) + 4 − 𝑘 𝐿.

Proof. We start with a lower bound in the proof of Lemma 3.2.10 with ambiguity on 𝜖 where

𝑘1 + 𝑘2 − 𝜖 = 𝑘 𝐿. Then 𝐶 (𝐾1) + 𝐶 (𝐾2) − 2 + 𝜖 ≤ 𝐶 (𝐿). Let 𝐷 (𝐾1) and 𝐷 (𝐾2) be the diagrams of

𝐾1 and 𝐾2 that arise from cutting 𝐿 as in Theorem 1.1.2. Notice that 𝑡𝑤(𝐷 (𝐾𝑖)) ≥ 2 for 𝑖 = 1, 2 as

𝑇𝑖 is strongly alternating, and tangle diagrams with twist number 1 will have a non prime closure.
Then by Lemma 3.3.4 we find for 𝑖 ∈ {1, 2} that (cid:108) 𝑡𝑤(𝐷 (𝐾𝑖))

+ 2 − 𝑘𝑖 ≤ 𝐶 (𝐾𝑖) where 𝐾𝑖 has 𝑘𝑖 link

(cid:109)

3

components. So:

(cid:24)𝑡𝑤(𝐷 (𝐾1))
3

(cid:25)

(cid:24)𝑡𝑤(𝐷 (𝐾2))
3

(cid:25)

+

+ 2 + 𝜖 − 𝑘1 − 𝑘2 ≤ 𝐶 (𝐿).

By Lemma 3.3.3 𝑡𝑤(𝑇𝑖) − 2 ≤ 𝑡𝑤(𝐷 (𝐾𝑖)) and from Lemma 3.3.2 we know 𝑡𝑤(𝑇1) + 𝑡𝑤(𝑇2) =

𝑡𝑤(𝐷 (𝐿)), combining the two lemmas shows

(cid:24)𝑡𝑤(𝐷 (𝐿))
3

(cid:25)

− 2 ≤

≤

(cid:25)

(cid:24)𝑡𝑤(𝑇1) − 2
3
(cid:24)𝑡𝑤(𝐷 (𝐾1))
3

(cid:25)

+

+

(cid:25)

(cid:24)𝑡𝑤(𝑇2) − 2
3
(cid:24)𝑡𝑤(𝐷 (𝐾2))
3

(cid:25)

.

(3.1)

(3.2)

Finally substituting in 𝑘1 + 𝑘2 = 𝑘𝑙 + 𝜖, we find:

(cid:24)𝑡𝑤(𝐷 (𝐿))
3

(cid:25)

− 𝑘 𝐿 ≤ 𝐶 (𝐿).

Now we consider the upper bound. Similar to the lower bound we start with a step from

Lemma 3.1.4, 𝐶 (𝐿) ≤ min{𝐶 (𝐾1𝑁 ) + 𝐶 (𝐾2𝑁 ) + 𝜖, 𝐶 (𝐾1𝐷) + 𝐶 (𝐾2𝐷) + 𝜖 }. By Lemma 3.3.3 we

see that 𝑡𝑤(𝐷 (𝐾𝑖𝑁 )) ≤ 𝑡𝑤(𝑇𝑖) and 𝑡𝑤(𝐷 (𝐾𝑖𝐷)) ≤ 𝑡𝑤(𝑇𝑖) and then by Theorem 3.3.4,

𝐶 (𝐿) ≤ 𝑡𝑤(𝑇1) + 𝑡𝑤(𝑇2) + 4 + 𝜖 − 𝑘1 − 𝑘2.

34

Then substituting for 𝑘1 + 𝑘2 = 𝑘 𝐿 + 𝜖 and considering Lemma 3.3.2,

𝐶 (𝐿) ≤ 𝑡𝑤(𝐷 (𝐿)) + 4 − 𝑘 𝐿.

Hence, showing the claim.

□

3.3.2

Jones Polynomial Bounds

From here we work to find bounds in terms of 𝑇𝐿, but first we have to generalize Theorem 1.6

in [19] which will allow us to relate the twist number of the diagram of a link 𝐿 to 𝑇𝐿. Theorem

1.6 from [19] only considers knots but we need a similar result for links. We start by recalling

some necessary vocabulary from Chapter 2.

Definition 3.3.6. Recall that if 𝐷 (𝐿) is a diagram of the link 𝐿 then the A resolution and B resolution

for a crossing are as shown in Figure 2.1. Then a Kauffman state for 𝐷 (𝐿) is a choice of resolution

for all crossings in the diagram.

Next, we recall the construction of a all A (resp. all B) state graph G𝐴 (𝐷 (𝐿)) (resp. G𝐵 (𝐷 (𝐿)))

by adding edges where we performed resolutions and then contracting the simple closed curves to

vertices. Let 𝑒 𝐴 (resp. 𝑒𝐵) be the number of edges in G𝐴 (𝐷 (𝐿)) (resp. G𝐵 (𝐷 (𝐿))). Further, if we

identify all parallel edges we find the reduced state graph G′

𝐴 (𝐷 (𝐿)) (resp. G′

𝐵 (𝐷 (𝐿))). Let 𝑒′
𝐴

(resp. 𝑒′

𝐵) be the number of edges in G′

𝐴 (𝐷 (𝐿)) (resp. G′

𝐵 (𝐷 (𝐿))).

Now we restate the definition of link adequacy.

Definition 3.3.7. We call a link diagram A-adequate (resp. B-adequate) if the A state (resp. B

state) graph of the diagram has no one edge loops. A link diagram is called adequate if it is both

A-adequate and B-adequate, and a link is adequate if it has a diagram which is adequate.

Here we recall some terminology from [19].

Definition 3.3.8. Let 𝐷 (𝐿) be the link diagram obtained by taking the Conway sum of strongly

alternating tangles 𝑇1, ..., 𝑇𝑛. Let ℓin(𝐷 (𝐿)) denote the loss of edges in G𝐴 (𝐷 (𝐿)) and G𝐵 (𝐷 (𝐿))

as we pass from 𝑒 𝐴 + 𝑒𝐵 to 𝑒′

𝐵 which come from equivalent crossings in the same tangle 𝑇𝑖.
Then let ℓext(𝐷 (𝐿)) be the number of edges we lose from identification when we take the Conway

𝐴 + 𝑒′

sum. It follows that ℓin(𝐷 (𝐿)) + ℓext(𝐷 (𝐿)) = 𝑒 𝐴 + 𝑒𝐵 − 𝑒′

𝐵.
𝐴 − 𝑒′

35

For an alternating tangle diagram 𝑇, notice that the vertices of G𝐴 (𝑇) and G𝐵 (𝑇) are in 1-1

correspondence with the regions of 𝑇. Note that for the state graph of a tangle, if we consider the

tangle lying within a disk, we have four exterior regions bounded by the disk. This means our state

graphs have interior vertices those whose region lie entirely within the interior of the disk and two

exterior vertices with corresponding region, with sides on the boundary of the disk.

For a tangle 𝑇𝑖, a bridge of G𝐴 (𝑇𝑖) (respectively G𝐵 (𝑇𝑖)) is a subgraph consisting of an interior

vertex 𝑣, and edges 𝑒′, 𝑒′′ which connect 𝑣 to the exterior vertices 𝑣′ and 𝑣′′. We call the bridge

inadmissible if the exterior vertices become identified in G𝐴 (𝐷 (𝐿)) (respectively G𝐵 (𝐷 (𝐿))), see

Figure 3.7 for reference.

Figure 3.7 Top: Two strongly alternating tangles. Middle: The all A-resolution diagrams for the
tangles. Bottom Left: The all A-state diagram for the Conway sum of two of the left tangles which
has all admissible bridges. Bottom Right: The all A-state diagram for the Conway sum of one of
both tangles, here the bridges are all inadmissible.

Now we find an upper bound for ℓext(𝐷 (𝐿)) in terms of the twist number. This work will largely

36

follow the proof of Lemma 5.4 in [19].

Lemma 3.3.9. Let 𝑇1 and 𝑇2 be strongly alternating tangles whose Conway sum is a link diagram

𝐷 (𝐿). Let 𝑘 𝐿 be number of link components in 𝐿. Then:

ℓext(𝐷 (𝐿)) ≤

𝑡𝑤(𝐷 (𝐿))
2

+ 𝑘 𝐿 + 4

Proof. For 𝑇 ∈ {𝑇1, 𝑇2} let 𝑏 𝐴 (𝑇), 𝑏𝐵 (𝑇) be the number of bridges in G𝐴 (𝑇) and G𝐵 (𝑇) re-

spectively. Then the contribution of 𝑇 to ℓext will be at most 𝑏 𝐴 (𝑇) + 𝑏𝐵 (𝑇). Any other edge

identification from moving to the reduced graph will still be counted by ℓint.

If 𝑏 is a bridge there are two possibilities:

i. The edges 𝑒′, 𝑒′′ do not come from the resolutions of a single twist region.

ii. The edges 𝑒′, 𝑒′′ come from the resolutions of a single twist region.

Notice that for type ii bridges the two crossings which result in the edges are the only two in their

respective twist region. Otherwise the two edges will not be adjacent to the exterior vertices or this

would no longer constitute a twist region.

For type i bridges notice that when we pass from G𝐴 (𝑇) and G𝐵 (𝑇) to G′

𝐵 (𝑇) the
contributions to ℓext is half the number of twist regions involved in such bridges. Unlike in [19]

𝐴 (𝑇) and G′

we can have more than one type ii bridge, as each additional type ii bridges creates a new link

component.

Case 1: Suppose that 𝑏 𝐴 (𝑇) ≥ 3 or 𝑏𝐵 (𝑇) ≥ 3. Without loss of generality let 𝑏 𝐴 (𝑇) ≥ 3,

then 𝑏𝐵 (𝑇) = 0. If 𝑏𝐵 (𝑇) were not zero then the 𝐵 state bridge would cross the 𝐴 state bridges,

implying two internal vertices which is not a bridge.

There can be any number of type ii bridges, but we notice each bridge beyond the first will add

a new link component. If we have only type ii bridges 𝑏 𝐴 (𝑇) ≤ 𝑘𝑇 where 𝑘𝑇 is the number of
tangle components. On the other hand if we only have type i bridges 𝑏 𝐴 (𝑇) ≤ 𝑡𝑤(𝑇)
mix of bridges we find that 𝑏 𝐴 (𝑇) + 𝑏𝐵 (𝑇) ≤ 𝑡𝑤(𝑇)

2 + 𝑘𝑇 .
Case 2: In this case we will consider 𝑏 𝐴 (𝑇) = 𝑏𝐵 (𝑇) = 2. Then 𝑘 must be at least 2, as the

. Then for any

2

37

bridges in G𝐴 (𝑇) and G𝐵 (𝑇) will create a square resulting in a tangle second component. Also

notice that as G𝐴 (𝑇) has two bridges there are at least two twist regions in 𝑇. Then 𝑏 𝐴 (𝑇) + 𝑏𝐵 (𝑇) ≤

𝑡𝑤(𝑇)

2 + 𝑘𝑇 + 1.
Case 3: Either 𝑏 𝐴 (𝑇) ≤ 2 and 𝑏𝐵 (𝑇) ≤ 1 or 𝑏 𝐴 (𝑇) ≤ 1 and 𝑏𝐵 (𝑇) ≤ 2. Without loss

of generality consider the first possibility. Then 𝑏 𝐴 (𝑇) + 𝑏𝐵 (𝑇) is at most three.

If it’s less

than three we see that 𝑘𝑇 + 1 ≥ 2 so we need only consider when they sum to three. But as

with the previous case we will have at least two twist regions as G𝐴 (𝑇) has two bridges. Thus,
𝑏 𝐴 (𝑇) + 𝑏𝐵 (𝑇) ≤ 𝑡𝑤(𝑇)

2 + 𝑘𝑇 + 1.

Then by Lemma 3.3.2 we know that the twist number is additive over Conway sums. By

Lemma 3.2.1 𝑘1 + 𝑘2 ≤ 𝑘 𝐿 + 2. Then we find the following bound;

ℓext ≤

2
∑︁

𝑖=1

𝑏 𝐴 (𝑇𝑖) + 𝑏𝐵 (𝑇𝑖) ≤

𝑡𝑤(𝐷 (𝐿))
2

+ 𝑘 𝐿 + 4.

□

Now we have the tools necessary to prove the main lemma needed to find bounds for 𝐶 (𝐿) in

terms of 𝑇𝐿.

Lemma 3.3.10. Let 𝑇1, ..., 𝑇𝑛 be strongly alternating tangles whose Conway sum is a link diagram

𝐷 (𝐿) for a link 𝐿. Then letting 𝛽𝐿 and 𝛽′

𝐿 be the second and second-to-last coefficients of the

Jones polynomial of 𝐿, 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 |, and 𝑘 𝐿 the number of link components, we have

tw(𝐷 (𝐿))
2

− 𝑘 𝐿 − 2 ≤ 𝑇𝐿 ≤ 2tw(𝐷 (𝐿)).

Proof. We start by noting that we can mutate the link 𝐿 in such a way that it either is alternating or

the sum of 𝑇 and 𝑇 ′ where 𝑇 is a positive strongly alternating tangle and 𝑇 ′ is a negative strongly

alternating tangle, without changing its Jones polynomial [32]. Where the positive and negative

refer to whether the northwest strand originates from and overcrossing or an undercrossing. In the

former case we have a stronger result by Dasbach and Lin [9] that 𝑇𝐿 = 𝑡𝑤(𝐿).

We will assume that 𝐿 is not alternating. Then work by Lickorish and Thistlewaite [33] shows

that 𝐷 (𝐿) is adequate. Further, by propositions 1 and 5 of [33] we have 𝑣 𝐴 + 𝑣 𝐵 = 𝑐 where 𝑣 𝐴 is the

38

number of vertices in G𝐴 (𝐷 (𝐿)), 𝑣 𝐵 the same in G𝐵 (𝐷 (𝐿)) and 𝑐 the number of crossings in 𝐷 (𝐿).

Every edge we lose when passing from G𝐴 (𝐷 (𝐿)) and G𝐵 (𝐷 (𝐿)) to G′

𝐵 (𝐷 (𝐿))
comes from edges beyond the first in a twist region or a type (i) inadmissible bridge. By Lemma

𝐴 (𝐷 (𝐿)) and G′

5.2 in [19] we see that the number of edges lost from twist regions is 𝑐 − tw(𝐷 (𝐿)). On the other

hand type (i) inadmissible bridges for strongly alternating tangles will be precisely those which we

lose from identification when taking the Conway sum, ℓext(𝐷 (𝐿)). Then the first part of the move

from (3.4) to (3.5) below will come from the equality 𝑒′

𝐴 + 𝑒′

𝐵 − 𝑒 𝐴 − 𝑒𝐵 = −(𝑐 − 𝑡𝑤(𝐷 (𝐿)) + ℓ𝑒𝑥𝑡).

Work by Stoimenow shows that for an adequate link diagram, (see [9] for a proof)

𝑇𝐿 = 𝑒′

𝐴 + 𝑒′

𝐵 − 𝑣 𝐴 − 𝑣 𝐵 + 2

= (𝑒′

𝐴 + 𝑒′

𝐵 − 𝑒 𝐴 − 𝑒𝐵) + 𝑒 𝐴 + (𝑒𝐵 − 𝑣 𝐴 − 𝑣 𝐵) + 2

= −(𝑐 − 𝑡𝑤(𝐷 (𝐿)) + ℓ𝑒𝑥𝑡) + 𝑐 + (𝑐 − 𝑣 𝐴 − 𝑣 𝐵) + 2

≥ 𝑡𝑤(𝐷 (𝐿)) − ℓ𝑒𝑥𝑡 + 2

≥ 𝑡𝑤(𝐷 (𝐿)) −

𝑡𝑤(𝐷 (𝐿))
2

− 𝑘 𝐿 − 4 + 2 =

𝑡𝑤(𝐷 (𝐿))
2

− 𝑘 𝐿 − 2.

The upper bound on 𝑇𝐿 was shown by Futer, Kalfagianni and Purcell in [15].

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

□

Theorem 1.1.1. Let 𝑇1 and 𝑇2 be non-splittable, twist reduced, strongly alternating tangles whose

Conway sum is a link 𝐿. If 𝐶 (𝐿) is the crosscap number of 𝐿, 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 | and 𝑘 𝐿 is the number

of link components in 𝐿 we find that,
(cid:25)

(cid:24)𝑇𝐿
6

− 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑘 𝐿 + 8.

Proof. This follows immediately from Theorem 3.3.5 and Lemma 3.3.10.

□

The following corollary is an immediate result from setting further restrictions on the twist

numbers of our tangles.

Corollary 3.3.11. Let 𝑇1 and 𝑇2 be twist reduced, non-splittable, strongly alternating tangles whose

Conway sum is a link 𝐿. Assume that 𝑡𝑤(𝑇𝑖) = 𝑡𝑤(𝐾𝑖 𝑁) = 𝑡𝑤(𝐾𝑖 𝐷). If 𝐶 (𝐿) is the crosscap

number of 𝐿, 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 | and 𝑘 𝐿 is the number of link components in 𝐿, we have,
(cid:24)𝑇𝐿
6

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑘 𝐿 + 8.

(cid:25)

39

3.4 Generalizing to Larger Conway Sums of Tangles

Our goal in this section is to generalize Theorem 1.1.1 to Conway sums of more than two

tangles. A Conway sum of more than 2 tangles is a closure where we connect diagrams of the

tangles 𝑇1, 𝑇2, ..., 𝑇𝑙 linearly west to east shown in Figure 3.2. As with the case of the sum of two

tangles, if we let 𝐿 be our Conway sum and 𝑆 a spanning surface, cutting 𝑆 along a Conway sphere

intersecting 𝑇𝑖 will result in a spanning surface for either 𝐾𝑖𝑁 or 𝐾𝑖𝐷, dictating the closure for the

tangle. When we cut 𝐿, we position 𝑙 Conway spheres such that Σ𝑖 intersects 𝐿 at the directional

strands of 𝑇𝑖. We note that 𝑆3\ (cid:208)𝑖 Σ𝑖 will not be a sphere, but we are concerned with the surfaces

within the interior of each Σ𝑖. 𝑆/∪𝑖Σ𝑖 will be a collection of bands and tubes which we consider in

the Euler characteristic change. This section will have similar results to the previous sections but

with a factor for the number of tangles. We start with the following lemma which is a generalization

of Lemma 3.2.1.

Lemma 3.4.1. Let 𝐿 be the Conway sum of the tangles 𝑇1, 𝑇2, ..., 𝑇𝑙, and 𝐾1, 𝐾2, ..., 𝐾𝑙 are closures

of the tangles. Then if 𝑘 𝐿 is the number of link components for 𝐿, and 𝑘1, 𝑘2, ..., 𝑘𝑙 the number of
link components for each link respectively then 𝑘 𝐿 = (cid:205)𝑙

𝑖=1 𝑘𝑖 − 𝑙 + 𝜖 for 𝜖 = 0, 1, 2.

Theorem 3.4.2. Let 𝑇1, 𝑇2, ..., 𝑇𝑙 be non-splittable, strongly alternating tangles, and let 𝐿 be the

Conway sum that results from the 𝑙 tangles. If 𝐾𝑖 is the closure of 𝑇𝑖 resulting from cutting the

crosscap realizing spanning surface for 𝐿 for all 𝑖 ∈ {1, 2, ..., 𝑙} and 𝐾𝑖𝑁 is the numerator closure

of 𝑇𝑖 and 𝐾𝑖𝐷 the denominator closure, then we have:

𝑙
∑︁

𝐶 (𝐾𝑖) − 𝑙 ≤ 𝐶 (𝐿) ≤ min{

𝑙
∑︁

𝐶 (𝐾𝑖𝑁 ) + 𝑙,

𝑙
∑︁

𝐶 (𝐾𝑖𝐷) + 2}.

𝑖=1
Proof. This proof will largely follow the work we did in Lemma 3.2.10 and Lemma 3.1.4. We will

𝑖=1

𝑖=1

start by considering the upper bound.

First we consider the case where we have 𝐾𝑖𝑁 for all 𝑖 ∈ {1, 2, ...., 𝑙}, and spanning surfaces 𝑆𝑖

for each 𝐾𝑖𝑁 such that 𝐶 (𝑆𝑖) = 𝐶 (𝐾𝑖𝑁 ). Unlike in Lemma 3.1.4 the NW and NE strands connect to

different tangles and we will find the same for the SW and SE strands. Then the spanning surface

resulting from the Conway sum will have northern and southern disks attached to each of the 𝑆𝑖 by

a band as seen in figure 3.8. Let this resulting surface be 𝑆.

40

Figure 3.8 We see here that a surface would have to fill the shaded areas to connect the numerator
closures of the tangles since the western and eastern boundaries of the 𝑆𝑖 connect to separate
tangles.

By this construction 𝜒(𝑆) = (cid:205)𝑙

𝑖=1 𝜒(𝑆𝑖) − 2𝑙 + 2 where the 2𝑙 comes from the bands connecting
each surface to the disks, and the 2 from the disks themselves. Then by Lemma 3.4.1 we see that:

𝑘 𝐿 = (cid:205)𝑙

𝑖=1 𝑘𝑖𝑁 − 𝑙 + 𝜖. Then
𝑙
∑︁

𝐶 (𝑆) = 2 − (

𝜒(𝑆𝑖) − 2𝑙 + 2) − (

𝑙
∑︁

𝑘𝑖𝑁 − 𝑙 + 𝜖) =

𝑖=1
Then we see the weakest upperbound is when 𝜖 = 0, so

𝑖=1

𝐶 (𝐿) ≤ 𝐶 (𝑆) =

𝑙
∑︁

𝑖=1

𝐶 (𝐾𝑖𝑁 ) + 𝑙.

𝑙
∑︁

𝑖=1

𝐶 (𝐾𝑖𝑁 ) + 𝑙 − 𝜖 .

Meanwhile the all denominator closure case will be similar to when 𝑙 = 2.

In particular

𝜒(𝑆) = (cid:205)𝑙

Lemma 3.4.1 and similar computations to the numerator closure case we find 𝐶 (𝐿) ≤ (cid:205)𝑙

𝑖=1 𝜒(𝑆𝑖) − 𝑙 as we add bands to connect each of the 𝑆𝑖 to their neighboring surfaces. By
𝑖=1 𝐶 (𝐾𝑖𝐷)+
2. We have a 2 instead of an 𝑙 as we added half the number of bands in constructing 𝑆. Then we

take the minimum of the denominator and numerator bounds to find an upperbound for 𝐶 (𝐿).

Now we consider the lower-bound. Let 𝑆 be a spanning surface for 𝐿 such that 𝐶 (𝑆) = 𝐶 (𝐿).

Similar to Lemma 3.2.10 we will be considering the surfaces 𝑆1, 𝑆2, ..., 𝑆𝑙 that result from cutting

along the Conway spheres Σ𝑖. By a similar argument to the one for Lemma 3.2.8, 𝑆 can be

chosen so that Σ𝑖 ∩ 𝑆 contains at most one closed curve for all 𝑖. Notice that if any of the steps in

Lemma 3.2.8 were to disconnect the surface outside Σ𝑖, then for some other Σ 𝑗 , 𝑖 ≠ 𝑗, 𝑇𝑗 would

span a disconnected surface which is a contradiction to Lemma 3.2.6. Then when we cut along the

Conway spheres we see that we are at most cutting along two arcs and a closed curve.

41

We know from Lemma 3.4.1 that 𝑘 𝐿 = (cid:205)𝑙

𝑖=1 𝑘𝑖 − 𝑙 + 𝜖 for 𝜖 = 0, 1, 2. If we have closed curves
along a Σ𝑖, when we cut we will have to add disks to both resulting boundary components which

increases the Euler charactersitic by 2. For any surface resulting from cutting that is orientable we

will have to add in a half twist band to make it non-orientable. Each such half twist band reduces

the Euler characteristic by 1. Then (cid:205)𝑙

𝑖=1 𝜒(𝑆𝑖) = 𝜒(𝑆) + 𝑡 + 𝑐 − 𝑏 where 𝑡 = 𝑙 or 𝑡 = 2𝑙 − 2 depending
on if the 𝐾𝑖 are the denominator or numerator closures, 𝑐 the number of closed curves on the Σ𝑖

which can be as large as 𝑙 and 𝑏 the number of twist bands added to make the 𝑆𝑖 non-orientable

which also has maximum 𝑙. The value of 𝑡 arises from the bands which sit in 𝑆3\ (cid:208)𝑖 Σ𝑖.

Now we see that

𝑙
∑︁

𝑖=1

(𝐶 (𝑆𝑖)) = 2𝑙 −

𝑙
∑︁

𝑖=1

𝜒(𝑆𝑖) −

𝑙
∑︁

𝑖=1

𝑘𝑖 = 2𝑙 − 𝜒(𝑆) − 𝑡 − 𝑐 + 𝑏 − 𝑘 𝐿 − 𝑙 + 𝜖 .

Notice that the weakest upperbound for (cid:205)𝑙

𝑖=1(𝐶 (𝑆𝑖)) is when 𝑡 and 𝑐 are minimal and 𝑏 and 𝜖 are
maximal. So this will be when we do not have any closed curves and each of the resulting 𝑆𝑖 need

half twist bands to make them non orientable. So:

𝑙
∑︁

𝑖=1

(𝐶 (𝑆𝑖)) = 2𝑙 − 𝜒(𝑆) − 𝑙 + 𝑙 − 𝑘 𝐿 − 𝑙 + 2 = 𝐶 (𝐿) + 𝑙.

Then moving the 𝑙 to the other side we see that (cid:205)𝑙

𝑖=1 𝐶 (𝐾𝑖) − 𝑙 ≤ 𝐶 (𝐿), showing the claim.

□

As in section 3 we will now use Theorem 3.3.4 to find bounds for 𝐶 (𝐿) in terms of 𝑡𝑤(𝐷 (𝐿))

where 𝑡𝑤(𝐷 (𝐿)) is the twist number for a diagram of 𝐿.

Theorem 3.4.3. Let 𝑇1, 𝑇2, ..., 𝑇𝑙 be non-splittable, twist reduced, strongly alternating tangles and

let 𝐷 (𝐿) be the link diagram for the link 𝐿 which results from taking the Conway Sum of the tangles.

Let 𝑡𝑤(𝐷 (𝐿)) denote the twist number of 𝐷 (𝐿) and 𝐶 (𝐿) the crosscap number for 𝐿, then

(cid:24)𝑡𝑤(𝐷 (𝐿))
3

(cid:25)

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 𝑡𝑤(𝐷 (𝐿)) + 𝑙 + 2 − 𝑘 𝐿.

Proof. We start with the bounds from Theorem 3.4.2. From here we use Lemma 3.3.4 and

Lemma 3.3.3 to get bounds on 𝐶 (𝐿) with respect to the twist numbers of diagrams of 𝑇1, ..., 𝑇𝑙.

42

Combining these two statements we find

𝑙
∑︁

𝑖=1

(cid:24)𝑡𝑤(𝑇𝑖) − 2
3

(cid:25)

+ 𝑙 −

𝑙
∑︁

𝑖=1

𝑘𝑖 ≤ 𝐶 (𝐿) ≤

𝑙
∑︁

𝑖=1

𝑡𝑤(𝑇𝑖) + 2𝑙 + 2 −

𝑙
∑︁

𝑖=1

𝑘𝑖,

where 𝑘𝑖 is the number of link components for each 𝐾𝑖.

We know that the twist number of strongly alternating tangles is additive over a Conway sum

by Lemma 3.3.2. So the only detail left to consider is the relationship between 𝑘 𝐿 and (cid:205)𝑙

𝑖=1 𝑘𝑖. By

Lemma 3.4.1 and a similar argument to that in Theorem 3.3.5 we find the claim:

(cid:24)𝑡𝑤(𝐷 (𝐿)) − 2𝑙
3

(cid:25)

+ 2 − 𝑘 𝐿 ≤ 𝑐(𝐿) ≤ 𝑡𝑤(𝐷 (𝐿)) + 𝑙 + 2 − 𝑘 𝐿.

□

The final piece of our puzzle is to find bounds in terms of 𝑇𝐿. To do this we use Lemma 3.3.10

and Theorem 3.4.3 and the result follows.

Theorem 1.1.3. Let 𝑇1, 𝑇2, ..., 𝑇𝑙 be non-splittable, twist reduced, strongly alternating tangles and

let 𝐿 be the link which results from taking the Conway Sum. Then let 𝐶 (𝐿) be the crosscap number

and 𝑘 𝐿 the number of link components in 𝐿, then

(cid:25)

(cid:24)𝑇𝐿 − 2𝑙
6

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑙 + 6 + 𝑘 𝐿.

With an additional constraint on our tangles we find the following corollary:

Corollary 3.4.4. Let 𝑇1, 𝑇2, ..., 𝑇𝑙 be non-splittable, twist reduced, strongly alternating tangles such

that 𝑡𝑤(𝑇𝑖) = 𝑡𝑤(𝐷 (𝐾𝑖𝑁 )) = 𝑡𝑤(𝐷 (𝐾𝑖𝐷)) for all 𝑖 ∈ {1, ..., 𝑙}. Let 𝐿 be the link which results from

taking the Conway Sum, 𝐶 (𝐿) the crosscap number, and 𝑘 𝐿 the number of link components in 𝐿,

then

(cid:25)

(cid:24)𝑇𝐿
6

+ 2 − 𝑘 𝐿 ≤ 𝐶 (𝐿) ≤ 2𝑇𝐿 + 𝑙 + 6 + 𝑘 𝐿.

3.5 Families where 𝑇𝐿 and the Crosscap Number are Independent

We begin by recalling the following theorem from [24] which gives linear bounds for the

crosscap number of an alternating link in terms of 𝑇𝐿, where 𝑇𝐿 = |𝛽𝐿 | + |𝛽′

𝐿 | and 𝛽𝐿 and 𝛽′

𝐿 are

the second and second to last coefficients of the Jones polynomial of 𝐿 respectively.

43

Theorem 3.5.1. Let 𝐿 be a non-split, prime alternating link with 𝑘-components and with crosscap

number 𝐶 (𝐿). Suppose that K is not a (2, 𝑝) torus link. We have

(cid:25)

(cid:24)𝑇𝐿
3

+ 2 − 𝑘 ≤ 𝐶 (𝐿) ≤ 𝑇𝐿 + 2 − 𝑘

where 𝑇𝐿 is as above. Furthermore, both bounds are sharp.

In the previous sections we showed Theorem 3.5.1 generalizes to Conway sums of strongly

alternating tangles. In this section we will show that Theorem 3.5.1 does not generalize to arbitrary

knots.

Theorem 1.1.4. We have the following;

(a) There exists a family of links for which 𝑇𝐿 ≤ 2, but 𝐶 (𝐿) is arbitrarily large.

(b) There exists a family of links for which 𝐶 (𝐿) ≤ 3, but 𝑇𝐿 is arbitrarily large.

3.5.1 Part (a) of theorem 1.1.4

In this section we will consider the following family of torus knots; 𝑇 ( 𝑝, 𝑞), where 𝑞 = 𝑗 and

𝑝 = 2 + 2 𝑗 𝑘 for odd 𝑗 > 1 and all natural numbers 𝑘. This family will allow us to prove part(a) of

Theorem 1.1.4. We start with the following definition from Teragaito [45].

Definition 3.5.2. We define the value 𝑁 ( 𝑝, 𝑞) from [45] for fractions 𝑝
to begin write 𝑝

𝑞 as a continued fraction,

𝑞 , where 𝑝 and 𝑞 are coprime,

𝑝
𝑞

= [𝑎0, 𝑎1, 𝑎2, ..., 𝑎𝑛] = 𝑎0 +

1

𝑎1 +

1
𝑎2+ 1
1
·+
𝑎𝑛

,

where the 𝑎𝑖 are integers, 𝑎0 ≥ 0, 𝑎𝑖 > 0 for 1 ≤ 𝑖 ≤ 𝑛, and 𝑎𝑛 > 1. A continued fraction of

this form is unique (cf [21]). Now we recursively define 𝑏𝑖 as follows:

𝑏0 = 𝑎0

𝑎𝑖

if 𝑏𝑖−1 ≠ 𝑎𝑖−1 or if

0

if 𝑏𝑖−1 = 𝑎𝑖−1 and

𝑏𝑖 =






44

𝑖−1
(cid:205)
𝑗=0
𝑖−1
(cid:205)
𝑗=0

𝑏 𝑗 is odd,

𝑏 𝑗 is even.

Then, 𝑁 ( 𝑝, 𝑞) = 1
2

(cid:205)𝑛

𝑖=1 𝑏𝑖. We say a torus knot 𝐾 is even if the product of 𝑝 and 𝑞 is even and

we say 𝐾 is odd otherwise. Using these definitions we can state Theorem 1.1 from [45].

Theorem 3.5.3. Let 𝐾 be the non-trivial torus knot of type ( 𝑝, 𝑞), where 𝑝, 𝑞 > 0 and let 𝐹 be a

non-orientable spanning surface of 𝐾 with 𝐶 (𝐹) = 𝐶 (𝐾).

(1) If 𝐾 is even, then 𝐶 (𝐾) = 𝑁 ( 𝑝, 𝑞) and the boundary slope of 𝐹 is 𝑝𝑞.

(2) If 𝐾 is odd, then 𝐶 (𝐾) = 𝑁 ( 𝑝𝑞 − 1, 𝑝2) (resp. 𝑁 ( 𝑝𝑞 + 1, 𝑝2)) and the boundary slope of 𝐹

is 𝑝𝑞 − 1 (resp. 𝑝𝑞 + 1) if 𝑥𝑞 ≡ −1 (mod 𝑝) has an even (resp. odd) solution 𝑥 satisfying

0 < 𝑥 < 𝑝.

We take advantage of (1) from Theorem 3.5.3 to both construct our family of torus link and

prove Proposition 3.5.5 below. We also need the following lemma, which gives an explicit formula

for the Jones polynomial of a torus knot originally given in Proposition 11.9 of [23], which allows

us to calculate 𝑇𝐿 for torus knots.

Lemma 3.5.4. The Jones polynomial for a torus knot 𝑇 ( 𝑝, 𝑞) is given by;

𝑉 (𝑇 ( 𝑝, 𝑞)) = 𝑡 ( 𝑝−1)(𝑞−1)/2 1 − 𝑡 𝑝+1 − 𝑡𝑞+1 + 𝑡 𝑝+𝑞

1 − 𝑡2

.

Proposition 3.5.5. Let 𝐿 = 𝑇 ( 𝑝, 𝑞) be the family of torus knots where 𝑞 > 1 is odd and 𝑝 = 2 + 2𝑞𝑘

for 𝑘 ∈ N, then 𝑇𝐿 ≤ 2 but 𝐶 (𝑇 ( 𝑝, 𝑞)) can be made arbitrarily large.

Proof. Let 𝑞 > 1 be odd, and 𝑝 = 2 + 2𝑞𝑘 where 𝑘 is a natural number. To show that for all such

torus knots, 𝐿 = 𝑇 ( 𝑝, 𝑞), 𝐶 (𝐿) does not have a universal upper bound with respect to 𝑇𝐿, we will

show that as 𝑘 goes to ∞, 𝐶 (𝐿) also goes to ∞, but 𝑇𝐿 ≤ 2. We start by computing the crosscap

number of 𝑇 ( 𝑝, 𝑞) using Theorem 3.5.3.

First we notice that 𝑝

𝑞 =

2+2𝑞𝑘
𝑞

= 2𝑘 + 1
1+ 1
2

𝐵 = [2𝑘, 0, 2]. Finally, as 𝑝𝑞 is even,

. Then 𝐴 = [2𝑘, 1, 2]. Then by definition 3.5.2

𝐶 (𝐿) = 𝑁 ( 𝑝, 𝑞) =

2𝑘 + 0 + 2
2

= 𝑘 + 1.

45

Then as 𝑘 → ∞, 𝐶 (𝐿) also goes to ∞.

Next by Lemma 3.5.4 we know that

𝑉 (𝐿) = 𝑡 ( 𝑝−1)(𝑞−1)/2 1 − 𝑡 𝑝+1 − 𝑡𝑞+1 + 𝑡 𝑝+𝑞

1 − 𝑡2
= 𝑡 ((2+2𝑞𝑘)𝑞−(2+2𝑞𝑘)−𝑞+1)/2 1 − 𝑡2+2𝑞𝑘+1 − 𝑡𝑞+1 + 𝑡2+2𝑞𝑘+𝑞

1 − 𝑡2

= 𝑡 ((2+2𝑞𝑘)𝑞−(2+2𝑞𝑘)−𝑞+1)/2(−𝑡2𝑞𝑘+𝑞 − 𝑡2𝑞𝑘+𝑞−2−

· · · − 𝑡2+2𝑞𝑘+1 + 𝑡𝑞−1 + · · · + 𝑡2 + 1).

The last step arises from taking the polynomial division. Therefore, given our choices of 𝑝 and 𝑞

we see that 𝑇𝐿 ≤ 2.

□

Then Proposition 3.5.5 shows part (a) of Theorem 1.1.4.

3.5.2 Part (b) of theorem 1.1.4

In this section we will work to prove part (b) of Theorem 1.1.4 and the following theorem:

Theorem 3.5.6. There does not exist a universal linear lower bound on 𝐶 (𝐿) for all links, 𝐿, in

terms of 𝑇𝐿.

To prove this we will introduce a family of links for which 𝐶 (𝐿) is uniformly bounded but

𝑇𝐿 can be made arbitrarily large. These links will be constructed by using the Whitehead double

defined here:

Definition 3.5.7. The Whitehead double of a knot 𝐿 is the satellite of the unknot clasped inside

of the torus. We call it a positive Whitehead double if the clasp is as in Figure 3.9 and a negative

Whitehead double if not.

The particular family is defined in this next theorem:

Theorem 3.5.8. Let 𝐾1, 𝐾2, ..., 𝐾𝑛 be alternating knots such that 𝛽′
𝐾𝑖

≠ 0. Then we let 𝐾 be the

connect sum of 𝐾1, 𝐾2, ..., 𝐾𝑛 such that 𝐾 is alternating and 𝑊−(𝐾) be the negative Whitehead

double of 𝐾 using the blackboard framing. Then 𝐶 (𝐿) ≤ 3 and |𝛽′

𝐿 | ≥ 𝑛.

46

Figure 3.9 Here we see the unknot with a clasp contained inside the torus, then we see the resulting
positive Whitehead double with the blackboard framing when map the torus to the trefoil.

Lemma 3.5.9. If a link is B-adequate then the negative Whitehead double of the link using the

blackboard framing is also B-adequate.

A similar statement was proven in [6] as Proposition 7.1. They show it for the untwisted

negative Whitehead double of a knot with non-negative writhe. The writhe of the knot introduces

extra twists into the diagram of the untwisted Whitehead double, which can interfere with adequacy

around the clasp.

Proof. We start by showing that the blackboard 2-cabling will be B-adequate. This is shown by

Lickorish in [32] for 𝑛-cablings. Let 𝐷 be a B-adequate diagram for our link and 𝐷2 the 2 cabling.

Notice in 𝐷2 there will be four copies of each crossing in 𝐷. Then when we have the all B-resolution

state we will end up with four parallel strands instead of two as we did in 𝐷. If we were to have a

one edge loop, then two of the strands are part of the same state circle. But these state circles are

copies of the state circles for 𝐷 so this would contradict that 𝐷 is adequate.

Now we want to look at the negative Whitehead double of 𝐷 using the blackboard framing. If

we let the Whitehead double be 𝑊−(𝐷) we will see that 𝐺′(𝑊−(𝐷)) will be the same as 𝐺′(𝐷2)

but with an additional vertex and 2 new edges as we see in Figure 3.10. As resolving the clasp does

not create a one edge loop we see that 𝑊−(𝐷) is B-adequate.

□

Here we remind the reader that 𝐺′

𝐵 (𝐷 (𝐿)) for a link diagram 𝐷 (𝐿) is the reduced all B-state

graph. We continue with the following lemma:

Lemma 3.5.10. If 𝑊−(𝐷 (𝐿)) is the negative Whitehead double of a B-adequate link dia-

gram 𝐷 (𝐿) using the blackboard framing, and 𝑏(𝐺) the first Betti number for a graph, then

47

Figure 3.10 The result of the B-resolution on the clasp of a negative Whitehead double.

𝑏(G′

𝐵 (𝑊−(𝐷 (𝐿)))) = 𝑏(G′

𝐵 (𝐷 (𝐿))) + 1.

Proof. Dasbach and Lin [10] showed in Lemma 2.5 that if 𝐷2 is the two cabling of a B-adequate

link diagram then 𝑏(G′

𝐵 (𝐷2)) = 𝑏(G′

𝑒 is the number of edges and 𝑣 the number of vertices.

𝐵 (𝐷 (𝐿))). For a graph 𝐺, 𝑏(𝐺) = 𝑒 − 𝑣 + 1, where
In the reduced graph when we take

the two cabling every parallel copy of a state circle will also produce a new edge. Hence, the

change in 𝑣 and 𝑒 will be the same between G′

𝐵 (𝐷 (𝐿)). Then when we move to
𝑊−(𝐷 (𝐿)) the clasp will add 2 edges and 1 vertex as we see in figure 3.10. Then we see that

𝐵 (𝐷2) and G′

𝑏(G′

𝐵 (𝑊−(𝐷 (𝐿)))) = 𝑏(G′

𝐵 (𝐷2)) + 1 = 𝑏(G′

𝐵 (𝐷 (𝐿))) + 1.

□

The two previous lemmas allow us to see that the blackboard framing of the negative Whitehead

double of an alternating link will be B-adequate. Also, we have a formula for the Betti number of

the Whitehead double in relation to the first Betti number of the original link. The only remaining

piece of the puzzle is to get from the first Betti number of the reduced B state graph to the second to

last coefficient of the Jones polynomial. This comes from the following result proven by Stoimenow

in Proposition 3.1 of [42].

Lemma 3.5.11. If 𝐷 (𝐿) is a B-adequate, connected diagram for a link, then in the representation

of the Jones polynomial, 𝑉 (𝐷 (𝐿)), we have 𝛼′

𝐷 (𝐿) = ±1, 𝛼′

𝐷 (𝐿)

𝐷 (𝐿) ≤ 0, and
𝛽′

|𝛽′

𝐷 (𝐿) | = 𝑒′ − 𝑣′ + 1 = 𝑏(G′

𝐵 (𝐷 (𝐿))),

where G′

graph G′

𝐵 is the reduced all B state graph and 𝑒′ and 𝑣′ are the number of edges and vertices of the
𝐵 (𝐷 (𝐿)), respectively.

48

We now have the tools to prove Theorem 3.5.8. But first we show a more specific example of a

family which satisfies Theorem 1.1.4 part (b).

Proposition 3.5.12. Let 𝑊−(𝐾𝑚) be the negative Whitehead double using the blackboard framing

of the connect sum of 𝑚 trefoils as in Figure 3.11. Then for all 𝑚, 𝐶 (𝑊−(𝐾𝑚)) ≤ 3 and 𝑇𝑊− (𝐾𝑚)
grows with 𝑚. Therefore, 𝑇𝑊− (𝐾𝑚) can be made arbitrarily large across the family of knots.

Figure 3.11 The negative Whitehead double of the connect sum of 𝑚 trefoil knots.

Proof. The first part of the lemma is a direct result of [7] by Clark where he shows that 𝑐(𝐾) ≤

2𝑔(𝐾) + 1 where 𝑔(𝐾) is the genus of the knot. For any Whitehead double we can find an oriented

spanning surface with genus exactly one by taking the annulus with a double twisted band at the

clasp. Then 𝐶 (𝑊−(𝐾𝑚)) ≤ 3 as 𝑔(𝑊−(𝐾𝑚)) = 1.

Now we will compute 𝛽′

𝐵 (𝑊−(𝐾𝑚)). By Lemma 3.5.11 we only need to
find the number of vertices and edges as 𝑊−(𝐾𝑚) is B-adequate. By Lemma 3.5.11 and the graph

by finding G′

𝑊− (𝐾𝑚)

G′

𝐵 (𝑊−(𝐾𝑚)) shown in Figure 3.12

|𝛽′

𝑊− (𝐾𝑚) | = 𝑒(G′

𝐵 (𝑊−(𝐾𝑚))) − 𝑣(G′

𝐵 (𝑊−(𝐾𝑚))) + 1

= (5𝑚 + 3) − (4𝑚 + 3) + 1 = 𝑚.

Hence, showing that 𝑇𝑊− (𝐾𝑚) ≥ 𝑚 for all 𝑘, proving the claim.

(3.8)

(3.9)

□

Here we will introduce a more general family of knots for which Theorem 1.1.4 part (b) holds

true:

49

Figure 3.12 Left: The all B-state circle diagram. Right: The reduced state graph G′
the disjoint dots are not nodes for the graph but represent that we have 𝑘 copies of the subgraph on
the left.

𝐵 (𝐿). Notice

Theorem 3.5.8. Let 𝐾1, 𝐾2, ..., 𝐾𝑛 be alternating knots such that 𝛽′
𝐾𝑖

≠ 0. Then let 𝐾 be the connect

sum of 𝐾1, 𝐾2, ..., 𝐾𝑛 such that 𝐾 is alternating, and let 𝑊−(𝐾) the negative Whitehead double of

𝐾 using the blackboard framing. Then 𝐶 (𝑊−(𝐾)) ≤ 3 and |𝛽′

𝑊− (𝐾) | ≥ 𝑛.

Proof. As in Lemma 3.5.12 for a Whitehead doubles such as 𝑊−(𝐾), 𝐶 (𝑊−(𝐾)) ≤ 3. Now we

𝑖=1 ±𝛽′
𝐾𝑖

will work to compute 𝑇𝑊− (𝐾). From [32] we know that the Jones polynomial for 𝐾 will be the
product of the Jones polynomials of the 𝐾𝑖. Then as all of the 𝐾𝑖 are alternating, 𝛼′
= ±1 so
𝐾𝑖
𝐾 = (cid:205)𝑛
𝛽′
and 𝛽′
𝛼′
𝐾𝑖
𝐾𝑖
𝐾 = (cid:205)𝑛
hypothesis |𝛽′
𝐾𝑖

𝑖=1(−1)𝑚±1|𝛽′
𝐾𝑖
| > 0 for all 𝑖, hence |𝛽′

. From Lemma 3.5.11 we know that 𝛼′
𝐾𝑖

do not match. If we let 𝑚 be the number of the 𝛼′

|. Hence, in our sum the signs match so |𝛽′

𝑖 which are negative, then we see

≤ 0 which tells us that the signs of

𝐾 | = (cid:205)𝑛

|. By our

𝑖=1 |𝛽′
𝐾𝑖

𝐾 | ≥ 𝑛.

that 𝛽′

𝛽′
𝐾𝑖

By Lemma 3.5.9 we know that 𝑊−(𝐾) will be B-adequate as 𝐾 is alternating and therefore

B-adequate. Then by Lemma 3.5.10 and Lemma 3.5.11 we see that |𝛽′

is at least as large as the number of knots in the connect sum so is |𝛽′

𝑊− (𝐾) | = |𝛽′

𝐾 | + 1 and as |𝛽′
𝐾 |
𝑊− (𝐾) | and further 𝑇𝑊− (𝐾).

Then 𝑇𝑊− (𝐾) will grow with 𝑛 showing that it is unbounded across the family.

□

Lemma 3.5.12 and Theorem 3.5.8 both show that Theorem 1.1.4(b).

50

CHAPTER 4

CROSSING NUMBER OF CABLE KNOTS

In this chapter we will calculate the crossing number for an infinite family of cable knots. Similar

to the previous chapter we will do this using a quantum invariant, in this case we will be using the

colored Jones polynomial. The work and figures in this chapter come from a preprint by Kalfagianni

and McConkey [26].

4.1

Jones Diameter

We will start this chapter by introducing some notation we will use throughout and introducing

some tools we will need. We previously defined the colored Jones polynomial and in this chapter

we will be using it to make calculations with respect to the crossing number of a family of cable

knots. In particular we will be using a property of the colored Jones polynomial known as the Jones

diameter which we define shortly. Afterwards we will introduce some results related to the Jones

diameter which give us the tools we need for the main part of this chapter.

Here we state notation we will use in this chapter, some of which we have seen before. This is

for the convenience of a reader whose interest lies in this chapter.

• 𝑐(𝐷) is the number of crossings of the knot diagram 𝐷.

• With an orientation on 𝐷, 𝑐+(𝐷) and 𝑐−(𝐷) are respectively the number of positive crossings

and the number of negative crossings in the knot diagram 𝐷 following the conventions of

Figure 4.1.

• 𝑣 𝐴 (𝐷) is the number of state circles in the all-𝐴 state, and 𝑣 𝐵 (𝐷) is the number of state

circles in the all-𝐵 state.

• The state graphs G𝐴 (𝐷) and G𝐵 (𝐷) have vertices the state circles of the all-𝐴 and all-𝐵 state

respectively, and edge segments recording the original location of the crossing. In Figure

2.1, these segments are indicated in dashed line.

• The writhe of a knot diagram 𝐷, denoted by wr(𝐷) := 𝑐+(𝐷) − 𝑐−(𝐷).

51

• The Turaev genus of 𝐷 is defined by 2𝑔𝑇 (𝐷) := 2 − 𝑣 𝐴 (𝐷) − 𝑣 𝐵 (𝐷) + 𝑐(𝐷) [47, 11].

+1

−1

Figure 4.1 A positive crossing and a negative crossing.

Recall from definition 2.1.4 that a knot diagram 𝐷 = 𝐷 (𝐾) is 𝐴-adequate (resp. 𝐵-adequate)

if G𝐴 (𝐷) (resp. G𝐵 (𝐷)) has no one-edged loops. A knot is adequate if it admits a diagram

𝐷 = 𝐷 (𝐾) that is both 𝐴- and 𝐵-adequate [33, 32].

Notice that if 𝐷 = 𝐷 (𝐾) is an adequate diagram, then quantities 𝑐(𝐷), 𝑐±(𝐷) [32, 28, 37, 46]

as well as the Turaev genus 𝑔𝑇 (𝐷) [1] are minimal over all diagrams representing 𝐾. As a result of

this one can see that the writhe wr(𝐷) for adequate diagrams of a given knot 𝐾 is always constant.

Thus, these are all invariants of 𝐾 and we will denote them by 𝑐(𝐾), 𝑐±(𝐾), 𝑔𝑇 (𝐾), and wr(𝐾)

respectively.

Given a knot 𝐾 let 𝐽𝐾 (𝑛) denote its 𝑛-th colored Jones polynomial with variable in 𝑡. Let

𝑑+ [𝐽𝐾 (𝑛)] and 𝑑− [𝐽𝐾 (𝑛)] denote the maximal and minimal degree of 𝐽𝐾 (𝑛) in 𝑡 and set

𝑑 [𝐽𝐾 (𝑛)] := 4𝑑+ [𝐽𝐾 (𝑛)] − 4𝑑− [𝐽𝐾 (𝑛)].

Lemma 4.1.1. [32] Given a knot diagram 𝐷 = 𝐷 (𝐾), for all 𝑛 ∈ N, we have the following.

(a) 𝑑+ [𝐽𝐾 (𝑛)] ≤ 𝑐+ (𝐷)

2 𝑛2 + 𝑂 (𝑛) and we have equality if 𝐷 is 𝐵-adequate.

(b) 𝑑− [𝐽𝐾 (𝑛)] ≥ − 𝑐− (𝐷)

2 𝑛2 + 𝑂 (𝑛) and we have equality if 𝐷 is 𝐴-adequate.

(c) 𝑑 [𝐽𝐾 (𝑛)] ≤ 2𝑐(𝐷)𝑛2 + (4 − 4𝑔𝑇 (𝐷) − 2𝑐(𝐷))𝑛 + (4𝑔𝑇 (𝐷) − 4), and we have equality if 𝐷 is

adequate.

The set of cluster points (cid:8)𝑛−2𝑑 [𝐽𝐾 (𝑛)](cid:9)′

𝑛∈N

is known to be finite and the point with the largest

absolute value, denoted by 𝑑 𝑗𝐾, is called the Jones diameter of 𝐾.

The following, Theorem 4.1.2, gives us an essential relationship between the crossing number

of 𝐾 and its Jones diameter. The part of the theorem we will use for our main results is the

52

non-adequate part as this allows us to directly calculate the crossing number for a family of cable

knots.

Theorem 4.1.2. [25]Let 𝐾 be a knot with Jones diameter 𝑑𝑗𝐾 and crossing number 𝑐(𝐾). Then,

𝑑𝑗𝐾 ≤ 2 𝑐(𝐾),

with equality 𝑑 𝑗𝐾 = 2 𝑐(𝐾) if and only if 𝐾 is adequate.

In particular, if 𝐾 is a non-adequate knot admitting a diagram 𝐷 = 𝐷 (𝐾) such that 𝑑𝑗𝐾 =

2(𝑐(𝐷) − 1), then we have 𝑐(𝐷) = 𝑐(𝐾).

The next couple of results we will state give formulae for the extreme degrees of the colored

Jones polynomials for knots where the degrees 𝑑± [𝐽𝐾 (𝑛)] are quadratic polynomials.
Proposition 4.1.3. [27, 5] Suppose that 𝐾 is a knot such that 𝑑+ [𝐽𝐾 (𝑛)] = 𝑎2𝑛2 + 𝑎1𝑛 + 𝑎0 and

𝑑− [𝐽𝐾 (𝑛)] = 𝑎∗

2𝑛2 + 𝑎∗
1 ≥ 0 and that
Then for 𝑛 large enough,

𝑎1 ≤ 0, 𝑎∗

1𝑛 + 𝑎∗
𝑞 < 4𝑎2 and −𝑝
𝑝

𝑞 < 4𝑎∗
2.

0 are quadratic polynomials for all 𝑛 > 0. Suppose, moreover, that

4𝑑+ [𝐽𝐾 𝑝,𝑞 (𝑛)] = 𝑞24𝑎2𝑛2 + (𝑞4𝑎1 + 2(𝑞 − 1) ( 𝑝 − 4𝑞𝑎2))𝑛 + 𝐴,

4𝑑− [𝐽𝐾 𝑝,𝑞 (𝑛)] = −𝑞24𝑎∗

2𝑛2 − (𝑞4𝑎∗

1 + 2(𝑞 − 1) ( 𝑝 − 4𝑞𝑎∗

2))𝑛 + 𝐴∗,

where 𝐴, 𝐴∗ ∈ Q depend only on 𝐾 and 𝑝, 𝑞.

Proof. The first equation is shown in [27] (see also [5]). To obtain the second equation we use

the fact that, since 𝐾 ∗

−𝑝,𝑞 = (𝐾𝑝,𝑞)∗, we have 𝑑− [𝐽𝐾 𝑝,𝑞 (𝑛)] = −𝑑+ [𝐽𝐾 ∗

− 𝑝,𝑞 (𝑛)] and apply the first

equation to 𝐾 ∗

−𝑝,𝑞.

□

Here is a second result which handles the case not covered in Proposition 4.1.3.

Lemma 4.1.4. [27, 5]Let the notation and setting be as in Proposition 4.1.3.

53

If

𝑝
𝑞 > 4𝑎2, then

4𝑑+ [𝐽𝐾 𝑝,𝑞 (𝑛)] = 𝑝𝑞𝑛2 + 𝐵,

where 𝐵 ∈ Q depends only on 𝐾 and 𝑝, 𝑞.

Similarly, if −𝑝

𝑞 > 4𝑎∗

2, then

4𝑑− [𝐽𝐾 𝑝,𝑞 (𝑛)] = −𝑝𝑞𝑛2 + 𝐵∗,

where 𝐵∗ ∈ Q depends only on 𝐾 and 𝑝, 𝑞.

4.2 Lower bounds and admissible knots

We will say that a knot 𝐾 is admissible if there is a diagram 𝐷 = 𝐷 (𝐾) such that we have

𝑑𝑗𝐾 = 2 (𝑐(𝐷) − 1).

Our interest in admissible knots comes from the fact that if 𝐾 is admissible and non-adequate, then

by Theorem 4.1.2, 𝐷 is a minimal diagram (i.e. 𝑐(𝐷) = 𝑐(𝐾)).

Theorem 4.2.1. Let 𝐾 be an adequate knot and let 𝑐(𝐾), 𝑐±(𝐾) and 𝑤(𝐾) be as above.

(a) For any coprime integers 𝑝, 𝑞, we have

𝑐(𝐾𝑝,𝑞) ≥ 𝑞2 · 𝑐(𝐾).

(4.1)

(b) The cable 𝐾𝑝,𝑞 is admissible if and only if 𝑞 = 2 and 𝑝 = 𝑞 𝑤(𝐾) ± 1.

Proof. Since 𝐾 is adequate, by Lemma 4.1.1,

4𝑑+ [𝐽𝐾 (𝑛)] − 4𝑑− [𝐽𝐾 (𝑛)] = 2𝑐(𝐾)𝑛2 + (4 − 4𝑔𝑇 (𝐾) − 2𝑐(𝐾))𝑛 + 4𝑔𝑇 (𝐾) − 4,

(4.2)

for every 𝑛 ≥ 0.

We distinguish three cases.

Case 1. Suppose that 𝑝

𝑞 < 2𝑐−(𝐾). Then, 𝑑+ [𝐽𝐾 (𝑛)] satisfies the hypothesis
of Proposition 4.1.3 with 4𝑎2 = 2𝑐+(𝐾) > 0 and 𝑑− [𝐽𝐾 (𝑛)] = −𝑑+ [𝐽𝐾 ∗ (𝑛)], where 𝑑+ [𝐽𝐾 ∗ (𝑛)]

𝑞 < 2𝑐+(𝐾) and −𝑝

satisfies that hypothesis of Proposition 4.1.3 with 4𝑎∗

2 = 2𝑐+(𝐾 ∗) = 2𝑐−(𝐾). The requirement that

54

𝑎1 ≤ 0 is satisfied since for adequate knots the linear terms of the degree of 𝐽∗

𝐾 (𝑛) are multiples
of Euler characteristics of spanning surfaces of 𝐾. See [27, Lemmas 3.6, 3.7]. Now Proposition

4.1.3 implies that for sufficiently large 𝑛 we have that 𝑑± [𝐽𝐾 𝑝,𝑞 (𝑛)] is a quadratic polynomial and

the Jones diameter of 𝐾𝑝,𝑞 is 𝑑 𝑗𝐾 = 𝑞2𝑐(𝐾). Hence by Theorem 4.1.2 we get 𝑐(𝐾𝑝,𝑞) ≥ 𝑞2 · 𝑐(𝐾)

which proves part (a) of Theorem 1.1.7 in this case.

For part (b), we recall that a diagram 𝐷 𝑝,𝑞 of 𝐾𝑝,𝑞 is obtained as follows: Start with an adequate

diagram 𝐷 = 𝐷 (𝐾) and take 𝑞 parallel copies to obtain a diagram 𝐷𝑞.

In other words, take

the 𝑞-cabling of 𝐷 following the blackboard framing. To obtain 𝐷 𝑝,𝑞 add 𝑡-twists to 𝐷𝑞, where

𝑡 := 𝑝 − 𝑞 𝑤𝑟 (𝐾) as follows: If 𝑡 < 0 then a twist takes the leftmost string in 𝐷𝑞 and slides it

over the 𝑞 − 1 strings to the right; then we repeat the operation |𝑡|-times. If 𝑡 > 0 a twist takes the

rightmost string in 𝐷𝑞 and slides it over the 𝑞 − 1 strings to the left; then we repeat the operation

|𝑡|-times. Now

𝑐(𝐷 𝑝,𝑞) = 𝑞2 𝑐(𝐾) + |𝑡|(𝑞 − 1) = 𝑞2 𝑐(𝐾) + | 𝑝 − 𝑞 𝑤𝑟 (𝐾)|(𝑞 − 1),

while 𝑑𝑗𝐾 = 2𝑞2 𝑐(𝐾). Now setting 2𝑐(𝐷 𝑝,𝑞) − 2 = 𝑑𝑗𝐾, we get | 𝑝 − 𝑞 𝑤𝑟 (𝐾)|(𝑞 − 1) = 1

which gives that 𝑞 = 2 and 𝑝 = 𝑞 𝑤𝑟 (𝐾) ± 1. Similarly, if we set 𝑝 = 𝑞 𝑤𝑟 (𝐾) ± 1 we find that

2𝑐(𝐷 𝑝,𝑞) − 2 = 𝑑 𝑗𝐾 must also be true. Hence in this case both (a) and (b) hold.

Figure 4.2 Left: 3 positive twists on four strands Right: 3 negative twists on four strands.

Case 2. Suppose that 𝑝

𝐵 ∈ Q depends only on 𝐾 and 𝑝, 𝑞. Since 𝑝
hand, since −𝑝

𝑞 < 0, we clearly have −𝑝

𝑞 > 2𝑐+(𝐾). Then by Lemma 4.1.4, 4𝑑+ [𝐽𝐾 𝑝,𝑞 (𝑛)] = 𝑝𝑞𝑛2 + 𝐵, where
𝑞 > 2𝑐+(𝐾), we get 𝑝𝑞 > 2𝑐+(𝐾)𝑞2. On the other
− 𝑝,𝑞 (𝑛)]

𝑞 < 2𝑐−(𝐾), and Proposition 4.1.3 applies to 𝑑+ [𝐽𝐾 ∗

55

for 4𝑎∗

2 = 2𝑐−(𝐾). Then

4𝑑+ [𝐽𝐾 (𝑛)] − 4𝑑− [𝐽𝐾 (𝑛)] = 𝑑+ [𝐽𝐾 (𝑛)] + 4𝑑+ [𝐽𝐾 ∗

− 𝑝,𝑞 (𝑛)] > 𝑞2 · 𝑐(𝐾),

as desired. This finishes the proof for part (a) of the theorem.

In this case, we don’t get any

admissible knots: first note that 𝑝 > 2𝑞𝑐+(𝐾) > 𝑞 𝑤(𝐾). As in Case 1 we get a diagram 𝐷 𝑝,𝑞 of

𝐾𝑝,𝑞 with

𝑐(𝐷 𝑝,𝑞) = 𝑞2 𝑐(𝐾) + ( 𝑝 − 𝑞 𝑤(𝐾)) (𝑞 − 1),

while 𝑑𝑗𝐾 = 2𝑞2 𝑐−(𝐾) + 𝑝 𝑞. Now setting 2𝑐(𝐷 𝑝,𝑞) − 2 = 𝑑𝑗𝐾, and after some straightforward

algebra, we find that in order for 𝐾𝑝,𝑞 to be admissible we must have

2(𝑞2 − 𝑞) 𝑐−(𝐾) + 2𝑞 𝑐+(𝐾) + 𝑝 (𝑞 − 2) − 2 = 0.

However, since 𝑝, 𝑐(𝐾) > 0 and 𝑞 ≥ 2, above equation is never satisfied.

Case 3. Suppose that −𝑝

𝑞 > 2𝑐−(𝐾) > 0, in which case 𝑝

𝑞 < 0 ≤ 2𝑐−(𝐾). This case is similar

to Case 2 above.

Remark 4.2.2. In [43] inequality (4.1) is also verified, for some choices of 𝑝 and 𝑞, using crossing

number bounds obtained from the ordinary Jones polynomial in [44] and also from the 2-variable

Kauffman polynomial. Theorem 1.1.7 shows that the colored Jones polynomial and the results

of [25] provide better bounds for crossing numbers of satellite knots, allowing in particular exact

□

computations for infinite families.

4.3 Non-adequacy results

To prove the stronger version of inequality (4.1), stated in Theorem 1.1.7, we need to know that

the cables 𝐾𝑝,𝑞 are not adequate. This is the main result in this section.

Theorem 4.3.1. Let 𝐾 be an adequate knot with crossing number 𝑐(𝐾) > 0 and suppose that
𝑞 < 2𝑐+(𝐾) and −𝑝
𝑝

𝑞 < 2𝑐−(𝐾). Then, the cable 𝐾𝑝,𝑞 is non-adequate.

To prove Theorem 4.3.1 we need the following lemma:

56

Lemma 4.3.2. Let 𝐾 be an adequate knot with crossing number 𝑐(𝐾) > 0 and suppose that
𝑞 < 2𝑐+(𝐾) and −𝑝
𝑝
𝑞 < 2𝑐−(𝐾). If 𝐾𝑝,𝑞 is adequate, then 𝑐(𝐾𝑝,𝑞) = 𝑞2 𝑐(𝐾).

Proof. By our earlier discussion, for 𝑛 large enough,

4𝑑+ [𝐽(𝐾 𝑝,𝑞 (𝑛)] − 4𝑑− [𝐽𝐾 𝑝,𝑞 (𝑛)] = 𝑑2𝑛2 + 𝑑1𝑛 + 𝑑0,

with 𝑑𝑖 ∈ Q. By Proposition 4.1.3, we compute 𝑑2 = 𝑞2(4𝑎2 + 4𝑎∗

adequate, since 𝑑2 = 2𝑐(𝐾𝑝,𝑞), we must have 𝑐(𝐾𝑝,𝑞) = 𝑞2𝑐(𝐾).

We now give the proof of Theorem 4.3.1:

2) = 2𝑞2𝑐(𝐾). Now if 𝐾𝑝,𝑞 is
□

Proof. First, we let 𝐾, 𝑝, and 𝑞 such that 𝑡 := 𝑝 − 𝑞 𝑤(𝐾) < 0.

Recall that if 𝐾 has an adequate diagram 𝐷 = 𝐷 (𝐾) with 𝑐(𝐷) = 𝑐+(𝐷) + 𝑐−(𝐷) crossings

and the all-𝐴 (rep. all-𝐵) resolution has 𝑣 𝐴 = 𝑣 𝐴 (𝐷) (resp. 𝑣 𝐵 = 𝑣 𝐵 (𝐷)) state circles, then

4 𝑑− [𝐽𝐾 (𝑛)] = −2𝑐−(𝐷)𝑛2 + 2(𝑐(𝐷) − 𝑣 𝐴 (𝐷))𝑛 + 2𝑣 𝐴 (𝐷) − 2𝑐+(𝐷),

4 𝑑+ [𝐽𝐾 (𝑛)] = 2𝑐+(𝐷)𝑛2 + 2(𝑣 𝐵 (𝐷) − 𝑐(𝐷))𝑛 + 2𝑐−(𝐷) − 2𝑣 𝐵 (𝐷).

(4.3)

(4.4)

Equation (4.3) holds for 𝐴-adequate diagrams 𝐷 = 𝐷 (𝐾). Thus in particular the quantities

𝑐−(𝐷), 𝑣 𝐴 (𝐷) are invariants of 𝐾 (independent of the particular 𝐴-adequate diagram). Similarly,

Equation (4.4) holds for 𝐵-adequate diagrams 𝐷 = 𝐷 (𝐾) and hence 𝑐+(𝐷), 𝑣𝐵 (𝐷) are invariants

of 𝐾. Recall also that 𝑐(𝐷) = 𝑐(𝐾) since 𝐷 is adequate.

Now we start with a knot 𝐾 that has an adequate diagram 𝐷 then wr(𝐷) = wr(𝐾). Hence we

have 𝑐+(𝐷) = 𝑐−(𝐷) + wr(𝐾). Since 𝐷 is 𝐵-adequate and 𝑡 < 0, the cable 𝐷 𝑝,𝑞 is a 𝐵-adequate
diagram of 𝐾𝑝,𝑞 with 𝑣 𝐵 (𝐷 𝑝,𝑞) = 𝑞𝑣 𝐵 (𝐷) and 𝑐+(𝐷 𝑝,𝑞) = 𝑞2𝑐+(𝐷). See Figure 4.3. Furthermore,

since as said above these quantities are invariants of 𝐾𝑝,𝑞, they remain the same for all 𝐵-adequate

diagrams of 𝐾𝑝,𝑞.

57

Figure 4.3 Left: the -1,2 cabling of the figure eight knot. Right: the B-state graph showing that
the cabling is B-adequate.

Now assume, for a contradiction, that 𝐾𝑝,𝑞 is adequate: Then, it has a diagram ¯𝐷 that is
both 𝐴 and 𝐵-adequate. By above observation we must have 𝑣 𝐵 ( ¯𝐷) = 𝑣 𝐵 (𝐷 𝑝,𝑞) = 𝑞𝑣 𝐵 (𝐷) and
𝑐+( ¯𝐷) = 𝑐+(𝐷 𝑝,𝑞) = 𝑞2𝑐+(𝐷).

By Lemma 4.3.2, 𝑐( ¯𝐷) = 𝑐(𝐾𝑝,𝑞) = 𝑞2𝑐(𝐾).

Write

4 𝑑+ [𝐽𝐾 𝑝,𝑞 (𝑛)] = 𝑥𝑛2 + 𝑦𝑛 + 𝑧,

for some 𝑥, 𝑦, 𝑧 ∈ Q.

For sufficiently large 𝑛 we have two different expressions for 𝑥, 𝑦, 𝑧. On one hand, because ¯𝐷 is

adequate, we can use Equation (4.4) to determine 𝑥, 𝑦, 𝑧.

On the other hand, using 4 𝑑+ [𝐽𝐾 ∗

− 𝑝,𝑞 (𝑛)], 𝑥, 𝑦, 𝑧 can be determined using Proposition 4.1.3 with

𝑎2 and 𝑎1 coming from Equation (4.4) applied to 𝐷.

We will use these two ways to find the quantity 𝑦. Applying Equation (4.4) to ¯𝐷 we obtain

𝑦 = 2(𝑣 𝐵 ( ¯𝐷 − 𝑐( ¯𝐷))) = 2𝑞𝑣 𝐵 (𝐷) − 2𝑞2𝑐(𝐷)

(4.5)

On the other hand, using Proposition 4.1.3 with 𝑎2 and 𝑎1 coming from Equation (4.4) we have:

4𝑎2 = 2𝑐+(𝐷) = 𝑐(𝐷) + wr(𝐾). Also, we have 4𝑎1 = 2𝑣 𝐵 (𝐷) − 2𝑐(𝐷).

We obtain

𝑦 = 𝑞(4𝑎1)−2𝑞(𝑞−1)(4𝑎2)+2(𝑞−1) 𝑝 = 2𝑞𝑣 𝐵 (𝐷)−2𝑞2𝑐(𝐷)+2(𝑞−1) 𝑝−2𝑞(𝑞−1)wr(𝐾). (4.6)

58

It follows for the two expressions derived for 𝑦 from Equations (4.5) and (4.6) to agree we must

have

2𝑞((𝑞 − 1)2wr(𝐾) + 𝑝) − 2𝑝 = 0.

However this is impossible since 𝑞 > 1 and 𝑝, 𝑞 are coprime. This contradiction shows that 𝐾𝑝,𝑞

is non-adequate.

To deduce the result for 𝐾𝑝,𝑞, with 𝑡 (𝐾, 𝑝, 𝑞) := 𝑝 − 𝑞 𝑤(𝐾) > 0, let 𝐾 ∗ denote the mirror

image of 𝐾. Note that f (𝐾𝑝,𝑞)∗ = 𝐾 ∗

under taking mirror images, it is enough to show that 𝐾 ∗

−𝑝,𝑞 and since being adequate is a property that is preserved
−𝑝,𝑞 is non-adequate. Since 𝑡 (𝐾 ∗, −𝑝, 𝑞) :=
□

−𝑝 − 𝑞 𝑤(𝐾 ∗) = −𝑡 (𝐾, 𝑝, 𝑞) < 0, the later result follows from the argument above.

4.3.1 Proof of Theorem 1.1.7 and Corollary 1.1.8

By Theorem 4.2.1, we have

𝑐(𝐾𝑝,𝑞) ≥ 𝑞2𝑐(𝐾).

𝑝

We need to show that this inequality is actually strict. Recall that by the proof of Theorem 4.2.1, if
𝑞 > 2𝑐+(𝐾) or −𝑝
𝑞 < 2𝑐+(𝐾) and −𝑝
again we have 2𝑐(𝐾𝑝,𝑞) ≠ 𝑑 𝑗𝐾 and the strict inequality follows.

𝑞 > 2𝑐−(𝐾), then the above inequality is strict so we need to only consider when
𝑞 < 2𝑐−(𝐾). By Theorem 4.3.1, 𝐾𝑝,𝑞 is non-adequate. Hence by Theorem 4.1.2
□

𝑝

Next we discuss how to deduce Corollary 1.1.8:

Proof. If 𝑞 = 2 and 𝑝 = 𝑞 𝑤(𝐾) ± 1, then by Theorem 4.2.1 𝐾𝑝,𝑞 is admissible. Thus by

Theorem 4.1.2, the diagram 𝐷 𝑝,2 constructed in the proof of Theorem 4.2.1 is minimal. That is

𝑐(𝐾𝑝,2) = 𝑐(𝐷 𝑝,2) = 4 𝑐(𝐾) + 1.

□

4.4 Composite non-adequate knots

Here we give an application of Theorem 1.1.7 to the question on additivity of crossing numbers

under the connected sum of knots. [29, Problems 1.67]. As already mentioned, for adequate knots

the crossing number is additive under connected sum. The next result proves additivity for families

of knots where one summand is adequate while the other is not.

59

Theorem 1.1.10. Suppose that 𝐾 is an adequate knot and let 𝐾1 := 𝐾𝑝,2, where 𝑝 = 2 𝑤(𝐾) ± 1.

Then for any adequate knot 𝐾2, the connected sum 𝐾1#𝐾2 is non-adequate and we have

𝑐(𝐾1#𝐾2) = 𝑐(𝐾1) + 𝑐(𝐾2).

Before we proceed with the proof of the theorem we need some preparation. Given a knot

𝐾, such that for 𝑛 large enough the degrees of the colored Jones polynomials of 𝐾 are quadratic

polynomials with rational coefficients, we will write

4 𝑑+ [𝐽𝐾 (𝑛)] = 𝑥(𝐾)𝑛2 + 𝑦(𝐾)𝑛 + 𝑧(𝐾) and − 4 𝑑− [𝐽𝐾 (𝑛)] = 𝑥∗(𝐾)𝑛2 + 𝑦∗(𝐾)𝑛 + 𝑧∗(𝐾).

We also write

4𝑑+ [𝐽𝐾 (𝑛)] − 4𝑑+ [𝐽𝐾 (𝑛)] = 𝑑2(𝐾)𝑛2 + 𝑑1(𝐾)𝑛 + 𝑑0(𝐾).

Now let 𝐾1, 𝐾2 be as in the statement of Theorem 1.1.10. By assumption and Proposition 4.1.3,

for 𝑛 large enough the degrees of the colored Jones polynomials of both 𝐾1 and 𝐾2 are quadratic

polynomials. For the proof we need the following well known lemma:

Lemma 4.4.1. [32] For large enough 𝑛, the degrees 𝑑± [𝐽𝐾1#𝐾2 (𝑛)] are polynomials, and we have

the following.

(a) 𝑥(𝐾1#𝐾2) = 𝑥(𝐾1) + 𝑥(𝐾2) and 𝑥∗(𝐾1#𝐾2) = 𝑥∗(𝐾1) + 𝑥∗(𝐾2).

(b) 𝑦(𝐾1#𝐾2) = 𝑦(𝐾1) + 𝑦(𝐾2) − 2 and 𝑦∗(𝐾1#𝐾2) = 𝑦∗(𝐾1) + 𝑦∗(𝐾2) − 2.

(c) 𝑑2(𝐾1#𝐾2) = 𝑑2(𝐾1) + 𝑑2(𝐾2).

The second ingredient we need for the proof of Theorem 1.1.10 is the following lemma.

Lemma 4.4.2. Let 𝐾 be a non-trivial adequate knot, 𝑝 = 2 𝑤(𝐾) ± 1 and let 𝐾1 := 𝐾𝑝,2. Then for

any adequate knot 𝐾2, the connected sum 𝐾1#𝐾2 is non-adequate.

Proof. The claim is proven by applying the arguments applied to 𝐾1 = 𝐾𝑝,2 in the proofs of Lemma

4.3.2 and Theorem 4.3.1 to the knot 𝐾1#𝐾2 and using the fact that the degrees of the colored Jones

polynomial are additive under connected sum.

60

First we claim that if 𝐾1#𝐾2 were adequate then we would have

𝑐(𝐾1#𝐾2) = 4𝑐(𝐾) + 𝑐(𝐾2)

(4.7)

Note that as 𝑝 = 2 𝑤(𝐾) ± 1, we have 𝑝

2 < 2𝑐+(𝐾) and −𝑝

2 < 2𝑐−(𝐾). Hence Proposition 4.1.3

applies to 𝐾1. Now write

4𝑑+ [𝐽𝐾1#𝐾2 (𝑛)] − 4𝑑− [𝐽𝐾1#𝐾2 (𝑛)] = 𝑑2(𝐾1#𝐾2)𝑛2 + 𝑑1(𝐾1#𝐾2)𝑛 + 𝑑0(𝐾1#𝐾2).

Since we assumed that 𝐾1#𝐾2 is adequate, we have 𝑑2(𝐾1#𝐾2) = 2𝑐(𝐾1#𝐾2). On the other

hand by Lemma 4.4.1, 𝑑2(𝐾1#𝐾2) = 𝑑2(𝐾1) + 𝑑2(𝐾2) = 2 · 4𝑐(𝐾) + 2𝑐(𝐾2) which leads to (4.7).

Case 1. Suppose that 𝑝 − 2 𝑤(𝐾) = −1 < 0.

Start with 𝐷 = 𝐷 (𝐾) an adequate diagram and let 𝐷1 := 𝐷 𝑝,2 be constructed as in the proof

of Theorem 4.2.1. Also let 𝐷2 be an adequate diagram of 𝐾2. As in the proof of Theorem 4.3.1

conclude that 𝐷1#𝐷2 is a 𝐵-adequate diagram for 𝐾1#𝐾2 and that the quantities 𝑣 𝐵 (𝐷1#𝐷2) =

2𝑣 𝐵 (𝐷) + 𝑣 𝐵 (𝐷2) − 1 and 𝑐+(𝐷1#𝐷2) = 4𝑐+(𝐷) + 𝑐+(𝐷2) are invariants of 𝐾1#𝐾2.

Let ¯𝐷 be an adequate diagram. Then

𝑣 𝐵 ( ¯𝐷) = 𝑣 𝐵 (𝐷1#𝐷2) = 2𝑣 𝐵 (𝐷) + 𝑣 𝐵 (𝐷2) − 1 and 𝑐+( ¯𝐷) = 4𝑐+(𝐷) + 𝑐+(𝐷2).

Next we will calculate the quantity 𝑦(𝐾1#𝐾2) of Lemma 4.4.1 in two ways: Firstly, since we

assumed that ¯𝐷 is an adequate diagram for 𝐾1#𝐾2, applying Equation (4.4), we get

𝑦(𝐾1#𝐾2) = 2(𝑣 𝐵 ( ¯𝐷) − 𝑐( ¯𝐷)) = 2(2𝑣 𝐵 (𝐷) + 𝑣 𝐵 (𝐷2) − 1 − 4𝑐(𝐷) − 𝑐(𝐷2)).

Secondly, using by Proposition 4.1.3 we get 𝑦(𝐾1) = 2(2𝑣 𝐵 (𝐷) − 4𝑐(𝐷) + 𝑝 + 2 wr(𝐾)).

Then by Lemma 4.4.1,

𝑦(𝐾1#𝐾2) = 𝑦(𝐾1) + 𝑦(𝐾2) − 1 = 2(2𝑣 𝐵 (𝐷) − 4𝑐(𝐷) + 𝑝 − 2 wr(𝐾) + 𝑣 𝐵 (𝐷2) − 𝑐(𝐷2)) − 1.

Now note that in order for the two resulting expressions for 𝑦(𝐾1#𝐾2) to be equal we must have

2( 𝑝 − 2 wr(𝐾)) = 1 which contradicts our assumption that 𝑝 − 2 wr(𝐾) = −1. We conclude that

𝐾1#𝐾2 is non-adequate.

61

Case 2. Assume now that 𝑝 − 2 wr(𝐾) = 1. Since (𝐾𝑝,2)∗ = 𝐾 ∗

is a property that is preserved under taking mirror images, it is enough to show that 𝐾 ∗

−𝑝,2 and since being adequate
2 is
non-adequate. Since −𝑝 − 2 wr(𝐾 ∗) = −( 𝑝 − 2wr(𝐾))) = −1, the later result follows from the

−𝑝,2#𝐾 ∗

argument above.

Now we give the proof of Theorem 1.1.10.

□

Proof. Note that if 𝐾 is the unknot then so is 𝐾𝑝,2 and the result follows trivially. Suppose that 𝐾

is a non-trivial knot. Then by Lemma 4.4.2 we obtain that 𝐾1#𝐾2 is non-adequate.

As discussed above 𝑑 𝑗𝐾1 = 2(4𝑐(𝐾)) = 2(𝑐(𝐷±1,2) − 1). On the other hand, 𝑑𝑗𝐾2 = 2𝑐(𝐷2) =

2𝑐(𝐾) where 𝐷2 is an adequate diagram for 𝐾2. Hence, by Lemma 4.4.1, 𝑑𝑗𝐾1#𝐾2 = 2(𝑐(𝐷1#𝐷2) −

1), where 𝐷1 = 𝐷±1,2. By Theorem 4.1.2,

𝑐(𝐾1#𝐾2) = 𝑐(𝐷1#𝐷2) = 𝑐(𝐷1) + 𝑐(𝐷2) = 𝑐(𝐾1) + 𝑐(𝐾2),

where the last equality follows since, by Theorem 1.1.7, we have 𝑐(𝐾1) = 𝑐(𝐷1) = 𝑐(𝐷 𝑝,2).

□

62

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