Generalised knot groups of connect sums of torus knots · Introduction Knot theory is the study of mathematical knots, which are quite di erent from regular knots tied in string

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GENERALISED KNOT GROUPS OF CONNECT SUMSOF TORUS KNOTS

A thesis presented in partial fulfilment of the requirements for the Degreeof

Master of Sciencein

Mathematicsat Massey University, Manawatu,

New Zealand

Howida AL Fran2012

Abstract

Kelly (1990) and Wada (1992) independently identified and defined the generalised knot groups(Gn). The square (SK) and granny (GK) knots are two of the most well-known distinct knotswith isomorphic knot groups. Tuffley (2007) confirmed Lin and Nelson’s (2006) conjecture thatGn(SK) and Gn(GK) were non-isomorphic by showing that they have different numbers ofhomomorphisms to suitably chosen finite groups. He concluded that more information about Kis carried by generalised knot groups than by fundamental knot groups. Soon after, Nelson andNeumann (2008) showed that the 2-generalised knot group distinguishes knots up to reflection.The goal of this study is to show that for certain square and granny knot analogues, thedifference can be detected by counting homomorphisms into a suitable finite groups. This studyextends Tuffley’s work to analogues SKa,b and GKa,b of the square and granny knots formedfrom connect sums of (a, b)-torus knots. It gives further information about the generalised knotgroups of the connect sum of two torus knots, which differ only in their orientation.

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Acknowledgments

First of all, I would like to thank my god, Allah, who gave me the strength and power toaccomplish this thesis.I begin by expressing my humble gratitude and sincere thanks to my thesis supervisor, DrChristopher Tuffley, for his encouragement, guidance and his assistance in the improvement ofthis thesis. No amount of thanks can repay him for his efforts. Your advice, availability todiscuses challenges, and endless encouragement has been invaluable. I hope that one day I willbe a role-model to future mathematicians just as you have been to me.I also would like to thank all the professors and doctors in the mathematics department atMassey University for all the knowledge they gave me.

On a personal level, I would like to acknowledge the on-going support and sacrifice of mychildren throughout this research process. Your curiosity about my work and your indepen-dence have greatly helped me be successful. Thank you for being proud of me, as I too, am soproud of you.To my husband, your support has been essential in helping me achieve my goals. You haveencouraged me not to give up and have been by my side in this and all things.To my family, especially my mother, thank you for your continuing support and belief in meas I studied so far away. You are always in my thoughts.

I also acknowledge the financial support from the Saudi government, which has allowed meto undertake this research in New Zealand. New Zealand has opened my eyes to the person Ican be, and I look forward to advancing my studies further here in the future.

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Contents

Abstract 1

Acknowledgments 2

1 Introduction 71.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Goals of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Fundamental Concepts of Knot Theory 102.1 Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Knot diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Oriented knots and mirror images . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Torus knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 The connect sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Knot Groups 193.1 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Knot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 The presentation of knot groups . . . . . . . . . . . . . . . . . . . . . . . 213.3 Torus knots and the van Kampen theorem . . . . . . . . . . . . . . . . . . . . . 243.4 Expressing the van Kampen generators in terms of the Wirtinger generators . . 263.5 The meridian and the longitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.1 The effect of reflection on the meridian and the longitude . . . . . . . . . 313.5.2 The meridian and the longitude of a connect sum . . . . . . . . . . . . . 31

4 Generalised Knot Groups 334.1 The group Gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 The granny and square knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 The granny and square knot analogues . . . . . . . . . . . . . . . . . . . 36

5 Group Theory 375.1 Dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 The construction of Dp,q;θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3.1 Properties of Dp,q;θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Wreath products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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CONTENTS 4

5.5 The construction of Wh,k;τs,t;% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 The Main Result 476.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 The cycle product and applications . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3.1 The meridian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3.2 The longitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.4 The proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.1 Case 1: Trivial induced maps . . . . . . . . . . . . . . . . . . . . . . . . 536.4.2 Case 2: Nontrivial induced maps . . . . . . . . . . . . . . . . . . . . . . 536.4.3 Realisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Summary 58

List of Figures

2.1 Examples of knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The 51 and 52 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Triple points and tangencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 A knot diagram of 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Multiple diagrams for the trivial and the figure eight knots . . . . . . . . . . . . 122.6 The Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 An oriented trefoil knot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Positive and negative crossings. They are also known as right- and left- handed

crossing respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 A crossing change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.10 An example of a chiral knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.11 An example of an achiral knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.12 Examples of links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.13 A torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.14 The construction of a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.15 A (2, 3)-torus knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.16 The connect sum of two (2, 5)-torus knots . . . . . . . . . . . . . . . . . . . . . 18

3.1 A homotopy from f to f ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 The product f · g of two paths f and g. . . . . . . . . . . . . . . . . . . . . . . . 203.3 Multiplication in the knot group . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 The Wirtinger relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Generator for the knot group of K0. . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Generators for the knot group of a (2, 5)-torus knot. . . . . . . . . . . . . . . . 233.7 The union of two paths-connected spaces with path-connected intersection . . . 253.8 A (2,3)-torus knot on the surface of a torus . . . . . . . . . . . . . . . . . . . . . 263.9 Generators for the knot group of a (3, 5)-torus knot. . . . . . . . . . . . . . . . 273.10 The meridian and longitude of a solid torus. . . . . . . . . . . . . . . . . . . . . 283.11 The meridian and longitude of a trefoil knot. . . . . . . . . . . . . . . . . . . . . 293.12 The longitude of a (2, 2k + 1)-torus knot . . . . . . . . . . . . . . . . . . . . . . 293.13 The figure eight knot and its mirror image . . . . . . . . . . . . . . . . . . . . . 313.14 The meridian and the longitude of a connect sum . . . . . . . . . . . . . . . . . 32

4.1 A Gn relation at a crossing of a diagram . . . . . . . . . . . . . . . . . . . . . . 344.2 The figure eight knot, with labelled generators and relations. . . . . . . . . . . . 344.3 The granny and square knots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 The elements of D3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5

LIST OF FIGURES 6

5.2 A pentagon with labelled vertices realises D5 as a subgroup of S5 . . . . . . . . 395.3 Cycle structure of element of order t = 5 . . . . . . . . . . . . . . . . . . . . . . 445.4 Visualising elements and group operations in the wreath product G oD5. . . . . 46

6.1 Diagram of the cycle of ψ containing f in the case where t = 5. . . . . . . . . . 57

Chapter 1

Introduction

Knot theory is the study of mathematical knots, which are quite different from regular knotstied in string. In mathematics, a piece of rope with a knot tied and the two ends glued togethermakes a mathematician’s knot. The two ends of a knot must be joined together so that itcannot be undone, because in knot theory, the knot must be a continuous loop to allow theknot to be deformed into new knot formations. The essential question in knot theory is howwe prove whether two knots are the same or are different. This is difficult because knots canbe drawn in many different ways by stretching the knot, deforming the knot or twisting theknot; it is still the same knot, but it could look extremely different. The primary concern ofknot theory is to classify and distinguish between knots.

There is a family of methods to distinguish knots which are called invariants. A knot in-variant is a mapping from the set of all knots to some other set, where the value of the mapdoes not change, even if the picture of the knot does. In other words, a knot invariant is atest to distinguish one type of knot from another. One such invariant is the knot group (thefundamental group). This knot group is a powerful invariant; however, it has limitations. Forexample, it cannot distinguish between the square knot (SK) and granny knot (GK), two well-known distinct knots with isomorphic knot groups. Currently, no practical invariant is knownthat will always succeed at distinguishing all knots.

Kelly [13] and Wada [30] independently introduced several methods to define generalisedknot groups (Gn). These may be defined by a Wirtinger presentation (see Section 3.2.1), withconjugation by the generator ai replaced by conjugation by ani for all i. A second definitionof Gn shows that π1(K), which is the fundamental group of a knot (K), is a subgroup ofGn(K) for each n. Generalised knot groups (Gn) were the focus of three key studies, startingwith the conjecture of Lin and Nelson [16] who used granny and square knots to test Gn,continuing to Tuffley [29], who proved Lin and Nelson’s conjecture. In the third study, Nelsonand Neumann [21] took a topological view to show that generalised knot groups distinguishknots up to reflection.

1.1 Previous work

Generalised knot groups were first presented and introduced independently by Wada [30] andKelly [13]. Wada’s work searched for homomorphisms of the braid group Bn into Aut(Fn),

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CHAPTER 1. INTRODUCTION 8

where Fn is the free group on n generators, while Kelly was working with knot racks or quan-dles and the Wirtinger presentation.

Lin and Nelson [16] challenged themselves to show that Gn(GK) and Gn(SK) are notisomorphic for all n > 1. They introduced generalised knot groups through the language ofquandles. They suspected generalised knot groups held additional information about knot typesthat are not present in the usual fundamental group. They used the square knot (SK) and thegranny knot (GK) as a test case. As π1(SK) and π1(GK) are isomorphic, they wanted to checkwhether Gn(GK) and Gn(SK) are isomorphic to each other for n > 1 by using a computerprogram to calculate the number of homomorphisms of Gn(GK) and Gn(SK) into chosen finitegroups H. They did not succeed in detecting a difference, but, nevertheless, conjectured thatGn(GK) and Gn(SK) are not isomorphic to each other for all n ≥ 2.

Tuffley [29] proved Lin and Nelson’s conjecture. He also used the square and granny knotsfor his testing case, distinguishing Gn(GK) and Gn(SK) for all n ≥ 2 by counting the numberof homomorphisms into a chosen finite group. He proved the following theorem:

Theorem 1.1 (Tuffley [29]). For each n ≥ 2, there is a finite group H such that

|Hom(Gn(GK), H)| < |Hom(Gn(SK), H)|.

Consequently, Gn(GK) and Gn(SK) are not isomorphic to each other for all n ≥ 2.

His target groups were wreath products over PSL(2, p),

Hq,rp = Dq,r o PSL(2, p) = (Zr−1q o Zr)P

1(Fp) o PSL(2, p),

where p, q and r are distinct primes and PSL(2, p) acts on P 1(Fp), the projective line over thep-element field. This showed that the isomorphism types of the generalised knot groups carrymore information than the isomorphism types of the fundamental group itself.

Nelson and Neumann [21] showed that, in fact, the 2-generalised knot group distinguishesknots up to reflection. They proved the following theorem from a topological perspective:

Theorem 1.2 (Nelson and Neumann [21]). The 2-generalized knot group G2(K) determinesthe knot up to reflection.

In addition, they sketched the proof that the result also holds for Gn(K) for n > 2.

1.2 Goals of this study

The main goal of this thesis is to show that the difference between generalised knot groups forthe analogues SKa,b and GKa,b of the square and granny knots made from (a, b)-torus knotscan be detected by counting homomorphisms into suitably finite groups. This shows that, inprinciple, the difference between the groups can be detected algorithmically. We will use thewreath product of two different groups of the form Dp,q;θ, which are described in Chapter 5, asthe target groups and generalise Tuffley’s strategy to provide the result.

CHAPTER 1. INTRODUCTION 9

The structure of this thesis is as follows. In Chapter 2, we will give a detailed overviewof knot theory and provide basic definitions and examples of knots. Then, we will outline thefundamental group and knot groups in Chapter 3. In Chapter 4, we will define the generalisedknot groups and look at the Gn for the granny knot and square knot and their analogues.Next, we will provide a concise summary of some of the concepts of group theory which areneeded and related to the study in terms of dihedral groups, semidirect products and wreathproducts in Chapter 5. Then, we will describe the strategy and show our main result of thisstudy which extends Tuffley’s technique by using different target groups in Chapter 6. Finally,we will provide a summary of the thesis in Chapter 7.

Chapter 2

Fundamental Concepts of Knot Theory

In this chapter, we will review some basic concepts that have to do with knot theory. We willgive a number of definitions and examples that will be used throughout the study. Knot theoryis the mathematical study of knots. Mathematical knots are different from normal knots thatwe know and use in our daily lives. In mathematical knots, the two ends are joined together,so the knot cannot be undone. Knots can be defined in many ways; however, simply, a knot isa tangled piece of string in R3 which is a closed loop.

The definitions, theorems and lemmas in this chapter and chapter 3 are taken from: Adams [1],Alexander [2], Burde and Zieschang [5], Cromwell [7], Hatcher [9], Kauffman [11], Kawauchi [12],Lickorish [15], Livingston [17], Manturov [18], Murasugi [20], Reidemeister [22], Rolfsen [23] andStillwell [28]. Most of the pictures in this chapter were created with KnotPlot [26].

2.1 Knots

Definition 2.1. A knot is a non-self-intersecting closed curve in R3.

In other words, a knot is a subset of R3 that is homeomorphic to a circle. Some examplesof knots are shown in Figure 2.1. We now describe when two knots are considered to be thesame.

Definition 2.2. A continuous function F : X × I → X is said to be an isotopy if F (·, t) is ahomeomorphism for all t.

Figure 2.1: Examples of knots. (i) The unknot (trivial knot). (ii) The trefoil knot. (iii) The 52

knot.

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CHAPTER 2. FUNDAMENTAL CONCEPTS OF KNOT THEORY 11

Figure 2.2: (i) and (ii) refer to 51 and 52, the first and second of the two 5-crossing knots.However, this ordering is completely arbitrary, being inherited from the earliest tables compiled.

Two knots K1 and K2 in R3 are equivalent if there is an isotopy of R3 that carries K1 to K2.Such an isotopy is said to be an ambient isotopy from K1 to K2.

Definition 2.3. An unknot (trivial knot) is a knot that is equivalent to the knot in Fig-ure 2.1(i).

Some knots have specific names such as (i) the unknot and (ii) the trefoil knot (Figure 2.1);however, most are referred to by their numbers in the standard tables (Figure 2.2).

2.2 Knot diagrams

In order to study knots, it is useful to draw pictures of them called projections. The problemwith projections is that they do not show over and under crossings, and, therefore, do notcontain enough information to reconstruct the original knot. To solve this problem, we can usewhat is called a knot diagram. We will discuss these concepts in more detail.

Definition 2.4. A knot diagram is a way to picture and manipulate knots by projecting theknot on to a plane, so that the projection has no triple points or tangencies (Figure 2.3). Sucha projection is called a regular projection. During the projection, crossings (double points)may occur. At each crossing, it is important to distinguish the over strand from the understrand by breaking the strand that goes underneath, as shown in Figure 2.4. So by givingthis information, the original knot can be reconstructed. Knots can be represented by manypossible diagrams, as seen in Figure 2.5. That leads to the essential question in knot theory:when do two diagrams represent the same knot?

A regular diagram of a knot has a finite number of crossing points, which is called thecrossing number of the diagram. Now, it is relevant to define the crossing number of knots.

Definition 2.5. The crossing number c(K) of a knot K is the minimal number of crossingsin any diagram of that knot. A minimal diagram of K is one with c(K) crossings.

A knot diagram without any crossing is a trivial knot (unknot). There are no knots withcrossing number one or two. The two trefoils are the only knots with crossing number three.However, there are tables of knots up to about 16 crossings.


Figure 2.3: This picture shows a triple point and a tangency. Any projection of a knot can beperturbed to eliminate triple points and tangencies, so every knot has a regular projection.

Figure 2.4: A knot diagram of 51, where the crossings are circled.

Figure 2.5: (i) and (ii) are two different diagrams of the unknot. (iii) and (iv) present differentdiagrams of the figure eight knot.


Figure 2.6: Pictures of the three types of Reidemeister moves.

2.3 The Reidemeister moves

Kurt Reidemeister (1927) developed what are known as the Reidemeister moves. A Reidemeis-ter move is an operation that can be performed on the diagram of a knot. Simply, there arethree types of Reidemeister moves that can be used to modify a knot diagram. Figure 2.6 showsthe typical pictures that are used to define the Reidemeister moves: type (1) is the additionor removal of a twist; type (2) is the moving of a strand through a tangency; and type (3) isthe moving of a strand through a triple point. These three moves can change one diagram of aknot to another. The three moves can be used together or just one or two moves can be useddepending on what is needed.

The following theorem shows that these three moves plus planar isotopy are enough totransform any diagram of a knot into any other. See Reidemeister [22] for a proof.

Theorem 2.1 (Reidemeister [22]). Two knot diagrams K1 and K2 represent equivalent knotsif and only if there is a sequence of Reidemeister moves taking K1 to K2.

2.4 Oriented knots and mirror images

It is important to consider how a knot is oriented. In this section, we will discuss orientationand mirror images of knots.

Definition 2.6. An oriented knot is one with a chosen direction of circulation along thestring.

Orientation may be specified by putting an arrow somewhere on the knot to designate thedirection (Figure 2.7). Oriented knots are important for many applications in knot theory. Forexample, giving knots orientations allows us to define the sum of oriented knots by taking theconnect sum of the knots as oriented manifolds. Also, the orientation of the knot can be used todetermine whether a crossing is a negative or positive crossing, and this can be easily identifiedusing the right hand rule. We can point the thumb of our right hand along the over-strand in


Figure 2.7: An oriented trefoil knot.

Figure 2.8: Positive and negative crossings. They are also known as right- and left- handedcrossing respectively.

the direction of the arrow and curl our fingers into a letter-C shape. If our fingers curl in thedirection of the under-strand, then it is a positive or right-handed crossing; if our fingers curlagainst the direction, then it is a negative or left-handed crossing. Figure 2.8 pictures how canwe establish the sign of a crossing. Note that reversing the orientation of a knot reverses theorientation of both the over and under strand, and this means that the sign of a crossing doesnot depend on the choice of orientation.

We will define the reverse and mirror image of a knot. These involve changing the orientationof the knot, or of the ambient space R3.

Definition 2.7. The reverse rK of an oriented knot K has the same projection, but with theopposite orientation.

Before defining the mirror image, we will define crossing changes. As we can see in Figure 2.9to change the crossing, we have to change the over-strand to be the under-strand.

Definition 2.8. The mirror image K of the knot K is obtained by reflecting it in a planein R3. If the mirror is placed behind the knot and parallel to the plane of the projection, then

Figure 2.9: A crossing change. The portion of the knot outside the picture is unchanged.


Figure 2.10: The trefoil and its mirror image. The rectangle represents the plane of reflection.

Figure 2.11: This picture shows that the figure eight knot can be deformed into its mirrorimage. (i) The figure eight knot with one arc coloured blue. (ii) Rearrange the blue arc. (iii)Rotate the figure eight 180◦. (iv) Smooth deformation to get the mirror image of the figureeight.

the effect is to change all the crossings. The inverse rK is the composition of the reversal andmirror-image.

It is now relevant to define knot chirality (handedness).

Definition 2.9. A knot is said to be chiral (handed) if it is not equivalent to its mirror image.An example is the trefoil and its mirror image in Figure 2.10. The right-handed trefoil can notbe continuously deformed into the left-handed one, see Dehn [8] or Stillwell [28, pages 218-225].Therefore, the trefoil knot is called topologically chiral. However, if a knot is equivalent toits mirror image, it is called amphichiral or achiral. The figure eight knot is an example ofan amphichiral knot (Figure 2.11).

2.5 Links

In this section, we will briefly introduce links.

Definition 2.10. A link is a collection of disjoint closed curves in R3; each curve is called acomponent of the link.


Figure 2.12: This figure shows different links. (1) is said to be split, because there is a plane inR3 that separates the components. (2) is a two component link, where both of the componentsare unknots. It is called the Hopf link. (3) is also a two component link of unknots. It is calledthe Whitehead link. (4) is a two component link, where one component is 51 and one is anunknot.

Figure 2.13: A torus.

A knot is a one-component link. It is important to notice that individual components mayor may not be unknots. Some examples of links are shown in Figure 2.12.

2.6 Torus knots

There are many types of knots, for example, torus knots, satellite knots, hyperbolic knots andalmost alternating knots. However, torus knots are being used for the purpose of this study,so will be the type outlined in more detail. The main reason for looking at torus knots is theflexibility of the presentation of their knot groups.

The concept of torus can be defined geometrically and represented topologically. From ageometric perspective, a torus is a surface of revolution generated by revolving a circle in R3

about an axis coplanar with the circle. Most of the time it is assumed that the axis does nottouch the circle, resulting in a ring shape called a ring torus. Topologically, a ring torus ishomeomorphic to the product of two circles: S1 × S1.

Definition 2.11. A torus is a topological product of two circles (Figure 2.13).

Simply, a torus is a surface similar to that of a doughnut. It is a connected surface or shape.A torus can be constructed from a rectangle by gluing both pairs of opposite edges togetherwith no twists (Figure 2.14).

Definition 2.12. A torus knot is a knot that lies on the surface of an unknotted torus in R3.


Figure 2.14: (i) is a rectangle. (ii) is after gluing the opposite vertical edges together. (iii) isthe final shape of a torus after gluing the horizontal edges together with no twists.

Figure 2.15: An example of a (2, 3)-torus knot (trefoil).

Every torus knot can be described with two integers (p, q), where p and q are co-prime. A(p, q)-torus knot is a simple closed curve on the torus that winds p times around the first S1

factor and q times around the second. Let us denote this torus knot by Tp,q. The simplestnontrivial example of two co-prime integers is p = 2, q = 3 or p = 3, q = 2. In both of thesecases, we obtain the trefoil knot which can be written as (2, 3)-torus knot or T2,3 (Figure 2.15).If p and q are not relatively prime, then there will be a torus link with more than one component.

The following remark shows some properties of torus knots.

Remark 2.1. A (p,−q)-torus knot is the mirror image of a (p, q)-torus knot. A (−p,−q)-torusknot is equivalent to a (p, q)-torus knot. Also, a (p, q)-torus knot is equivalent to a (q, p)-torusknot.

2.7 The connect sum

The connect sum or knot sum is an operation that connects two knots together to make a singleknot.

Definition 2.13. Two oriented knots K1 and K2 can be connected by breaking the two knotsand joining them with straight bars, so that the orientation of both knots is retained (Fig-ure 2.16). The operation is called a connect sum and is denoted by #.


Figure 2.16: The knots in (A) are (2,5)-torus knots (the knot 51) that have the same handedness.(B) shows where the knots are broken to connect them together and (C) is a picture of theconnect sum of the two knots 51#51. This knot is analogous to the granny knot, which is theconnect sum 31#31 of two trefoils with the same handedness.

Remark 2.2. A knot can be connected with another copy of itself or its inverse or any otherknot.

Chapter 3

Knot Groups

3.1 The fundamental group

In order to develop a knot group, it is important to first understand key definitions related tothe fundamental group. The information in this section is taken from Hatcher [9].

Definition 3.1. Let X and Y be topological spaces. Let f and f ′ be continuous maps fromX into Y . Then f is homotopic to f ′ if there is a continuous map F : X× [0, 1]→ Y such that

F (x, 0) = f(x) and F (x, 1) = f ′(x).

F is called a homotopy between f and f ′.

In other words, two functions are homotopic if one can be continuously deformed into theother.

Definition 3.2. Let x0, x1 ∈ X. A path from x0 to x1 is a continuous map f : [0, 1] → Xsuch that

f(0) = x0, f(1) = x1.

A loop is a path that begins and ends at the same point.

Definition 3.3. Let f and f ′ be paths with fixed end points x0 and x1. The paths f and f ′

are homotopic rel endpoints if there is a continuous map F : [0, 1]× [0, 1]→ X such that

F (s, 0) = f(s), F (0, t) = x0,

F (s, 1) = f ′(s), F (1, t) = x1.

Figure 3.1 shows a homotopy from f to f ′. Homotopy rel endpoints is an equivalence relationon paths. We will write [f ] for the homotopy class of f .

Next, we will define a product operation on paths.

Definition 3.4. Let f be a path in X from x0 to x1 and let g be a path in X from x1 to x2.Then we can define the product f · g as the following;

(f · g)(s) =

{f(2s), 0 ≤ s ≤ 1

2

g(2s− 1), 12≤ s ≤ 1.

19

CHAPTER 3. KNOT GROUPS 20

Figure 3.1: A picture of a homotopy from f to f ′.

Figure 3.2: The product f · g of two paths f and g.

Geometrically, f · g is the concatenation of the paths f and g: in other words, it is the pathfrom x0 to x2 that follows f from x0 to x1, then g from x1 to x2 (Figure 3.2).

The product operation on paths is a well-defined operation on homotopy classes, definedby:

[f ] · [g] = [f · g]. (3.1)

Theorem 3.1 (Hatcher [9]). Let X be a topological space and let x0 be a point of X. The setof all homotopy classes of loops f based at x0 is a group under the operation (3.1). It is calledthe fundamental group of X relative to x0 and is denoted by π1(X, x0).

The following example illustrates the fact that the fundamental group of the circle S1 isisomorphic to the additive group of the integers Z.

Example 3.1. LetS1 = {x ∈ R2 : ‖x‖ = 1} = {z ∈ C : ‖z‖ = 1},

and use 1 as the base point.

Every loop is equivalent to a unique path of the form an(t) = e2nπit, for some n ∈ Z. Thereare three cases;

n > 0 (an(t) is a path that goes n times around S1 in the counterclockwise direction and returnsto 1),


n < 0 (an(t) is a path that goes n times around S1 in the clockwise direction and returns to1),

n = 0 (an(t) is the path that stays at 1).

Let n > 0 and m > 0. The path an travels n times around S1 and am travels m timesaround S1. Then an · am is a path that loops n times around S1 and then loops m more times.This is equivalent to the loop an+m. The equation [an · am] = [an+m] holds even when n,m arenot necessarily positive, and therefore π1(S

1, 1) ∼= Z.

Although the base point is required for the definition, the following theorem shows that theresulting group typically does not depend on the choice of base point:

Theorem 3.2 (Hatcher [9, page 28]). Let X be path-connected and let x0 and x1 be two pointsof X. Then π1(X, x0) is isomorphic to π1(X, x1).

The group π1(X, x0) can be written as π1(X), if X is path-connected and π1(X, x0) is upto isomorphism, independent of the choice of x0.

Generally, if a space is path-connected and has a trivial fundamental group, then it is calledsimply connected.

We conclude this section with the following theorem which shows the fundamental group ofa product of two path-connected spaces.

Theorem 3.3 (Hatcher [9, page 35]). Let X and Y be path-connected. Then

π1(X × Y ) ∼= π1(X)× π1(Y ).

The torus is an example of the theorem above. The fundamental group of the torus is justthe direct product of the fundamental group of the circle with itself:

π1(S1 × S1) = π1(S

1)× π1(S1) ∼= Z× Z = Z2.

3.2 Knot groups

In this section we will define the knot group and explain the presentation of the knot group.

Definition 3.5. Let K be a knot in R3. The fundamental group of the complement of K iscalled the knot group of K, and is denoted by π1(K).

Two elements a and b of the knot group π1(K) are paths in the complement of the knot Kthat begin and end at a fixed base point x0, up to deformation, and multiplied by concatenation.We can give the formula, but the picture is more important (Figure 3.3).

3.2.1 The presentation of knot groups

Wilhelm Wirtinger (1925) proved that for every diagram of a knot there is a presentation ofthe knot group that has one generator for each arc of the knot diagram and a relation for eachcrossing. In an oriented knot, the generators start at a base point above the plane of the knotdiagram, go around the specified arc in the positive direction, and return to the base point. The


Figure 3.3: A product ab of two elements a and b of the knot group can be calculated by passingfirst along a and then along b.

Figure 3.4: The Wirtinger relations at left- and right-handed crossings.

positive direction is determined by the right hand rule: if we hold the strand in our right handwith our thumb pointing in the direction of the orientation of the knot, then our fingers curl inthe positive direction. We get the following relation at each crossing, as shown in Figure 3.4:

a = bcb−1 (left-handed crossing),

a = b−1cb (right-handed crossing).

This gives a presentation which is called the Wirtinger presentation of the knot group.

A diagram with m ≥ 2 crossings will have m arcs. Then G will have the following presen-tation:

G = 〈g1, g2, g3, . . . , gm|r1, r2, r3, . . . , rm〉,

where G is the quotient of the free group on the Wirtinger generators g1, g2, g3, . . . , gm bythe smallest normal subgroup generated by the Wirtinger relations r1, r2, r3, . . . , rm. We cansee that these relations hold; the significance of Wirtinger’s work is that no other relationsare required. This is the content of following theorem, which is proved using van Kampen’stheorem; see Rolfsen [23] for the proof.

Theorem 3.4. Let K be a knot. Then the fundamental group π1(K) is generated by g1, . . . , gmand has presentation

π1(K) = 〈g1, . . . , gm|r1, . . . , rm〉.


Figure 3.5: Generator for the knot group of K0.

Figure 3.6: Generators for the knot group of a (2, 5)-torus knot.

Moreover, any one relation ri can be eliminated and the presentation of π1(K) still holds.

Abelianising the Wirtinger presentation makes all generators equal, leading to the followingtheorem.

Theorem 3.5. The abelianisation of π1(K) is isomorphic to Z.

The following example shows that the fundamental group of the trivial knot is isomorphicto Z.

Example 3.2. The trivial knot K0 has a diagram with one arc and no crossings. It has onegenerator a and no relations (Figure 3.5). So the knot group is given by

π1(K0) ∼= 〈a〉 ∼= Z.

We will show the Wirtinger presentation of (5, 2)-torus knot in the next example.

Example 3.3. Consider the (5, 2)-torus knot (Figure 3.6). Let a, b, c, d, e be the group gener-ators. Then the Wirtinger presentation gives

〈a, b, c, d, e | a = dbd−1, b = ece−1, c = ada−1, d = beb−1, e = cac−1〉.

We can use the relations a = dbd−1, b = ece−1, c = ada−1, d = beb−1 and e = cac−1 to write

a = dbd−1 ⇔ ad = db,

c = ada−1 ⇔ ca = ad,

b = ece−1 ⇔ be = ec,

d = beb−1 ⇔ db = be,

e = cac−1 ⇔ ec = ca.


We can rewrite the presentation as follows

〈a, b, c, d, e | ad = db = be = ec = ca〉.

Then we can rewrite ca = ec by using the above expressions for c, a and d to get

(ada−1)a = e(ada−1) (substitute c with ada−1)

ad = eada−1

ada = ead (multiply the right side by a)

(dbd−1)d(dbd−1) = e(dbd−1)d (substitute a with dbd−1)

dbdbd−1 = edb

dbdb = edbd (multiply the right side by d)

(beb−1)b(beb−1)b = e(beb−1)b(beb−1) (substitute d with beb−1)

bebe = ebebeb−1.

Then we rearrange bebe = ebebeb−1 to get bebeb = ebebe, and we also can express the otherrelations in terms of b and e in a similar way. Each relation leads to the same or a trivialrelation in terms of b and e, and we find

π1(T2,5) ∼= 〈b, e|bebeb = ebebe〉.

We can go from 〈b, e|bebeb = ebebe〉 to 〈x, y|x2 = y5〉 by writing x = bebeb and y = be, becauseb and e can also be written in terms of x and y as b = y−2x and e = b−1y = x−1y2y = x−1y3.

In the following section, we will derive the presentation for the torus knot π1(Ta,b) ∼=〈x, y|xa = yb〉 by using the van Kampen theorem.

3.3 Torus knots and the van Kampen theorem

In this section, we will compute the knot group of a torus knot by using the van Kampen the-orem (see Hatcher [9, page 43]). We will briefly give an overview of the van Kampen theorem.

The van Kampen theorem expresses the fundamental group of a path-connected unionA1 ∪A2 in terms of the fundamental groups of the path-connected spaces A1, A2 and A1 ∩A2.As seen in Figure 3.7, every element of A1 ∪ A2 can be expressed as a product of loops in A1

and A2. Consequently, π1(A1 ∪ A2) is generated by the generators of π1(A1) and π1(A2), andthe relations for these groups hold in π1(A1 ∪ A2) also. We get an additional relation for eachgenerator of π1(A1∩A2) reflecting the fact that it can be expressed as a word in the generatorsof each of π1(A1) and π1(A2).

The method for computing the fundamental group of the torus knot is as follows. Let Ta,bbe an (a, b)-torus knot on a standard torus T in S3. The torus T divides S3 into two regionsR1 and R2, such that the closures R1 and R2 are solid tori. If we set Ai to be Ri − Ta,b, thenS3\Ta,b can be written as A1 ∪ A2.


Figure 3.7: Given a loop a, break it into paths a1 . . . ak entirely in A1 or in A2; join the basepoint to the end of ai by a path pi:

a = a1a2a3a4 w a1p−11︸︷︷︸

loop in A1

p1a2p−12︸︷︷︸

loop in A2

p2a3p−13︸︷︷︸

loop in A1

p3a4︸︷︷︸loop in A2

,

which means that any loop in A1 ∪ A2 can be expressed as a product of loops in A1 and A2.

The intersection of T with the complement of Ta,b is a ribbon which turns around the torusa times in the direction of one factor and b times in the direction of the other. Let C denotethis ribbon and let x0 ∈ A1 ∩ A2 be a base point. The space A1 is essentially a solid torus, soits fundamental group is π1(A1) ∼= Z with the single generator x, corresponding to a path thatmakes one circle around the hole in the torus. The space A2 is also essentially a solid torus andits fundamental group is π1(A2) ∼= Z, with the single generator y corresponding to a path thatpasses once through the hole. The intersection A1 ∩ A2 = C is an annulus; the fundamentalgroup is also π1(C) ∼= Z with one generator c representing the path which travels once aroundthe annulus. From A1 and A2’s points of view, c passes a and b times respectively around thetorus, so it represents xa and yb respectively. Then, π(A1 ∪ A2) = Z ∗Z Z = 〈x, y|xa = yb〉.

We summarise the above result in the following theorem:

Theorem 3.6. The fundamental group π1 of Ta,b has the presentation 〈x, y|xa = yb〉.

Figure 3.8 shows a (2,3)-torus knot embedded on the surface of a torus. According to theabove discussion the fundamental group is π1(T2,3) = 〈x, y|x2 = y3〉.

We will define Ga,b as follows.

Definition 3.6. The group Ga,b is the knot group of (Ta,b) generated by x and y, with onerelation xa = yb.

The following theorem states that the knot groups of torus knots have centres. In fact, torusknots are the only knots whose knot groups have a nontrivial centre (see Stillwell [28]).

Theorem 3.7. The centre of the knot group of the (a, b)-torus knot is infinite cyclic generatedby xa = yb.


Figure 3.8: A (2,3)-torus knot on the surface of a torus. (i) and (ii) show that this knot cycles2 times around the blue curve of the torus and 3 times around the green curve. (iii) shows thegenerator c with red colour which turns once around the annulus.

3.4 Expressing the van Kampen generators in terms of

the Wirtinger generators

In the previous section we found that π1(Ta,b) = 〈x, y|xa = yb〉. In this section, we will findexpressions for the generators x and y in terms of Wirtinger generators.

Figure 3.9 shows a (3,5)-torus knot with Wirtinger generators ωi, i = 0, 1, 2, 3, 4. Any otherWirtinger generator can be expressed in terms of these; for instance ω′ = ω−10 ω4ω0. Let x bethe green core curve of the inner torus, and let y be the red core curve of the outer torus whichcan move around the diagram as shown. Then x and y can be written as

x = ω0ω1ω2ω3ω4,

y = ω0ω1ω2 = ω2ω3ω4 = ω4ω0ω1 = ω1ω2ω3 = ω3ω4ω0.

Then we can check that x3 = y5 as follows:

(ω0ω1ω2ω3ω4)3 = (ω0ω1ω2ω3ω4)(ω0ω1ω2ω3ω4)(ω0ω1ω2ω3ω4)

= (ω0ω1ω2)(ω3ω4ω0)(ω1ω2ω3)(ω4ω0ω1)(ω2ω3ω4)

= y5.

So x and y are generators giving the presentation

G3,5 = 〈x, y|x3 = y5〉,

which is the knot group of T3,5.

In general, for a, b relatively prime satisfying 0 < a < b, Ga,b has Wirtinger generatorsωi, i = 0, . . . , b− 1, and we may write x and y as follows

x = ω0 · · ·ωb−1,y = ω0 · · ·ωa−1 = ω1 · · ·ωa = · · · = ωb−1ω0 · · ·ωa−2.


Figure 3.9: Generators for a (3, 5)-torus knot.


Figure 3.10: The meridian and longitude of a solid torus.

3.5 The meridian and the longitude

In this section, we will define the meridian and longitude of a solid torus and give an expla-nation for the meridian and the longitude of a knot group. Also, we will briefly explain thepreferred longitude of a knot. Finally, we will identify the meridian of Ga,b in terms of x and y.

Let K be a knot, then thicken the knot to a solid torus L = S1 ×D2, where S1 is a circleand D2 is a disc (Figure 3.10). A simple closed curve on ∂D that bounds a disc in L is calleda meridian (µ),

µ = {x0} × ∂D.In other words, it is a simple closed curve encircling the width of L. A longitude (λ) is asimple closed curve that runs the entire length of the knot which follows the same orientationas the knot,

λ = S1 × {x0},where x0 ∈ ∂D.

Any two meridians are equivalent, because they can be slid around the knot so that they co-incide. However, two longitudes are not necessarily equivalent, because they can twist differentnumbers of times around the knot. The preferred longitude is the one that represents zero inthe abelianization of the π1(K). Figure 3.11 shows the meridian and a longitude of a trefoil knot.

The meridian and a longitude are oriented simple closed curves that intersect at one point.We can think of the meridian µ and a longitude λ as elements of the knot group π1(K) bychoosing a path from the base point to the point of intersection. They commute with eachother in every knot group, because they give a subgroup isomorphic to π(S1 × S1) = Z⊕ Z.

Remark 3.1. Any of the generators in the Wirtinger presentation can be used as a meridian ofthe knot. Consequently, Theorem 3.5 shows that the meridian generates the abelianization ofπ1(K); and it can be seen from the Wirtinger presentation that the meridian normally generatesπ1(K), since the remaining generators are conjugates of the chosen meridian.

The following example shows the way to find the meridian and the longitude of G2,2k+1.


Figure 3.11: The meridian and longitude of a trefoil knot.

Figure 3.12: Calculation of the longitude which begins and ends at the point a1.

Example 3.4. Let G2,2k+1 = 〈a1, . . . , a2k+1|aiai+1 = ai+1ai+2, ∀i〉 = 〈x, y|x2 = y2k+1〉. Let

x = a1a2a3 · · · a2ka2k+1,

y = a1a2 = a2a3 = · · · = a2ka2k+1 = a2k+1a1.

Then

x = a1(a2a3) · · · (a2ka2k+1),

x = a1yk =⇒ a1 = xy−k,

therefore µ = xy−k = a1. As we can see in Figure 3.12, the longitude can be taken as

λ = a2a4 · · · a2ka1a3 · · · a2k+1.

The preferred longitude is λa−(2k+1)1 which gives 0 in the abelianisation of the knot. Another

choice of longitude is obtained by starting at a1 and following the knot, twisting once aroundthe knot in the positive direction before each of the crossings, then

λ′ = a1a2 · · · a2k+1a1a2 · · · a2k+1.

Therefore, the longitude may be taken to be λ′ = x2 = y2k+1.

The following lemma shows the meridian of Ga,b, where a and b are integers and co-prime.

Lemma 3.1. Let Ga,b = 〈x, y|xa = yb〉 such that 0 < a < b, and let c and d be elements of Nsuch that bc−ad = 1. Then µ = xcy−d = ω0 is a meridian of Ta,b, with corresponding longitudeλ = xa = yb.


Note that we always can find a solution with c, d > 0.

Proof. Let

x = ω0ω1 · · ·ωb−1︸︷︷︸b

(which is of length b),

y = ω0ω1 · · ·ωa−1︸︷︷︸a

(which is of length a),

such that xa = yb. There are b representations of y of length a such that

y = ω0 · · ·ωa−1︸︷︷︸a

= ω1 · · ·ωa︸︷︷︸a

= ω2 · · ·ωa+1︸︷︷︸a

= · · · = ωb−1ω0 · · ·ωa−2︸︷︷︸a

.

Choose c and d ∈ N, such that bc− ad = 1. Then

xcy−d = (ω0ω1 · · ·ωb−1)cy−d

= (ω0ω1 · · ·ωb−1) · · · (ω0ω1 · · ·ωb−1)︸︷︷︸c times

y−d = (ω0 ω1 · · ·ωb−1 · · ·ω0ω1 · · ·ωb−1︸︷︷︸ad factors

)y−d.

By using the fact that y = ωiωi+1 · · ·ωi+a−1 for any i, we see that the product ω1 · · ·ωb−1 withad factors is equal to yd. Then

xcy−d = ω0ydy−d = ω0,

which is a meridian of the knot. Then we may take µ = xcy−d = ω0, as required.

We end this section with an example that illustrates the above lemma.

Example 3.5. Let G5,7 = 〈x, y|x5 = y7〉. Then

x = ω0ω1ω2ω3ω4ω5ω6,

and

y = ω1ω2ω3ω4ω5 = ω2ω3ω4ω5ω6 = ω3ω4ω5ω6ω0 = ω4ω5ω6ω0ω1

= ω5ω6ω0ω1ω2 = ω6ω0ω1ω2ω3 = ω0ω1ω2ω3ω4,

such that x5 = y7. We need to solve the equation bc− ad = 1, one solution is c = 3 and d = 4.Then

x3y−4 = (ω0ω1ω2ω3ω4ω5ω6)(ω0ω1ω2ω3ω4ω5ω6)(ω0ω1ω2ω3ω4ω5ω6)(ω1ω2ω3ω4ω5)−4,

= (ω0 ω1ω2ω3ω4ω5ω6ω0ω1ω2ω3ω4ω5ω6ω0ω1ω2ω3ω4ω5ω6︸︷︷︸y4

)(ω1ω2ω3ω4ω5)−4.

Then we get

x3y−4 = ω0y4y−4 = ω0,

µ = x3y−4 = ω0.


Figure 3.13: Figure eight knot and its mirror image. The rectangle represents the plane ofreflection.

3.5.1 The effect of reflection on the meridian and the longitude

In this section we will show by example that a knot and its mirror image have the same knotgroup and meridian, but inverted longitudes.

The figure eight knot has the following presentation

G = 〈ω0, ω3|ω0ω3ω−10 ω3ω0 = ω3ω0ω

−13 ω0ω3〉.

Figure 3.13 shows the figure eight knot and its mirror image. By reflecting in the givenplane, we can choose generators for the two knot groups, so that we get identical presentations.However, to do so we must orient the two knots as shown, so that the generators go around eachknot in the positive direction. The 0-framed longitude corresponding to ω0 of the figure eightis represented by λr = ω−13 ω0ω

−11 ω2, and the longitude of the mirror image of the figure eight is

represented by λl = ω−12 ω1ω−10 ω3. Note that ω−13 ω0ω

−11 ω2 and (ω−13 ω0ω

−11 ω2)

−1 = ω−12 ω1ω−10 ω3

represent longitudes corresponding to ω0. Both the figure eight and its mirror image have thesame meridian µ = ω0.

3.5.2 The meridian and the longitude of a connect sum

In this section we will demonstrate the meridian and the longitude of a connect sum in thefollowing remark.

Remark 3.2. Let K1 and K2 be knots, and let K1#K2 be the connect sum of these knots.Let µ1 and λ1 be the meridian and the longitude of K1 and let µ2 and λ2 be the meridianand the longitude of K2. The meridian (µ) of the connect sum will be exactly the same as themeridian of K1 and K2, therefore µ = µ1 = µ2. The longitude of the connect sum (λ) will bethe multiplication of the longitude of K1 and that of K2, λ = λ1 × λ2. Figure 3.14 shows themeridian and the longitude of the connect sum of two trefoil knots.

The following theorem calculates the presentation of the knot group of the connect sum oftwo knots, and the meridian and a longitude.

Theorem 3.8. Let

π1(K1) = 〈g1, g2, . . . , gl|r1, r2, . . . , rm〉, and π1(K2) = 〈h1, h2, . . . , hs|q1, q2, . . . , qt〉,


Figure 3.14: (i) shows the meridian and the longitude of two trefoil knots K1 and K2. (ii)shows the meridian and the longitude of the connect sum of K1 and K2.

with meridians µ1 and µ2 and longitudes λ1 and λ2 respectively. Then

π1(K1#K2) = 〈g1, g2, . . . , gl, h1, h2, . . . , hs|r1, r2, . . . , rm, q1, q2, . . . , qt, µ1 = µ2〉,

with meridian µ = µ1 = µ2 and longitude λ = λ1λ2.

Chapter 4

Generalised Knot Groups

In this chapter, we will define and explain the generalised knot groups Gn, which were indepen-dently defined by Wada [30] and Kelly [13]. Also, we will define the granny knot (GK) and thesquare knot (SK) and look at π1 for both of them. Then we will obtain Gn for the granny knotanalogues (GKa,b) and the square analogues (SKa,b) from π1(GKa,b) and π1(SKa,b) respectively.

4.1 The group Gn

Assuming K is a knot, the generalised knot groups Gn(K) can be defined in many differentways. We may define them by the Wirtinger presentation. The presentation of Gn(K) has agenerator for each arc of the knot diagram; and a relation ak = ani aja

−ni at a left- handed cross-

ing and ak = a−ni ajani at a right-handed crossing (Figure 4.1). Observe that if we substitute n

with 1, then we will get G1 = π1. The group Gn can be shown to be a knot invariant using theReidemeister moves.

If we set Ai = ani for all i, then we recover the Wirtinger presentation of π1, so we canthink of ai as an nth root of the corresponding meridian. We can show that ai commutes withthe corresponding longitude by using the Wirtinger presentation, as shown below for the figureeight knot in Example 4.1. Moreover, the Gn crossing relations can be recovered from this andthe π1 relations, leading to the following presentation: if π1(K) has presentation

π1(K) = 〈g1, g2, g3, . . . , gp|r1, r2, r3, . . . , rq〉,

and µ and λ are words in the generators representing the meridian and the longitude, thenGn(K) has a presentation

Gn(K) = 〈g1, g2, g3, . . . , gp, ν|r1, r2, r3, . . . , rq, νn = µ, λν = νλ〉.

Therefore, Gn(K) is obtained from π1 by adding a new generator ν which is an nth rootof the meridian and commutes with the corresponding longitude. This presentation can alsobe obtained directly using the van Kampen theorem from a topological definition of Gn (seeWada [30]).

The following example gives an explanation for the above discussion.

33

CHAPTER 4. GENERALISED KNOT GROUPS 34

Figure 4.1: A Gn relation at a crossing of a diagram

Figure 4.2: The figure eight knot, with labelled generators and relations.

Example 4.1. Let K be a figure eight knot (Figure 4.2). Then the generalised knot groupGn(K) has generators ai and relations γi for i = 1, 2, 3, 4 as follows:

γ1 : a1 = a−n4 a3an4

γ2 : a2 = an3a1a−n3

γ3 : a3 = an2a4a−n2

γ4 : a4 = a−n1 a2an1 .

By setting Ai = ani for i = 1, 2, 3, 4, then we can see that the Ai satisfy the knot group π1(K)relations:

γ′1 : A1 = A−14 A3A4

γ′2 : A2 = A3A1A−13

γ′3 : A3 = A2A4A−12

γ′4 : A4 = A−11 A2A1.

Now, we will show that the generator a1 commutes with the longitude λ = A−13 A1A−12 A4 as


follows:

a1λ = a1A−13 A1A

−12 A4

= A−13 a2A1A−12 A4

= A−13 A1a4A−12 A4

= A−13 A1A−12 a3A4

= A−13 A1A−12 A4a1

= λa1.

The generalised knot group relations can be written in terms of the generator a1 and thegenerators of π1(K) as we now show:

γ2 : a2 = A3a1A−13

γ3 : a3 = A2A−11 A3a1A

−13 A1A

−12

γ4 : a4 = A−11 A3a1A−13 A1.

Therefore the Wirtinger presentation of the group Gn can be written as follows:

Gn = 〈A1, A2, A3, A4, ν|γ′1, γ′2, γ′3, γ′4, νn = A1, νA−13 A1A

−12 A4 = A−13 A1A

−12 A4ν〉,

where ν is an nth root of the meridian A1 and commutes with the corresponding longitudeλ = A−13 A1A

−12 A4.

4.2 The granny and square knots

The granny and square knots are both connected sums of trefoil knots. So we can define themas follows:

Definition 4.1. The granny knot is the connect sum of two left- or two right-handed trefoilknots, while the connect sum of two knots with one left- and one right-handed trefoil knot isthe square knot (see Figure 4.3).

Figure 4.3: The granny and square knots.

Let K1 and K2 be two right-handed trefoils. If π1(K1) ∼= 〈x, y|x3 = y2〉, with µ1 = yx−1 andλ1 = x3; and π1(K2) ∼= 〈w, z|w3 = z2〉, with µ2 = zw−1 and λ2 = w3, then π1 for the grannyknot is

π1(GK) ∼= 〈x, y, w, z|x3 = y2, w3 = z2, yx−1 = zw−1〉,


which has meridian and longitude µ = µ1 = µ2 = yx−1 = zw−1, λ = x3w3. Similarly, let K1 andK2 be right- and left-handed trefoils respectively. If π1(K1) ∼= 〈x, y|x3 = y2〉, with µ1 = yx−1

and λ1 = x3; and π1(K2) ∼= 〈w, z|w3 = z2〉, with µ2 = zw−1 and λ2 = w−3, then π1 for thesquare knot is

π1(SK) ∼= 〈x, y, w, z|x3 = y2, w3 = z2, yx−1 = zw−1〉,

which has meridian and longitude µ = µ1 = µ2 = yx−1 = zw−1, λ = x3w−3. Thus, π1(GK) andπ1(SK) are both isomorphic to

〈x, y, w, z|x3 = y2, w3 = z2, yx−1 = zw−1〉.

Remark 4.1. The fundamental groups for the granny and square knots are both isomorphic:

π1(K1#K2) ∼= π1(K1#K2).

In other words, they have the same π1.

The presentation for Gn for GK and SK is obtained from π1 by adding a new generator νwhich is an nth root of µ as follows:

Gn(GK) ∼= 〈x, y, w, z, ν|x3 = y2, w3 = z2, νn = yx−1 = zw−1, x3w3ν = νx3w3〉,Gn(SK) ∼= 〈x, y, w, z, ν|x3 = y2, w3 = z2, νn = yx−1 = zw−1, x3w−3ν = νx3w−3〉.

We can see that the difference between the presentations for Gn(GK) and Gn(SK) is just inthe last relation, which states that the longitude commutes with an nth root of the meridian.

4.2.1 The granny and square knot analogues

The knots GKa,b and SKa,b are analogues of square and granny knots built from (a, b)-torusknots, where a and b are co-prime. We define GKa,b to be the connect sum of two right-handed(a, b)-torus knots or two left-handed (a, b)-torus knots, and SKa,b to be the connect sum of aright-handed (a, b)-torus knot and a left-handed (a, b)-torus knot. Recall from Lemma 3.1 thatthe meridian of Ga,b is equal to xcy−d, where c and d ∈ N are a solution to bc − ad = 1. Wecan write the presentations for π1(GKa,b) and π1(SKa,b) as follows:

π1(GKa,b) ∼= 〈x, y, w, z|xa = yb, wa = zb, xcy−d = wcz−d〉,π1(SKa,b) ∼= 〈x, y, w, z|xa = yb, wa = zb, xcy−d = wcz−d〉,

where both have the same meridian xcy−d = wcz−d, but different longitudes λGKa,b = xawa

and λSKa,b = xaw−a. As we can see both of GKa,b and SKa,b have the same knot group. Thepresentation for Gn(GKa,b) and Gn(SKa,b) can be obtained in exactly the same way as thosefor Gn(GK) and Gn(SK):

Gn(GKa,b) ∼= 〈x, y, w, z, ν|xa = yb, wa = zb, νn = xcy−d = wcz−d, xawaν = νxawa〉,Gn(SKa,b) ∼= 〈x, y, w, z, ν|xa = yb, wa = zb, νn = xcy−d = wcz−d, xaw−aν = νxaw−a〉.

By Nelson and Neumann [21], we know that Gn(GKa,b) is not isomorphic to Gn(SKa,b). Thegoal of this study is to show that the difference between these groups can be detected by countinghomomorphisms into suitably chosen finite groups. We will give the proof in Chapter 6.

Chapter 5

Group Theory

In this chapter, we will narrow our focus to the types of groups that will appear in our study.We will briefly introduce these groups: dihedral groups, direct products, semidirect productsand wreath products.

The definitions and theorems in this chapter are taken from: Aschbacher [3], Bogopol-ski [4], Chirikjian [6], Humphreys [10], Ledermann [14], Meldrum [19], Rose [24], Rotman [25],Scott [27] and Tuffley [29].

5.1 Dihedral groups

In this section, we will outline the dihedral groups with some examples.

Definition 5.1. For n ≥ 3 the dihedral group Dn is the group of symmetries of a regularpolygon with n sides. For n = 1 or n = 2, Dn is the group of symmetries of an interval orrectangle respectively. The order of the dihedral group Dn is 2n, for n ≥ 1.

There are two types of symmetries of the n-gon; exactly half of them are rotations and theother half are reflections as follows:

Rotations ρ0, ρ 2πn, ρ 4π

n, . . . , ρ 2(n−1)π

n

, where ρθ is an anti-clockwise rotation of angle θ.

Reflections µ0, µπn, µ 2π

n, . . . , µ (n−1)π

n

, where µθ is reflection about the line through the origin

meeting the vertical axis with an angle θ.

The dihedral group is generated by a rotation through 2π/n and a reflection. The dihedralgroup Dn has the presentation 〈a, b|bn = a2 = e, aba−1 = b−1〉, where a represents a reflectionand b a rotation.

We will clarify the above discussion in the following example.

Example 5.1. Let ρ0, ρ 2π3

and ρ 4π3

be the rotations of D3 and let µ0, µπ3

and µ 2π3

be thereflections of the group. Then the group of symmetries of the triangle can be written as

D3 = {ρ0, ρ 2π3, ρ 4π

3, µ0, µπ

3, µ 2π

3}.

As we can see in Figure 5.1 rotating the triangle gives three symmetries: ρ0 is the identityrotation, ρ 2π

3is the 120◦ anti-clockwise rotation and ρ 4π

3is the 240◦ anti-clockwise rotation.

37

CHAPTER 5. GROUP THEORY 38

Figure 5.1: The elements of D3, the symmetry group of an equilateral triangle. (i) is a diagramof the three rotations. (ii) is a diagram of the three reflections of D3.

The reflections across the diagonal and vertical lines of symmetry give three more symmetrieswhich are µ0, the reflection through the vertical line; µπ

3, the reflection through the diagonal

running northwest to southeast; and µ 2π3

, the reflection through the diagonal running northeastto southwest.

We will conclude this section with the following remark which shows Dn may be realised asa subgroup of Sn.

Remark 5.1. By labeling the vertices of the regular polygon with 0, 1, . . . , n − 1, Dn can berealised as a subgroup of Sn. It is generated by the rotation ρ(i) = i + 1 and the reflectionσ(0) = 0, σ(i) = n − i, for 1 6 i 6 n − 1. Also, both ρ and σ can be written in cycle form asfollows:

ρ = (0 1 . . . n− 1)

σ = (0)(1 n− 1) · · · (bn/2c dn/2e).

Note that (bn/2c dn/2e) represents the two cycle (k k + 1) when n = 2k + 1 is odd, and theone cycle (k) when n = 2k is even.

Let us illustrate the above remark by an example.

Example 5.2. The group S5 has a subgroup that is isomorphic to D5 (Figure 5.2), which isgenerated by ρ and σ, where

ρ = (0 1 2 3 4),

σ = (0)(1 4)(2 3).


Figure 5.2: A pentagon with labelled vertices which realises D5 as a subgroup of S5.

5.2 Semidirect products

In this section, we will describe the semidirect product and some of its properties. To be ableto define the semidirect product, the direct product needs to be defined first.

Definition 5.2. Let G be a group. Then G is to said to be a direct product of two groupsH and K if and only if they are normal subgroups of G such that H ∩K = {e} and G = HK.

Definition 5.3. G is said to be the semidirect product of H and K if and only if H and Kare subgroups of G, which satisfy the following conditions:

H C G,

H ∩K = {e},HK = G.

LetG = HK, whereH andK are subgroups ofG and satisfy the conditions in Definition 5.3.Each element of G can be uniquely expressed in the form hk, since hk = h′k′ implies (h′)−1h =k′k−1 ∈ H ∩ K = {e}. Since H C G, for each k ∈ K there is an automorphism of H whichis the inner automorphism of k restricted to H given by h 7→ khk−1. Moreover, the mapK → Aut(H) given by k 7→ (h 7→ khk−1) is a group homomorphism, as we now show. Supposeϕk is the inner automorphism of k restricted to H. Then, for t, k ∈ K and h ∈ H,

ϕtk(h) = tkh(tk)−1

= tkhk−1t−1

= t(khk−1)t−1

= ϕt(khk−1)

= ϕt(ϕk(h))

= (ϕt ◦ ϕk)(h).

For h ∈ H and k ∈ K we associate the ordered pair (h, k) ∈ H × K with the elementhk ∈ G = HK. For hk, h′k′ ∈ G and the homomorphism ϕ, the multiplication on G can be


written as follows:

(h, k)(h′, k′) = (hkh′k−1, kk′)

= (hϕk(h′), kk′).

This is what called a semidirect product, and it is denoted by H oϕK, or it can be written asH oK, when ϕ is understood.

We now give an example of the semidirect product.

Example 5.3. Let H = 〈x|xn = 1〉 ∼= Zn, and K = 〈y|y2 = 1〉 ∼= Z2. Let ϕ : K → Aut(H)be the homomorphism such that ϕy(x) = x−1. Then the semidirect product of the normalsubgroup H and the subgroup K is

G = H oϕ K.

We now show that G ∼= Dn, the dihedral group of order 2n. The element (x, 1) has order nand (1, y) has order 2. So,

(1, y)(x, 1)(1, y) = (ϕy(x), y)(1, y)

= (ϕy(x), y2)

= (ϕy(x), 1)

= (x−1, 1)

= (x, 1)−1.

Hence if we set v = (x, 1) and u = (1, y) then v and u satisfy vn = u2 = 1 and uvu = v−1.We can see that v and u generate G and |G| = |K||H| = 2n. So the relations of the group Gare exactly the same as those of the dihedral group Dn and that implies G is isomorphic to Dn.

Remark 5.2. The example above shows that the dihedral group Dn is isomorphic to thesemidirect product G = Zn o Z2.

Next, we will use the semidirect product to construct the group Dp,q;θ, which will form thebuilding blocks for our target groups in Chapter 6.

5.3 The construction of Dp,q;θ

We will begin with the definition of Tuffley’s Dp,q [29]. For p and q distinct primes the groupDp,q is a semidirect product

Dp,q = Zq−1p o Zq.

To define multiplication in Dp,q we regard Zq−1p as the additive group of the finite field Fpq−1 .The multiplicative group F×pq−1 is cyclic of order pq−1− 1, and so contains an element ζ of order

q, as q divides pq−1 − 1 by Fermat’s Theorem. The multiplication in Dp,q can be defined by theaction of i ∈ Zq on Zq−1p given by multiplication by ζ i. We remark that Dp,q may be regardedas a generalised dihedral group, in the sense that Dp,2

∼= Dp.

If F (x) = 1+x+x2+· · ·+xq−1 factors over Zp, then the isomorphism type of Dp,q depends onthe choice of root ζ, and therefore to avoid this ambiguity we need to define Dp,q;θ. To construct


the semidirect product Dp,q;θ, let θ(x) be an irreducible factor of F (x) = 1 +x+x2 + · · ·+xq−1

over Zp, and let ζ be a root of θ(x) in Fp;θ = Zp[x]/〈θ(x)〉 which is a finite field of order pdeg θ.The group Dp,q;θ is the semidirect product

Dp,q;θ = Vp;θ o Zq,

where Vp;θ = (Zp)deg θ is the additive group of Fp;θ. The multiplicative group FXp;θ is cyclic of

order pdeg θ − 1, and ζ is an element of multiplicative order q, because θ(x) divides 1 + x+ x2 +· · ·+xq−1, and so divides xq−1 = (x−1)(1+x+x2 + · · ·+xq−1). Therefore, the multiplicationin Dp,q;θ can be defined by

(v, i) · (u, j) = (v + ζ iu, i+ j).

Thus Dp,q;θ is a subgroup of Tuffley’s Dp,q [29] for a suitably chosen element ζ of order q, andDp,q;θ

∼= Dp,q when θ is irreducible over Zp.

Remark 5.3. In the special case where q = 2, we have F (x) = x+ 1 which is irreducible overZp. Then ζ = −1 and we see that Dp,2;x+1

∼= Dp, the dihedral group.

Since Vp;θ is normal, there is a map from Dp,q;θ into Zq, which we will use in Section 5.5. Iff = (v, i) ∈ Dp,q;θ, then we will write [f ] = i for its image in Zq.

The next section proves some properties of the group Dp,q;θ.

5.3.1 Properties of Dp,q;θ

In this section, we will introduce some properties of Dp,q;θ and show that it is generated byany element of order p together with any element of order q. Also, we will define cyclic andnoncyclic solutions to xa = yb, in a group G, and show when Dp,q;θ has a noncyclic solution toxa = yb.

The following lemma will present some facts about the elements of Dp,q;θ. For a proof seeTuffley [29].

Lemma 5.1 (Tuffley, [29, Lemma 3.2]). Let Dp,q;θ = Vp;θoZq, such that Vp;θ = (Zp)deg θ. Then

1. Elements in Dp,q;θ are of order 1, p, or q.

2. If two elements f and g ∈ Dp,q;θ commute with each other, then they belong to Vp;θ or thesame cyclic subgroup of order q.

3. If an element f = (v, 0) ∈ Vp;θ has order p, then the conjugacy class of f is (ζ iv, 0), where0 6 i 6 q − 1.

4. If f = (v, i) has order q, then the conjugacy class of f is {g : [g] = i} = {(ω, i) : ω ∈ Vp;θ}.

Remark 5.4. Since p and q are prime, as a consequence of Lemma 5.1 we note that if f is anontrivial element of Dp,q;θ, then f has an nth root in Dp,q;θ if and only if ord(f) is co-primeto n. Moreover, any such nth root belongs to 〈f〉.

We next will state our lemma which shows the generators of the group Dp,q;θ.


Lemma 5.2. Let α and β be elements of Dp,q;θ of orders p and q. Then the group Dp,q;θ isgenerated by α and β.

Proof. The group Dp,q;θ = Vp;θ o Zq. Suppose α is an element of order p such that 〈α〉 ={(ku, 0)|k = 0, . . . , p − 1}, and suppose β is an element of order q such that 〈β〉 = {(vj, j) =(v, 1)j|j = 0, . . . , q − 1}. Then letting 〈β〉 act on α = (u, 0) by conjugation we see that

βjαβ−j = (vj, j)(u, 0)(vj, j)−1

= (vj, j)(u, 0)(−ζ−jvj,−j) (as (vj, j)−1 = (−ζ−jvj,−j))

= (vj, j)(u− ζ−jvj,−j)= (vj + ζj(u− ζ−jvj), j − j)= (vj − vj + ζju, 0)

= (ζju, 0),

so (ζju, 0) belongs to 〈α, β〉 for j = 0, 1, . . . , q − 1.Let d = deg θ. We claim that (u, 0), (ζu, 0), . . . , (ζd−1u, 0) are linearly independent, and there-fore generate Vp;θ. We regard Vp;θ = (Zp)d as a vector space over Zp. Suppose that

c0u+ c1ζu+ · · ·+ cd−1ζd−1u = 0, (5.1)

where ci ∈ Zp for i = 0, . . . , d− 1. Factoring (5.1) we get[d−1∑i=0

ciζi

]u = 0,

so∑d−1

i=0 ciζi = 0 ∈ Fp;θ. But then ζ is a root of

∑d−1i=0 cix

i, a polynomial of degree lessthan the degree of θ, so we must have ci = 0 for all i since θ is irreducible over Zp. Then(ζ iu, 0), i = 0, . . . , d− 1 are linearly independent and therefore generate Vp;θ as claimed.

Since we are interested in homomorphisms from Ga,b, we are interested in solutions toxa = yb in a group G. The following definition gives more explanation.

Definition 5.4. If G has a solution to xa = yb, then the solution is cyclic if 〈x, y〉 is a cyclicsubgroup of Dp,q;θ. Otherwise, the solution is noncyclic.

Similarly, we will say that ρ : Ga,b → G is cyclic if 〈ρ(x), ρ(y)〉 is cyclic. Otherwise,ρ : Ga,b → G is noncyclic.

We now characterise when Dp,q;θ has a noncyclic solution to xa = yb.

Lemma 5.3. Let a and b be co-prime. Any noncyclic solution to xa = yb in Dp,q;θ satisfiesxa = yb = 1. Consequently, such a solution exists if and only if p|a and q|b, or q|a and p|b.

Proof. Suppose x and y are a solution to xa = yb such that xa = yb = g 6= 1. Then g has ordereither p or q. The element g belongs to 〈x〉 and 〈y〉, where 〈x〉 and 〈y〉 are groups of order p orq which are primes. Therefore, g generates 〈x〉 and 〈y〉, and 〈x〉 and 〈y〉 are cyclic subgroups of〈g〉. This implies that 〈x, y〉 = 〈g〉 is cyclic, so any noncyclic solution must satisfy xa = yb = 1.We show that a noncyclic solution exists only if p|a and q|b, or q|a and p|b. Let xa = yb be anoncyclic solution satisfying xa = yb = 1. Since a and b are co-prime and p and q are distinct


primes, either x is an element of order p and y is an element of order q or x is of order q and yis of order p. Then p|a and q|b, or q|a and p|b.We now prove a noncyclic solution exists if p|a and q|b, or q|a and p|b. Without loss of generality,assume p|a and q|b. In this case, choose x of order p and y of order q, and therefore xa = yb = 1and 〈x, y〉 is noncyclic by Lemma 5.2, since it equals Dp,q;θ.

Remark 5.5. By Lemma 5.3, if 〈x, y〉 is a noncyclic solution, then 〈x, y〉 = Dp,q;θ.

5.4 Wreath products

The wreath product in group theory is a product of two groups based on a semidirect product.It is important in the classification of permutation groups. For our purposes we may define thewreath product as follows.

Definition 5.5. Let G be a group, and let X be a set, then GX =∏

i∈X Gi, where Gi = G fori ∈ X. Suppose H is a group that acts on X on the right. Then there is a natural left actionof H on GX defined by

(h(g))i = gi·h,

where g = (gi)i∈X . Then the wreath product G oH is the semidirect product GXoϕH, whereϕ is the homomorphism ϕ : H → Aut(GX) defined as follows: for (gi)i∈X ∈ GX and h ∈ H, wehave (ϕh(g))i = gi·h.

We will illustrate the multiplication in the wreath product with a simple remark followedby an example.

Remark 5.6. Suppose (g1, g2, . . . , gn), (k1, k2, . . . , kn) ∈ GX , X = {1, . . . , n} and h1, h2 ∈ H.Then the multiplication rule in G oH = GX oϕ H can be defined by(

(g1, g2, . . . , gn), h1)(

(k1, k2, . . . , kn), h2)

= (g1k1·h1 , g2k2·h1 , . . . , gnkn·h1 , h1h2).

If g =((g1, . . . , gn), h

), then we define g = h. In general, let g, k ∈ G o H and g ∈ H.

Then (gk)i = giki·g, and the multiplication in H is given by gk = gk. The multiplication inpermutations is composed from left to right, as we will see in the following example.

Example 5.4. Let (g1, g2, g3), (k1, k2, k3) ∈ G3 and let (1 2), (1 3) ∈ S3. Then, the product ofg =

((g1, g2, g3), (1 2)

)and k =

((k1, k2, k3), (1 3)

)in G3 o S3 is

gk =((g1, g2, g3), (1 2)

)·((k1, k2, k3), (1 3)

),

=((g1k2, g2k1, g3k3), (1 2 3)

).

In the following section we will construct the group Wh,k;τs,t;% by modifying the construction

of Tuffley’s Hq,rp in [29].


Figure 5.3: When ω has order t = 5, its action on z has a single 1-cycle (p) and all other cyclesare of length 5.

5.5 The construction of Wh,k;τs,t;%

To construct our target group Wh,k;τs,t;% we will modify the construction of Tuffley’s Hq,r

p , whichis a wreath productDq,roPSL(2, p). We will do this by replacing PSL(2, p) withDs,t;% as follows.

Given distinct primes h, k, s and t and suitable polynomials τ ∈ Zk[x] and % ∈ Zt[x], thegroup Wh,k;τ

s,t;% is a wreath product of Dh,k;τ over Ds,t;%,

Wh,k;τs,t;% = Dh,k;τ oDs,t;% = (Dh,k;τ )

z oDs,t;%,

where z = {g ∈ Ds,t;%|[g] = 1} is a conjugacy class of Ds,t;% and we let Ds,t;% act on z byconjugation.

Elements of Wh,k;τs,t;% have the form

ω = ((ωi)i∈z, ω),

where ωi ∈ Dh,k;τ for each i ∈ z, so there will be |Vs;%| = sdeg % entries. The element ω ∈ Ds,t;%

has order 1, s or t.

Remark 5.7. We use Lemma 5.1 to describe the cycle structure of an element ω ∈ Ds,t;% actingon z by conjugation. Note that all elements of z have order t. If ω has order s, then ω ∈ Vs;%.Elements of order s and t do not commute, and therefore ω has no fixed point and all cyclesare of length s, because s is prime. If ω has order t, then it commutes only with 〈ω〉. There isa unique element ω ∈ 〈ω〉 such that ωr ∈ z, 0 6 r 6 t − 1. Then, ω has a unique fixed pointin z and therefore all other cycles have length t, because t is prime. Figure 5.3 pictures theabove discussion, with t = 5.

The map Dh,k;τ → Zk leads to a map Wh,k;τs,t;% → Zk oDs,t;% which is given by

[ω] = (([ωi])i∈z, ω).

It is convenient to factor the map Wh,k;τs,t;% → Zk oDs,t;% into two maps to distinguish a subgroup

of Wh,k;τs,t;% isomorphic to Zk oDs,t;%. Choose ξ in Dh,k;τ of order k such that [ξ] = 1, so 〈ξ〉 ∼= Zk.

We define Aks,t;% = 〈ξ〉 oDs,t;% ⊆Wh,k;τs,t;% .


Furthermore, we can get a well defined homomorphism Wh,k;τs,t;% → Zk oDs,t;% → Zk which is

given by

[[ω]] =∑i∈z

[ωi],

as Zk is abelian.

Similarly, the subgroup Vh;τ oDs,t;% has the homomorphism || · || : Vh;τ oDs,t;% → Vh;τ givenby

||(vi, v)i∈z|| =∑i∈z

vi.

To be able to use Wh,k;τs,t;% , we need to restrict h, k, s and t to be distinct primes which will

allow some flexibility in dealing with the processes in the following chapter in order to provethe main theorem.

We conclude this section with an example which illustrates the group operations in thewreath product G oD5.

Example 5.5. Consider the wreath product G o D5,2;x+1. Since D5,2;x+1 is isomorphic to D5,we therefore have G oD5. The group D5 is generated by ρ = (0 1 2 3 4) and σ = (0)(1 4)(2 3)(see example 5.2).Elements of GoDs may be described using diagrams such as those in Figures 5.4 (i) and (ii). Thearrows describe the permutation g and h, and they are labelled by the corresponding elementgi, hi. For example, let

g =((g0, g1, g2, g3, g4), (0 1 2 3 4)

), h =

((h0, h1, h2, h3, h4), (0)(1 4)(2 3)

).

Thengh =

((g0h1, g1h2, g2h3, g3h4, g4h0), (0 4)(1 3)(2)

),

which is represented by Figure 5.4(iii). As we can see in Figure 5.4 (iv), the inverse of g isfound by reversing all arrows and inverting all labels,

g−1 =((g0, g1, g2, g3, g4), (0 1 2 3 4)

)−1=((g−14 , g−10 , g−11 , g−12 , g−13 ), (0 4 3 2 1)

).

If g is raised to the third power: g · g · g, then

g3 =((g0g1g2, g1g2g3, g2g3g4, g3g4g0, g4g0g1), (0 1 2 3 4)3

)=((g0g1g2, g1g2g3, g2g3g4, g3g4g0, g4g0g1), (0 3 1 4 2)

).

This is represented by Figure 5.4 (v).


Figure 5.4: Visualising elements and group operations in G oD5.

Chapter 6

The Main Result

In this section we will describe the strategy of this study which extends the work of Tuffley [29],and we will state some definitions and lemmas which we will use in order to show that the dif-ference between Gn(GKa,b) and Gn(SKa,b) can be detected by counting homomorphisms intosuitable finite groups. For simplicity we will restrict our attention to values of n satisfying a cer-tain coprimality condition with respect to a and b. This avoids certain technical complicationsthat arise in Tuffley’s proof [29] that Gn(GK) and Gn(SK) are not isomorphic for n ≥ 2.

6.1 Strategy

Let K = GKa,b or SKa,b, and let µ, λ be the meridian and longitude of K. Then we will considertriples (W, ρ, η), where W is a finite group, ρ is a homomorphism which maps π1(K) into W,and η is an nth root of ρ(µ). If ηn = ρ(µ), then we will called (ρ, η) a map-root pair for Kin W. Then we will have a homomorphism ρ : Gn(K)→W, when it satisfies the compatibilitycondition ρ(λ)η = ηρ(λ). As we know π1(GKa,b) and π1(SKa,b) are both isomorphic to

Ha,b = 〈x, y, z, w|xa = yb, wa = zb, xcy−d = wcz−d〉,

where they have common meridian µ = xcy−d = wcz−d = ω0, so the map-root pairs forboth of them are the same. However, the compatibility conditions for GKa,b and SKa,b areρ(xawa)η = ηρ(xawa) and ρ(xaw−a)η = ηρ(xaw−a) respectively. As Ha,b = Ga,b ∗〈ω0〉 Ga,b, wecan think of ρ : Ha,b → W as two homomorphisms (ρ1, ρ2) mapping Ga,b into W such thatρ1(ω0) = ρ2(ω0).

The following is our main theorem.

Theorem 6.1. (Main Theorem) Suppose that n ≥ 2, and a and b are co-prime integers suchthat 1 < a < b. Choose primes s|a and t|b such that gcd(st, n) = 1. Let h be a prime dividing n,k a prime co-prime to 2nab, and let τ ∈ Zk[x] be an irreducible factor of F (x) = 1 + · · ·+ xk−1

over Zh, and % ∈ Zt[x] be an irreducible factor of F (x) = 1 + · · ·+ xt−1 over Zs. Then

|Hom(Gn(GKa,b),Wh,k;τs,t;% )| < |Hom(Gn(SKa,b),Wh,k;τ

s,t;% )|.

Therefore, Gn(GKa,b) is not isomorphic to Gn(SKa,b) for n ≥ 2 which satisfies the theoremhypotheses; in particular, this holds if gcd(ab, n) = 1.

47

CHAPTER 6. THE MAIN RESULT 48

Note that any nontrivial torus knot is equivalent up to reflection to a torus knot Ta,b satis-fying 1 < a < b.

The target group of this study is the wreath product Wh,k;τs,t;% defined earlier in Section 5.5.

Each one of the divisibility conditions that is described in the hypotheses of the above theoremwill play a crucial role in the proof of the theorem as we will see in the following sections. Now,let us describe the idea of the proof of Theorem 6.1 which is the same as that of Tuffley [29].We will show that:

A1. Every compatible pair (ρ, η) for GKa,b in Wh,k;τs,t;% is also compatible for SKa,b.

A2. Some pairs (ρ, η) are compatible for SKa,b in Wh,k;τs,t;% , but are incompatible for GKa,b.

Remark 6.1. Let a = 2 × 3 and b = 5 × 7, and let s and t be distinct primes such that s|aand t|b. Therefore, we have four possibilities for s and t. If we choose

• s = 2 and t = 5, then the possibilities for n will be 3, 7, 9, 11, 13, 17, 19, 21, . . . ;

• s = 2 and t = 7, then the possibilities for n will be 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, . . . ;

• s = 3 and t = 5, then the possibilities for n will be 2, 4, 7, 9, 11, 13, 16, 17, 19, 22, 23, . . . ;

• s = 3 and t = 7, then the possibilities for n will be 2, 4, 8, 11, 13, 16, 17, 19, 20, 22, 23, . . . .

6.2 The cycle product and applications

In this section we will state some lemmas and definitions which will play an important role in theproof of our Main Theorem 6.1. Although the lemmas and definitions were proved for Tuffley’sHq,rp = Dq,r oPSL(2, p) in Tuffley [29], the proofs still apply for our group Wh,k;τ

s,t;% = Ds,t;% oDh,k;τ .First we will define the cycle product which is an important tool for finding the results.

Definition 6.1 (Tuffley [29]). Let ω ∈Wh,k;τs,t;% , i ∈ z and let li(ω) be the length of the disjoint

cycle of ω that contains i. Then the cycle product of ω at i can be defined as follows:

πi(ω) =

li(ω)−1∏r=0

ωi·ωr = ωiωi·ω · · ·ωi·ωli(ω)−1 .

The cycle product is thus the ordered product, beginning at i, of ωj for j in the disjoint cycleof ω containing i. Notice that given a cycle, we can then see that the value of the cycle productis dependent on the beginning point i, while the conjugacy class is not, as πi·ω(ω) = ω−1i πi(ω)ωi.

Next, we will define elements in (reduced) standard form.

Definition 6.2 (Tuffley [29]). Let ω belong to Wh,k;τs,t;% . Then ω is said to be in standard form if

ωi·ω = ωi,

for each i. Moreover, ω is in reduced standard form if ω is in standard form and πi(ω) =

ωli(ω)i = 1 if and only if ωi = 1.


The next lemma implies that an element of Wh,k;τs,t;% is conjugate to an element in reduced

standard form.

Lemma 6.1 (Tuffley, [29, Lemma 3.3]). Let li(ω) be co-prime to the order of πi(ω) for all i.Then, ω is conjugate to an element α in reduced standard form such that α = ω and πi(α) isconjugate to πi(ω) for all i.

Note that in our case li(ω) ∈ {1, s, t} is co-prime to ord(πi(ω)) ∈ {1, h, k}, so the hypothesesof Lemma 6.1 are always satisfied. It follows that every element of Wh,k;τ

s,t;% is conjugate to anelement in reduced standard form.

Lemma 6.2 (Tuffley, [29, Lemma 3.4]). Suppose ω is an element of Wh,k;τs,t;% in standard form

such that ωi has order 1 or k for all i. Then ω is conjugate to an element α of Aks,t;% in standardform. Moreover, if ω is reduced then α may be chosen to be reduced.

The next lemma gives a condition for an element in reduced standard form to commutewith another element.

Lemma 6.3 (Tuffley, [29, Lemma 3.5]). Suppose ω is in reduced standard form and let γcommute with ω. If ωi is constant on orbits of γ, then γi commutes with ωi for each i and γiis constant on orbits of ω.

We conclude this section with the following lemma which gives a necessary condition for anelement ω of Wh,k;τ

s,t;% to be an nth power.

Lemma 6.4 (Tuffley, [29, Lemma 3.6]). If ω = γn belongs to Wh,k;τs,t;% , then γn = ω and

πi(ω) = (πi(γ))n/gcd(li(γ),n).

In particular, the conjugacy class of πi(ω) is constant on orbits of γ.

The proof of the main theorem requires the lemmas in the following section.

6.3 Images

In this section, we will identify the possibilities for images and roots of the meridian in the pair(ρ, η) by distinguishing up to conjugacy solutions to ηn = ω with ω 6= 1. Also, we will find thepossible values of the longitude. The lemmas and proofs in this section generalise those usedin Tuffley’s proofs [29].

6.3.1 The meridian

Lemma 6.5. Let h, k, s, t, τ and % be as given in Theorem 6.1. If ω ∈ Wh,k;τs,t;% is an nth power

such that ω 6= 1, then ω is a conjugate to an element of Aks,t;% that is in reduced standard form.Conversely, for such an element ω of Aks,t;% the solutions to ηn = ω are described by

1. η = ωr, where ωr is the unique nth root of ω in 〈ω〉;

2. ηi·ω = ηi and so also ηi·η = ηi for all i;


3. ηi = ω1ni ∈ 〈ξ〉 if ωi 6= 1, where

1

nis the multiplicative inverse of n in Zk; and

4. ηi ∈ Vh;τ if ωi = 1.

Consequently, the solutions to ηn = ω are parametrised by (Vh;τ )c, where c is the number of

cycles of ω on which ωi = 1.

Proof. Suppose ηn = ω. Then by Lemma 6.4, we get

πi(ω) = (πi(η))n/gcd(li(η),n).

We have gcd(li(η), n) = 1, because li(η) ∈ {1, s, t} and gcd(st, n) = 1. So πi(ω) = πi(η)n. Nowh divides n, so πi(ω) ∈ Dh,k;τ is an hth power in Dh,k;τ and therefore πi(ω) has order 1 or k forall i. Since li(ω) is co-prime to k for all i, Lemma 6.1 and Lemma 6.2 imply that ω is conjugateto an element of Aks,t;% in reduced standard form, as claimed.

To prove part (1), suppose ω is an element of Aks,t;% in reduced standard form, such thatω 6= 1. Then ω has order either s or t, which are distinct primes and co-prime to n. Thereforeω has a unique nth root α ∈ Ds,t;%, and α is of the form α = ωr for some r. We show that ωhas an nth root η such that η = α = ωr. Suppose that ω = ηn. The order of ω is prime, so theorbits of η = ωr and the orbits of ω coincide. Therefore ωi is constant on orbits of η, becauseω is in reduced standard form. Since ω = ηn commutes with η, Lemma 6.3 implies that ωicommutes with ηi, and therefore ηi·ω = ηi and also ηi·η = ηi as they have the same orbits, whichproves part (2). Since ηi·η = ηi, η is in standard form (Definition 6.2) and we therefore have

ωi = (ηn)i = (ηi)n

in Dh,k;τ . Because of our construction, n is divisible by h, but not k. Therefore, either ωi is oforder k and ηi is the unique nth root of ωi in 〈ωi〉 = 〈ξ〉, or ωi is of order 1 and then we canchoose any element ηi ∈ Vh;τ , which prove parts (3) and (4). These values do in fact define nthroots, so the nth roots of ω are parametrised by (Vh;τ )

c, where c is the number of cycles of ωon which ωi = 1.

6.3.2 The longitude

Lemma 6.6. Let ω belong to Aks,t;% in reduced standard form such that ω is nontrivial. Let

h, k, s, t, τ and % be as we described them earlier in Theorem 6.1. Assume ρ : Ga,b −→Wh,k;τs,t;% is

a homomorphism such that ρ(ω0) = ω and let ε = ρ(xa). If 〈ρ(x), ρ(y)〉 is cyclic then ε = ωab,

and if 〈ρ(x), ρ(y)〉 is noncyclic then

1. ε = 1;

2. εi is constant on orbits of ω;

3. the conjugacy class of εi is constant on z;

4. [εi] = absdeg %

[[ω]] for all i;

5. εi = ξab[[ω]]/sdeg %

if ωi 6= 1.


Proof. Define χ = ρ(x) and ψ = ρ(y), so ε = χa = ψb. We consider two cases:

1. 〈χ, ψ〉 is a noncyclic subgroup of Ds,t;%,

2. 〈χ, ψ〉 is a cyclic subgroup of Ds,t;%.

Case 1: noncyclic case. We consider first the case that 〈χ, ψ〉 is noncyclic. Since xa = yb

generates the centre of Ga,b (by Theorem 3.7), ε commutes with ω. To prove part (1), we have

ε = χa = ψb in Ds,t;%, and 〈χ, ψ〉 is noncyclic, so by Lemma 5.3 ε = 1. Therefore ωi is constanton orbits of ε (since each orbit is a singleton) and by applying Lemma 6.3, εi is constant onorbits of ω and commutes with ωi for all i, which proves part (2). Since ωi ∈ 〈ξ〉 for all i, byapplying Lemma 5.1 part (2) we have εi ∈ 〈ξ〉, whenever ωi 6= 1.

As we have εi = πi(ε) for all i and ε = χa = ψb, so by Lemma 6.4 the conjugacy class of εiis constant on orbits of χ and ψ. Since χ and ψ generate Ds,t;% (by Lemma 5.2) and Ds,t;% actstransitively on z, the conjugacy class of εi is constant on z, which proves part (3).

To prove part (4), since the conjugacy class of εi is constant on z, the value of [εi] is constanton z. So we can calculate the common value [εi] by using the abelianisation Wh,k;τ

s,t;% → Zk asfollows:

[[ε]] =∑r∈z

[εr] = sdeg %[εi]. (6.1)

Since ω generates the abelianisation of Ga,b (by Remark 3.1), we have

[[ε]] = [[χa]] = a[[χ]] = ab[[ω]]. (6.2)

From (6.1) and (6.2), we getsdeg %[εi] = ab[[ω]]. (6.3)

We can divide (6.3) by sdeg % mod k as sdeg % is co-prime to k. Then we have

[εi] =ab

sdeg %[[ω]], (6.4)

for all i as required.

When ωi 6= 1, we can use (6.4) and the fact that εi ∈ 〈ξ〉 to get

εi = ξab[[ω]]/sdeg %

, (6.5)

which proves part (5).

Case 2: cyclic case. Now, consider the case that χ and ψ are powers of ω. Thereforeρ maps Ga,b into the subgroup Dh,k;τ o 〈ω〉. Since ω generates the abelianisation of Ga,b (byRemark 3.1), we therefore have ε = ωab, so ωi is constant on orbits of ε. By applying Lemma 6.3,εi is constant on orbits of ω and commutes with ωi for all i, so εi ∈ 〈ξ〉 if ωi 6= 1. Let σ1, . . . , σmbe the disjoint cycles of ω of lengths l1, . . . , lm, then the subgroup Dh,k;τ o 〈ω〉 can be regardedas a subgroup of the direct product

m∏r=1

Dh,k;τ o 〈σr〉 =m∏r=1

(Dh,k;τ )lr o σr.


We will regard the map ρ as a product of maps ρr to each factor. In the rest of the proof,σr will denote both the cycle and the set of points in the orbit of this cycle of z. Forγ ∈

∏mr=1(Dh,k;τ )

lr o σr, we will write γ|σr for the projection of γ on the factor.

Since ωi and εi are constant on σr, the abelianisation [[·]] : Dh,k;τ o 〈σr〉 → Zk gives

[[ε|σr ]] = lr[εi], (6.6)

and[[ε|σr ]] = ab[[ω|σr ]]. (6.7)

From (6.6) and (6.7), we get

[[ε|σr ]] = lr[εi] = ab[[ω|σr ]] = ablr[ωi]. (6.8)

As li(ω) ∈ {1, s, t} is co-prime to k for all i, we have [εi] = ab[ωi] on σr. If ωi is nontrivial onσr, then we have

εi = ξ[εi] = ξab[ωi] = ωabi = (ωab)i.

If ωi is trivial on σr, then [ωi] = 0 so εi is in Vh;τ . We need to prove that εi is trivial. Weassume that εi = v ∈ Vh;τ , and that v 6= 1, to lead to a contradiction. We claim that ρr mapsGa,b into the subgroup Vh;τ o 〈σr〉. To prove this we need to show that the images of x and ylie in this subgroup, because x and y generate Ga,b (by Definition 3.6). As we know, ε and χcommute and εi is constant on orbits of χ, because the orbits of χ are contained in the orbitsof ω. Applying Lemma 6.3 we can conclude that χi commutes with εi for all i. By applyingLemma 5.1 part (2), this means χi ∈ Vh;τ on σr. We can apply the same argument to y, andthen we may conclude that ψi ∈ Vh;τ for all i; as required.

We have the abelianisation || · || : Vh;τ o 〈σr〉 → Vh;τ , and applying this we get

||ε|σr || = lrv = ab||ω|σr || = 0.

Since lr is co-prime to h, v = 1 is the identity. Therefore, εi = 1 = ωabi = (ωab)i on σr.

In both cases above we have εi = (ωab)i, so ε = ωab as required.

Let us now investigate Lemma 6.6 when [[ω]] = 0, and ε 6= ωab.

Lemma 6.7. Suppose ω, ε, ρ and h, k, s, t, τ and % are as described in Lemma 6.6, and let[[ω]] = 0 and ε 6= ωab. If ωj is nontrivial for some j, then ε is trivial.

Proof. Since ε 6= ωab, ε is described by the noncyclic case of Lemma 6.6, and εi ∈ Vh;τ for all i.If ωj is nontrivial, then εj = ξ0 = 1 by (5); and now by (3) the conjugacy class of εi is constant,so εi = 1 for all i. From statement (1), we have ε = 1, so ε = 1 .

Remark 6.2. When [[ω]] = 0, and ε 6= ωab, it follows from Lemma 6.5 and Lemma 6.6, ifωi = 1 for all i, then ηj and εj belong to Vh;τ for all j. Since the orbits of η coincide with theorbits of ω, εi is constant on orbits of η. Therefore

(ηε)i = ηiεi·η = ηiεi = εiηi = (εη)i.

So ε and η commute as ηε = η = εη.


6.4 The proof of the main theorem

We devote this section to the proof of our Main Theorem 6.1. To prove the theorem we needto show (ρ, η), which is a map-root pair for GKa,b and SKa,b in Wh,k;τ

s,t;% , is compatible for GKa,b

only if it is compatible for SKa,b. We will study the compatibility conditions for GKa,b and

SKa,b in the two cases: when ρ(ω0) = 1, which is a trivial induced map to Ds,t;%, and ρ(ω0) 6= 1,which is a nontrivial induced map to Ds,t;%.

6.4.1 Case 1: Trivial induced maps

Let ρ(ω0) = 1. The induced homomorphism ρ : Ha,b → Ds,t;% is trivial, because the conjugacyclass of ω0 generates Ha,b (by Remark 3.1). Then we may regard the homomorphism ρ as ahomomorphism Ha,b → Dz

h,k;τ which is a product of homomorphisms Ha,b → Dh,k;τ . As wehave chosen h and k to be co-prime to a and b, each map ρ : Ha,b → Dh,k;τ factors throughZ (by Lemma 5.3). As a result, ρ : Ha,b → Dz

h,k;τ factors through Z too. Then, we have

ρ(xa) = ρ(wa) = ρ(ω0)ab = ωab. As η commutes with ηn = ω, it commutes with both ρ(xa) and

ρ(wa). Therefore, the pair (ρ, η) is compatible for GKa,b and SKa,b.

6.4.2 Case 2: Nontrivial induced maps

In this case, let ρ(ω0) 6= 1. There is ω, α ∈Wh,k;τs,t;% such that ω = αρ(ω0)α

−1 ∈ Aks,t;% is in reducedstandard form (by Lemma 6.5). Then, we can prove statement (A.1) by assuming ρ(ω0) = ω,because the pair (ρ, η), where ρ maps Ga,b into Wh,k;τ

s,t;% , is compatible for either GKa,b or SKa,b

if and only if (ρ′, η′) = (αρα−1, αηα−1) is.

If ε = ρ(xa) and δ = ρ(wa), then we can rewrite the compatibility conditions as follows:

(SKa,b) : εδ−1η = ηεδ−1,

(GKa,b) : εδη = ηεδ.

Referring to the possibilities for ε and δ in Lemma 6.6, we can study the compatibilityconditions in two separate cases:

At least one of ε or δ equals ωab

We can easily show a pair is compatible for GKa,b if and only if it is compatible for SKa,b fromthe compatibility conditions by writing them in the form

δ∓1ηδ±1 = ε−1ηε.

As we can see at least one of ε and δ commutes with η, so we can conclude that each compati-bility condition is satisfied if and only if the other commutes with η as well.

Neither ε nor δ equals ωab

In this section we will restrict our attention to the case [[ω]] 6= 0, as when [[ω]] = 0 it is clearboth ε and δ commute with η (by Lemma 6.7 and Remark 6.2). Consequently, any such map-root pair is compatible for both GKa,b and SKa,b. Now, let us assume that [[ω]] 6= 0. Then


ωi is constant on orbits of η, and because η is a power of ω, we will investigate cycles of η onwhich ωi is nontrivial and trivial in two separate cases.In the first case, let ωi 6= 1. Then by using Lemma 6.5, ηi belongs to 〈ξ〉, and by Lemma 6.6both of εi and δi satisfy Equation (6.5). Therefore,

(ηεδ∓1)i = ηiεi·ηδ∓1i·η (using multiplication in wreath product)

= ηiεiδ∓1i (as εi = εi·η and δ∓1i = δ∓1i·η (by Lemma 6.5))

= εiδ∓1i ηi (both εi and δ∓1i belong to 〈ξ〉, and therefore they commute with ηi)

= (εδ∓1η)i (since ε = δ = 1).

In the second case, let ωi = 1. Then by Lemma 6.5 ηi belongs to Vh;τ . Therefore we canwrite the compatibility condition for SKa,b as follows:

εiδ−1i ηi = ηiεi·ηδ

−1i·η = ηiεiδ

−1i .

By Lemma 6.6, we get [εiδ−1i ] = [εi] − [δi] = ab

sdeg%[[ω]] − ab

sdeg%[[ω]] = 0, which implies εiδ

−1i

belongs to Vh;τ , and therefore εiδ−1i and ηi commute.

On the other hand, the compatibility condition for GKa,b is

εiδiηi = ηiεi·ηδi·η = ηiεiδi.

By Lemma 6.6, we get [εiδi] = [εi] + [δi] = 2ab[[ω]]/sdeg % 6= 0, because k is co-prime to 2ab.Thus, εiδi 6= Vh;τ ; and therefore εiδi and ηi do not commute (by Lemma 5.1) except whenηi = 1. Then we can conclude that the condition for SKa,b is satisfied, but the condition forGKa,b is not satisfied.

In the next section, we will complete the proof by showing that not all pairs that arecompatible for SKa,b in Wh,k;τ

s,t;% are compatible for GKa,b, by showing that we can realise thecase above where [[ω]] 6= 0, but ωi = 1 on some cycle of ω.

6.4.3 Realisation

Choose χ ∈ Ds,t;% of order s and ψ ∈ Ds,t;% of order t. By Lemma 3.1 we solved bc− ad = 1 to

get µ = χcψ−d, where c, d > 0, to be the meridian. Since s|a and t|b and bc− ad = 1, we havegcd(s, c) = 1. Furthermore, χc belongs to Vs;% and has order s. Because elements of order sand t do not commute, χ has no fixed point in z and all cycles are of length s. Then χc hasno fixed point (by Remark 5.7). We have gcd(t,−d) = 1 (because t|b and bc − ad = 1), andthen ψ−d has order t. Therefore, χcψ−d has order t, so µ has a fixed point. Choose f to be theunique fixed point of µ in z which is given by Remark 5.7. Note that the fixed point f is notthe fixed point of ψ, because otherwise, f would be a fixed point of χc = µψd. Define χj and

ψj in Wh,k;τs,t;% such that χj = ξb for all j, and

ψj =

ξa+1, if j = f ;

ξa−1, if j = f · ψ−1;ξa, otherwise.


Then it is seen (using Figure 6.1 (i) for the cycle of ψ containing f) that πj(ψ) = ξat, for j not

the fixed point of ψ. It follows that

(χa)j = (ψb)j = ξab

for all j. Since also χa = ψb = 1, we have χa = ψb. Hence ρ′(x) = χ, ρ′(y) = ψ defines ahomomorphism ρ′ : Ga,b → Wh,k;τ

s,t;% . We may then obtain a homomorphism Ha,b → Wh,k;τs,t;% by

setting ρ′(w) = χ, ρ′(z) = ψ also.We now compute ρ′(µ) = χcψ−d. To do this it will first be convenient to calculate ψ−d = (ψ−1)d.From above we obtain

(ψ−1)j =

ξ−a+1, if j = f ;

ξ−a−1, if j = f · ψ;

ξ−a, otherwise.

Let d = qt+r with 0 < r < t−1. Then ((ψ−1)d)j = (πj(ψ−1))q

∏r−1i=0 ψ

−1j·ψ−i = ξ−atq

∏r−1i=0 ψ

−1j·ψ−i .

Therefore, we will have the following cases:

On cycles that do not contain f

On such cycles (ψ−1)i = ξ−a, for all i, so

(ψ−d)j = ((ψ−1)d)j = ξ−ad.

On the cycle that contains fWe have (ψ−1)i = ξ−a except at two exceptional points f and f · ψ. Then the value dependson whether

∏r−1i=0 ψ

−1j·ψ−i involves zero, one or two of these.

When it involves zeror−1∏i=0

ψ−1j·ψ−i = (ξ−a)r = ξ−ar.

When it involves two, the exceptional points contribute ξ−a+1ξ−a−1 = ξ−2a and the others ξ−a.Therefore we get ξ−ar.When it involves one, the exceptional points are adjacent, so the calculation only involves oneif the product begins or ends at an exceptional point.When it begins at f , we get

(ψ−d)f = (πf (ψ−1))q

r−1∏i=0

ψ−1f ·ψ−i = ξ−atq

r−1∏i=0

ψ−1f ·ψ−i = ξ−aqtξ−arξ = ξ−a(qt+r)+1 = ξ−ad+1.

When it begins at f · ψd, we get

(ψ−d)f ·ψd = (πf ·ψd(ψ−1))q

r−1∏i=0

ψ−1(f ·ψd)·ψ−i = ξ−atq

r−1∏i=0

ψ−1(f ·ψd)·ψ−i = ξ−aqtξ−arξ−1 = ξ−a(qt+r)−1 = ξ−ad−1.

We can rewrite ((ψ−1)d)j (Figure 6.1(ii)) as follows:

((ψ−1)d)j =

ξ−ad+1, if j = f ;

ξ−ad−1, if j = f · ψd;ξ−ad, otherwise.


We can write the image of the meridian as follows:

(ρ′(µ))j = (χcψ−d)j

= (χc)j(ψ−d)j·χc

= (χj)c(ψ−d)j·χc

= ξbc(ψ−d)j·χc .

To calculate (ρ′(µ))i,i∈z, we use the fact that

f · µ = f

f · χcψ−d = f

f · χc = f · ψd.

For j · χc 6= f, f · ψd, we have(ψ−d)j·χc = ξ−ad.

Therefore,

(ρ′(µ))j = ξbc(ψ−d)j·χc

= ξbcξ−ad

= ξ.

The exceptional cases are j = f · ψdχ−c = f and j = f · χ−c.When j = f , we have

(ρ′(µ))f = ξbc(ψ−d)f ·χc

= ξbc(ψ−d)f ·ψd (as f · χc = f · ψd)= ξbcξ−ad−1 (substitute ((ψ−1)d)f ·ψd = ξ−ad−1)

= ξbc−ad−1

= ξ0 = 1 (as bc− ad = 1).

When j = f · χ−c, we have

(ρ′(µ))f ·χ−c = ξbc(ψ−d)(f ·χ−c)·χc

= ξbc(ψ−d)f ·χc−c

= ξbc(ψ−d)f (as χc−c = χ0 = 1)

= ξbcξ−ad+1 (substitute ((ψ−1)d)f = ξ−ad+1)

= ξbc−ad+1 = ξ2 (as bc− ad = 1).

Therefore (ρ′(µ))j can be written as

(ρ′(µ))j =

1, if j = f ;

ξ2, if j = f · χ−c;ξ, otherwise.


Figure 6.1: Diagram of the cycle of ψ containing f in the case where t = 5.

After conjugating ρ′ so that ω is in reduced standard form, observe that on the cycle (f)of ω we have ωf = 1. Since [[ω0]] = sdeg % 6= 0, this gives us the case where we have map-rootpairs that are compatible for SKa,b, but are not compatible for GKa,b. This shows that theinequality is strict.

Chapter 7

Summary

This thesis has generated a new theorem, which shows that the difference between Gn(GKa,b)and Gn(SKa,b) can be detected by counting homomorphisms into suitably chosen finite groups.This shows that the difference can be detected by an algorithm.

Main Theorem (Theorem 6.1). Suppose that n ≥ 2, and a and b are co-prime integers suchthat 1 < a < b. Choose primes s|a and t|b such that gcd(st, n) = 1. Let h be a prime dividingn, k a prime co-prime to 2nab, and let τ ∈ Zk[x] be an irreducible factor of F (x) = 1+· · ·+xk−1over Zh, and % ∈ Zt[x] be an irreducible factor of F (x) = 1 + · · ·+ xt−1 over Zs. Then

|Hom(Gn(GKa,b),Wh,k;τs,t;% )| < |Hom(Gn(SKa,b),Wh,k;τ

s,t;% )|.

Therefore, Gn(GKa,b) is not isomorphic to Gn(SKa,b) for n ≥ 2 which satisfies the theoremhypotheses; in particular, this holds if gcd(ab, n) = 1.

In order to prove the resulting theorem, we defined and understood knots and knot groups.For the purpose of the study, we gave our attention to torus knots Ta,b. We have explainedthat two curves, a meridian (µ) and a longitude (λ), represent two elements of a knot groupin a natural way. We have shown that the knot group Ga,b = 〈x, y|xa = yb〉 of the torus knotTa,b has a meridian µ = xcy−d = ω0, where c and d ∈ N are a solution to bc − ad = 1 (seeLemma 3.1), with a corresponding longitude λ = xa = yb.

We defined generalised knot groups which are knot invariants; and also we gave a briefdiscussion about the square (SK) and granny (GK) knots and their analogues (SKa,b) and(GKa,b) made of (a, b)-torus knots. We discussed previous results. Tuffley proved Gn(SK) andGn(GK) are non-isomorphic, then Nelson and Neumann showed that generalised knot groupsdistinguish knots up to reflection. We generalised and extended Tuffley’s result to the grannyand square knot analogues.

We also developed a subgroup Dp,q;θ from Tuffley’s Dp,q group, as we have found that ifF (x) = 1 + x + x2 + · · · + xq−1 factors over Zp, then the isomorphism type of Dp,q dependson the choice of root of F . We proved that the group Dp,q;θ is generated by elements of orderp and q and we also characterised when Dp,q;θ has cyclic and noncyclic solutions to xa = yb.

We constructed our target groups for this study, which are the wreath products Wh,k;τs,t;% of the

semidirect products Dh,k;τ over Ds,t;%. We put some restrictions on Wh,k;τs,t;% to provide some

flexibility in order to prove the result.

58

CHAPTER 7. SUMMARY 59

We generalised Tuffley’s strategy and identified the images and the roots of the meridian andthe images of the longitude which are important tools to prove the result. We considered pairs(ρ, η) where ρ is a homomorphism that maps the knot group of the analogues of the granny orsquare knots into the target group Wh,k;τ

s,t;% , and η is an nth root of ρ(µ). Then, we proved that

every compatible pair (ρ, η) for GKa,b in Wh,k;τs,t;% is also compatible for SKa,b; however, some

compatible pairs for SKa,b in Wh,k;τs,t;% are not compatible for GKa,b.

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Generalised knot groups of connect sums of torus knots · Introduction Knot theory is the study of mathematical knots, which are quite di erent from regular knots tied in string