a blueprint for equivalence: the quest for a complete knot invariant · 2015-12-28 · a blueprint...

A Blueprint for Equivalence: the quest for acomplete Knot invariant

24th May 2013

Undergraduate Awards 2013

Mathematical Science Category

Word Count: 19201 (main corpus) or 24314 (including bibliography andappendix )

1

A Blueprint for Equivalence: the quest for a complete Knot Invariant

Abstract

This project is intended to be a brief foray into the field of mathematicscalled knot theory. There is a particular focus upon special tools knownas knot invariants. First we shall provide a statement of the goal ofknot theory as a source of motivation, then the project begins with somebasic introductory theory. This includes the definition of terms and theReidemeister moves, together with a proof of the Reidemeister Theorem.

Once this grounding in knot theory has been established, we will en-deavour to investigate knot equivalence vís-a-vís a number of differentinvariants of varying complexity. These include colourability, the funda-mental group of the knot complement and some polynomial invariants.This culminates in the construction of the Alexander polynomial.

This project then concludes with some practical applications of knottheory, the identification of some open problems in the field and also someareas of ongoing and potential research. An overview of knot theory’sapplication in the study of DNA proteins will be illustrated. We will alsoexpose some of the shortcomings of the invariants we have described andsee reasons as to why they fail to be complete invariants.

In the Appendix section there are brief remarks on some more power-ful polynomial invariants. Here the reader will discover what has beeninferred from the Alexander polynomial. There is also a knot table thatdescribes some other knots and links.

A three component Brunnian link

Undergraduate Awards 2013 ii Mathematical Science


Contents

I Preliminaries 4

II Basic knot theory 9

1 A Mathematical Knot 9

2 Knots and Links 12

3 The Reidemeister Theorem 17

4 Basic Invariants 22

III Further Invariants 33

5 Algebraic Methods 33

6 Alexander Polynomial 58

IV Conclusion 67

V Bibliography 71

VI Appendix 77

A Other Polynomial invariants 77

B The Rolfson Knot Table 94

Undergraduate Awards 2013 iii Mathematical Science

Part I

Preliminaries“Turn him to any cause of policy,The Gordian knot of it he will unloose,Familiar as his garter ”.

William Shakespeare (1564–1616),

quoted from the Archbishop of Canterbury,

Henry V, act 1, sc. 1, l. 45-7.

Introduction

0.1 PrefaceIt is my intention that this project is accessible, in the most part, to someonewith minimal experience in knot theory. This is also balanced with the aim ofmaking this research project ‘self-contained’ and ‘self-sufficient’. The idea ofwriting a research project on the topic of invariants in knot theory, was entirelymy own and stemmed from a personal interest in this branch of mathematics.

Although I have read some articles and papers about elements of knot the-ory before embarking on this concentrated research of the topic, I have nevertaken any formal modules in this subject. Thus, we will construct ideas andconcepts essentially from the ground up, which will provide a platform for thereader to survey a panorama of knot theory and its invariants by the conclusion.Simultaneously, there will be a glimpse into applications and supplementary in-variants. Due to this, there is some further elaboration of mathematical ideas,as well as a detailed appendix with information on further and more recent knotinvariants.

In-depth discussions are used as much as possible and I have endeavoured toaugment concentrated descriptions with some intuitive analogies which run par-allel to the description in the text. The aesthetics of knot theory not only derivefrom the mathematics underpinning it, but also from the visual aspect. Hence,I have included numerous figures to aid and stimulate the reader’s visualisationof these abstract forms. I hope this results in really ‘bringing the subject mat-ter to life’ for the reader. In this regard it should be noted that I have createdmost of the diagrams and examples myself, unless otherwise stated. This wasdone using R. G. Scharein’s KnotPlot program, and proved to be one of theunexpected challenges of the project. Due to the abstract nature of the subjectmatter, I found that very few sources provided figures or diagrams of the con-cepts they explained. Thus, in endeavouring to achieve my aim of helping thereader to visualise the content, the crafting of figures of the abstract notionsin this document, particularly those present in algebraic topology, was deemednecessary. This took me a considerable amount of time and caused occasionalfrustration at times!

In my project I attempt to draw together information from a number of dif-ferent sources, some old and some contemporary. An analysis and explanation

Undergraduate Awards 2013 4 Mathematical Science

0.1 Preface

of statements and assumptions will be given in my own words, often juxtaposedwith commentaries from authors of other texts. During the course of the ma-terial I will supplement referenced material with explanations and digressionsof my own and of others on the subject in point. Due to the breadth of thissubject, I do not prove all statements in detail, choosing to focus on proofs thatdevelop our understanding of invariance. However, attempts to outline a proofare made where appropriate.

The astute reader will note that in the application of theories, and to main-tain a sense of continuity, many of the examples in this document centre on threeparticular knots: 31

1, 412 and 52

3. This is so that the reader becomes intimatelyfamiliar with these knots and their different mathematical perspectives, featuresand characteristics, which can consequently be compared easily.

Acknowledgement. Finally, I would like to thank my supervisor, who very kindlyagreed to supervise this project. Their assistance in communicating the variousconcepts in this project proved to be invaluable. It is this guidance in particularwith Part Three, for which I am especially grateful.

I would also like to thank the numerous proof readers, namely my familyand friends, who helped aid me in the rectification of mathematical and typeerrors.

1This is Alexander-Briggs notation, which the reader shall become familiar with later inthe document. This knot is also called the ‘Trefoil Knot’.

2Known as the ‘Figure of eight’ knot.3Also called the ‘Square knot’.


0.2 Motivation

0.2 MotivationKnots have fascinated and enthralled mankind even before Alexander the Greatcut the Gordian Knot4. Such tales illustrate the awakening of a deep intriguein the structures which we now call knots.

In each part of this document the reader will discover the pervasion of thisinterest in knots not only into mathematics, but also its permeation into literat-ure and culture. This provides the source of an impetus behind the motivationof our study. Thus, it can be rightly stated that over time this fascination hasmatured into a mathematical field in its own right.

There are many definitions ranging in succinctness, precision and technical-ity, however in light of these we shall define knot theory as a branch of topologythat studies the placement problem, or the embedding of one topological spaceinto another [Lomonaco, 1996, Fox, 1962]. Knot theory concerns itself with dis-joint circles embedded in three-dimensional space. Note that a mathematicalknot is different one’s first idea of a knot, which is usually a piece of string withtwo unconnected ends. The knots studied in knot theory are closed loops.

The main question of knot theory is determining which two knots are iso-topic5 and which are not [Manturov, 2012]. This problem is called the knotrecognition problem. Once we have an isotopy equivalence relation, one canstudy knot isotopy classes, or clusters of knots or indeed knot diagrams, whichare essentially equivalent to one another.

From this perspective, we shall use the term ‘knot’ when referring to the knotisotopy class, i.e. the collection of all potential diagrams of that knot that differvisually, but are merely only different visual manifestations of the same knot.We shall mainly discuss what are called knot invariants. These are functionson knot isotopy classes, or functions on knots, that remain invariant when anisotopy is applied. The identification and study of some knot invariants will bethe primary object of our focus.

The importance of invariants lies in their ability to determine equivalence.Two knots or links are equivalent if there exists an orientation-preserving homeo-morphism6 on R3 that maps one knot onto the other7[Hodorog, 2010]. Thiscondition is equivalent to that of isotopy. At this point we shall establish thatsevering a knot, or self-interesections along a particular knot are not permitted.

It remains quite enigmatic to determine if two given knots are indeed equi-valent. Nevertheless knot theory is centred upon developing techniques whichmake this question easier to answer: knot invariants are pivotal to this. Knotsthat are homeomorphic to polygonal paths in three-dimensional space are calledtame [Scharein, 1998]. This is in antithesis to other knots, known as wild , whichcontain points of self-intersections or singularities.

Knot theory usually revolves around the use of tame knots, and these arethe only knots that we will study. Knots that are homeomorphic to a planarpolygon are considered to be unknotted, or trivial. With the rudiments of this

4It was said that whoever undid the Gordian knot would rule Asia, and Alexander theGreat, who cut it with his sword, went on to great conquests.

5Note that this condition is equivalent to saying that one knot may be smoothly deformedinto the other.

6A homeomorphism is a continuous one-to-one mapping with a continuous inverse.7Note that this homeomorphism is a mapping from the ambient space into itself that

“carries” the knot along with it [Scharein, 1998]. Any two knots are obviously homeomorphicif the mapping is between the knots themselves.


0.3 Basic knot theory

field, and its motivation established, the aims of this project including a briefoverview will follow.

Project OverviewSummary. This project serves a two functions. The first, as stated earlier, isthat it intends to provide a brief overview of general knot theory. Logicallythis underpins the second function, which has its crux at the constructs of knotinvariance. This project introduces many facets and attempts to harmonisethem in one document, therefore to learn more about particular elements ofknot theory alluded to in this project, one might consult several excellent treat-ments of the subject: [Manturov, 2012, Lickorish, 1997, Murasugi, 1996]. Thesetexts discuss the topic, and indeed many avenues that are mentioned, but leftunexplored in this document, in greater detail.

Originally it was intended that all of the main polynomial invariants dis-covered in the last century would be discussed in turn, however this resulted inan extremely long main corpus of the project. Hence, it was thought best toleave this supplemental work that I conducted in the appendix section for thereader to consider if they wish to know what this work culminates in. Therewas an intention to include some research into an open problem in the area ofcolourability invariants, although time and document length constraints causedthe inclusion of just my brief thought on this problem. This can be found inthe appendix section.

Another aspect which has not made this final edition is the illustrious anddynamic history of knot theory. A section was formulated in the early stageswhich summarised this, but it was felt that this would exasperate the readerby the time they reached the mathematical content. Therefore the interestedreader is directed towards a comprehensive history of knot tabulations, whichcan be found in a paper by [Thistlewaithe, 1985]. What follows is a detailedoverview of the parts of this project.

0.3 Basic knot theoryIn the second part we begin our endeavour to discover simple properties andcharacteristics of knots. We will provide not one, but several definitions of aknot, which will endeavour to define their multi-faceted nature.

We will also look at links, which are the simple components of even themost trivial or complex knots. The reader will be exposed to various typesof knots, and will encounter a special sequence of ‘moves’ used to manipulateand transform them from one into another. The aforementioned idea of aninvariant and its mechanisms, which provides our established motivation behindthis investigation and indeed knot theory, will then be discussed in detail.

0.4 The InvariantsIn this third part we will introduce some basic knot invariants, like those ofcolourability (three colours and higher) and the unknotting, crossing and linkingnumbers. This part will conclude by discussing some useful elements of algebraictopology that have germinated development in knot theory. The principle notion


0.4 The Invariants

is that the fundamental group of the complement of a knot in R3 provides a greatdeal of information about the knot. The notation we need for this is in the formof presentations, which will lead us to a major theorem. We will investigatethe applications of the Van Kampen Theorem, which is serves as a method forcomputing fundamental groups of spaces. In the transition into part three, it willbe shown how methods in algebraic topology lead to the Alexander polynomial.

In the section on the Alexander polynomial, I shall summarise the investig-ation of different approaches to the computation of the Alexander polynomial.However, my approach will be based upon explaining a clear derivation of thepolynomial beginning with Seifert surfaces, transitioning into homolgy theoryand ultimately computing Alexander matrices.

In fact, in these approaches we shall see many elements of knot theory,moreover specifically elements discussed in this project, utilised and combinedtogether. This is my reasoning for its special treatment. In this regard, thissection serves almost as a ‘grand example’ that ties together previous basic andadvanced theory in harmony, with the goal of identifying intrinsic characteristicsof knots.

Conclusion. In the conclusion of this document the reader should have a well-grounded platform from which to retrospectively survey the landscape of equi-valence in knot theory that they have negotiated through the course of the pro-ject. Simultaneously, we will also look forward and speculate upon the futuredevelopment of knot theory and potential avenues of research. One importantaspect of this section will concern itself with some of the current and futureapplications of the material we have discussed.


1 A MATHEMATICAL KNOT

Part II

Basic knot theory“O time! thou must untangle this, not I;It is too hard a knot for me to untie!”

William Shakespeare,

Twelfth Night, Act II, Scene 2.

1 A Mathematical KnotWhat is a mathematical knot? When one thinks of the concept of a ‘knot’,this conjures the image of “a length of string or rope wound around itself withthe ends fastened so that it cannot be unravelled again” [Long, 2005]. In atopological manner, one prevents the knot from being unravelled by essentially‘gluing’ both ends of the string to one another, which forms a continuous loopin space. “We can also think of the string as having no thickness” [Long, 2005].

What remains is a curve situated in three-dimensional space that has noself-intersections. If working from a geometrical standpoint, we may interpretthe curve as being made up of a number of line segments, joined end to endat pivot points, which change the angle to approximate the curves of the knot.This creates what is called a polygonal knot, which shall be defined later. Thesecan be approximated so fine that one may casually think of this series of straightlines which approximating the knot, as a smooth knot for aesthetic purposes.

Following [Hodorog, 2010] in her section on some basics of knot theory, thefundamental idea of a mathematical knot will be introduced. We will utilisevarious definitions which prove more intuitive and accessible in different con-texts. In borrowing this scheme of equivalent definitions we will begin with ageneral definition of a knot, and then encounter a specific definition that allowsus to specify a particular knot.

Definition 1.1. A knot K ⊆ R3or S3 is a subset of points homeomorphic tothe circle, S1. This means that there is a continuous smooth diffeomorphism8

f : S1 → R3s.t f ′ 6= 0.

Note how this creates a smooth embedding of the circle in R3.

Example 1.1.

Figure 1.1:This is the Unknot or Trivial Knot

“The trivial knot is the simplest knot that can be represented as the boundary ofa disc embedded in R3” [Manturov, 2012](pg. 5).

8A diffeomorphism f : M → N , is a map between two manifolds M and N such that fand f−1 are differentiable.



Example 1.2.

Figure 1.2:The Trefoil or 31 Knot

Example 1.3.

Figure 1.3:This is the Valknut, or “Warrior’s Knot”, from Old Norse: valr, “slain warrior”+ knut, “knot”. It has occurred frequently in battle scenes depicted on Germanicstanding stones dating from the 7th century. Topologically, you will notice thatit is actually equivalent to figure 1.2, the trefoil knot.Source: [Wikipedia, 2012]

Remark 1.1. Note that there is no insistence upon how the circle is arranged inR3. This will be of importance later on when smooth and polygonal curves arediscussed.

It is natural to ask, how may the idea of equivalence amongst knots beenvisaged: i.e. the deformation of one knot into another in R3, or in otherwords, when are two knots identical? The definition above describes a knot asa collection of points, however we must to consider two knots equivalent even ifthey are not equivalent sets. Furthermore, we can think of equivalence betweentwo knots as the ability to manipulate, stretch, tangle and untangle one knotuntil it is identical to the other knot.

Formally, we describe two knots as being equivalent (or of the same knottype) if they are ambient isotopic [Long, 2005]. In a synopsis, homotopy allowsa knot to intersect itself, whereas isotopy allows a knot to contract down to apoint, and ambient isotopy allows a knot to be deformed through the space inR3 in which it sits. Using the definition of an ambient isotopy, we can begin todefine an ambient isotopy for two knots.

Definition 1.2. Let X,Y be topological spaces, and f, g : X −→ Y be continu-ous functions. Then f is homotopic to g, denoted f ' g if and only if

∃G : X × [0, 1] −→ Y

G(x, 0) = f(x) and G(x, 1) = g(x)

for all x ∈ X.



Here the map G is called a homotopy.

Definition 1.3. Let X,Y be two topological spaces, and f, g : X −→ Y be twohomeomorphisms. We say that f is isotopic with g, and we denote it f u g, ifand only if ∃H : X × [0, 1] −→ Y , a homotopy, such that Ht = H|X × {t} isa homeomorphism for all 0 ≤ t ≤ 1 .

We call H an isotopy.

Definition 1.4. Suppose K1,K2 ∈ R3are two knots. K1and K2are ambientisotopic if and only if there exists an isotopy h : R3 × [0, 1] −→ R3such that

h(K1, 0) := h0(K1) = K1

andh(K1, 1) := h1(K1) = K2

Ambient isotopic knots are denoted K1 ∼ K2. The strands of the knot maybe altered and displaced through space without self-intersecting each other, theentire knot may be contracted or expanded and we are not allowed to continu-ously ‘tighten’ the knot to the extent to which it unknots itself by contractinginto an infinitesimal point.

One must always remember that when we work with knots we are dealingwith their projections from R3 to R2 by means of a diagram of a knot. Theseprojections are, in most instances, easier to understand and to work with, al-though we are mainly concerned with three-dimensional topology. This leads usto a more specific definition of a knot that allows us to specify different typesof knot.

Definition 1.5. A knot is a continuous simple closed curve in R3.

The continuity in this definition allows the existence of infinitely knottedloops (wild knots). To eliminate the wild points in the wild knots (the pointwhere the small knots bunch up), we can introduce a condition of differentiab-ility. This eliminates the wild points, since there is no continuous way to definea tangent direction at this point.

In order to combat the difficulty of these wild points on wild knots, we imposethe condition of local flatness. The notion of local flatness dictates that at everypoint of the curve, within some arbitrarily small spherical neighbourhood of thepoint, the arc of the knot contained within the sphere is homeomorphic to adiameter of the sphere [Long, 2005].

Wild knots exhibit knot features in a similar way to the detail which can befound in fractal pictures: however much the picture is zoomed in on, the pictureexhibits increasing intricacy, which was previously unseen . In this instance,polygonal knots cannot be wild, so an alternative solution to this problem isto only consider the knot isotopy classes of polygonal knots. We can see anexample of a wild knot with a wild point below:


2 KNOTS AND LINKS

Figure 1.4:The Fox-Artin Wild Knot [Hodorog, 2010]

In an alternative manner, one can use polygonal curves to define knots. Aknot can be a collection of straight lines, with the coordinates of the endpoints,or corners, being given. Using the formal definition of a polygonal curve, wecan use an alternative definition of a knot.

Definition 1.6. Let p, q ∈ R3with p 6= q. Let [p, q] denote the line segmentjoining p and q. Let (p1, ..., pn) be an ordered set with each pi 6= qi for i, j ∈{1, ..., n}. Then P = ∪n−1

i=1 [pi, pi+1] ∪ [pn, p1] is a closed polygonal curve. P issimple if and only if each line segment intersects exactly two other line segmentsat their endpoints only.

This produces an augmented knot definition:

Definition 1.7. [Manturov, 2012]A knot is a closed polygonal curve in R3.

The line segments are called the edges of the knots, and the corners of arecalled the vertices of the knot. A knot is called tame if it has a polygonalrepresentation.

Knots are casually thought of and drawn as smooth curves, rather than poly-gonal curves. Naturally, a smooth curve may be approximated to a polygonalcurve in the manner we have specified. In this project knots are presented in asmooth manner for aesthetic reasons, but we should also think of them in theirpolygonal form.

2 Knots and LinksLinks generalise the idea of a knot to an entity with more than one component.Thus, they are knotted figures that may contain several loops, which cannot beclassified as knots on the whole. We call the number of components of a link itsmultiplicity, and so any knot is just a link of multiplicity 1.

Definition 2.1. A link is a finite disjoint union of knots, L = K0∪K1∪...∪Kn.Each knot, Ki is called a component of the link. The number of components ofa link is denoted mathematically by the linking number: lk(L).

With this established as our definition of a link, it clearly follows that:

Corollary 2.1. A knot is a link with one component.


2 KNOTS AND LINKS

Remark 2.1. Links can be oriented such that each component is assigned anorientation. This specifies a direction that we follow if we start at a chosenpoint on the link and continue along it until we return to that point.

Example 2.1. The name link indicates that the components of the link areconnected in such a manner that they are unable to be directly pulled aparthowever, as a knot need not necessarily be knotted, a link must not always be‘linked’, in our everyday understanding of the term. This can be observed in thecase of the unlink or the trivial link 2.1, which can be envisaged as the unionof unknots all lying in the same plane. A link that is n copies of the unknot iscalled the trivial link of multiplicity n.

Figure 2.1:The unlink or trivial link

Example 2.2. Two examples of simple 2-component links are the Hopf linkand the Whitehead link (this will appear later). Another interesting example isthat of the Borromean rings.

Figure 2.2:The Hopf link

Figure 2.3:The Borromean Rings

The Borromean Rings is a 3-component link with the property that if anyone of the rings (unknots) is removed then the remaining two rings becomeunlinked. Historically, this link appears as a heraldic symbol to represent thenotion that the strength of a group of people depends on each of the individualsand the loss of any one would undermine the strength of the whole [Long, 2005].

Just as we did for knots, we will define an ambient isotopy for links.

Definition 2.2. Two links L1, L2 are ambient isotopic if and only if there existsan isotopy h : R3 × [0, 1] −→ R3such that

h(L1, 0) := h0(L1) = L1


2 KNOTS AND LINKS 2.1 Projections & Diagrams

andh(L1, 1) := h1(L1) = L2

This ambient isotopy definition has to preserve the orientation of the links,without this condition there is a free choice of how to match the components ofL1 with those of L2. Without this condition, mirror images of knots would beambient isotopic to each other, despite having opposite orientations.Remark 2.2. This last condition is important, for example in the case of the leftand right trefoils, which are not ambient isotopic if the orientation condition istaken into account, but are if it is disregarded.

Figure 2.4:Here we see the left and right trefoil, which are not equivalent when our definitionof ambient isotopy for links is applied. However, if we remove the orientationpreservation condition, which produces a less stringent definition, they are indeedequivalent.

2.1 Projections & DiagramsWe now have a sufficient breadth of definitions for knots in R3, but in order tomake our computations with knots easier, we need to be able to describe themwith what is called a diagram. This is done by projecting the knot onto the2-dimensional plane, and then annotating it in some way which shows whetheran arc passes over or under another arc, when there is a crossing in the knot.These terms will be defined shortly.

We also require, to prevent confusion, that all singularities are double points,with the approaching arcs having distinct tangents. A diagram satisfying theseconditions is called a regular projection of the knot. [Long, 2005]

This idea can be used to answer the following question: how do we know thatthe trefoil knot and the unknot are different? Indeed, we require a representationof a knot in the form of a projection. There are many different projections ofthe same knot, and some are better, i.e. they give us more information, thanothers. This will be explained in the following definition.

Definition 2.3. Let p : R3 −→ R2 be the projection map from general 3-spaceto the xy-plane, with (x, y, z) 7−→ (x, y). If K is a knot, then the image of Kunder p is called the projection of K (i.e. K 7→ p(K)) [Hodorog, 2010].

This maps the points of the knot in R3 to R2, as if we see the shadow of theknot. Furthermore, a knot projection is called a regular projection if no threepoints on the knot are projected to the same point, and no vertex of the knotis projected to the same point as any other point on the knot. A double pointof a regular projection is called a crossing point of the projection.


2 KNOTS AND LINKS 2.1 Projections & Diagrams

Example 2.3.

Figure 2.5:A Projection K ′ of a knot K , with the crossing points marked in the zy-plane.Source: http://www.math.cornell.edu/ ~mec/2008-2009/HoHonLeung/page3_knots.htm

Regular projections have the advantage that they provide us with somevaluable information about neighbouring knots. If a knot does not have a regularprojection then, by definition, there exists an equivalent knot nearby that alsodoes not have a regular projection. Conversely, if a knot does have a regularprojection then all nearby knots are equivalent and have regular projections.

However, the disadvantage of regular projections is that they contain doublepoints (crossings), where it is not clear which of the two strands lies over orunder the other. Once a regular projective map of the knot or knots has beenestablished, we can then describe the projections by way of a diagram in R2.We shall see that a diagram is a planar representation of a knot.

Definition 2.4. A knot/link diagram is the image of a knot/link under regularprojection. This is coupled with information on each crossing that shows whichstrand passes under and which passes over. These are called over-crossings andunder-crossings.

Definition 2.5. An arc is the part of the diagram between two under-crossings.

Definition 2.6. We define a segment of a knot diagram or knot to be the portionof the knot projection or knot between crossings.

Definition 2.7. An oriented knot diagram is defined by determining a directionof travel around the knot. This direction is denoted by placing coherently directedarrows along the projection of the knot in the direction of choice. In this manner,we say that the knot is oriented. [Hodorog, 2010]

Definition 2.8. An alternating projection illustrates the under and over cross-ings of a knot.

Definition 2.9. An alternating projection with a fixed direction is called anoriented alternating projection.

Example 2.4. Here are some diagrams of the Figure of Eight (41) knot.


2 KNOTS AND LINKS 2.2 Arcs and Crossings

Figure 2.6:From left to right: A regular projection of a figure-of-eight knot diagram, analtered projection, an oriented alternating knot diagram and an alternatingprojection. [Hodorog, 2010]

2.2 Arcs and CrossingsEarlier we mentioned intuitive terms like arc and crossing. If one looks at theexample of the Trefoil knot, one can see that it features three arcs and threecrossings. One important definition for knots is that of the crossing number :

Definition 2.10. [Roberts, 2012] (1.2.7) The crossing number, denoted c(K),of a knot K is the minimal number of crossings in any diagram of that knot.This is a natural measure of complexity. A minimal diagram of K is one withc(K) crossings.

Remark 2.3. Currently one may find knot tables containing knots of up to 16crossings. There are tens of thousands of variations of these knots, according to[Roberts, 2012].

We can establish a correlation between arcs and crossings: it follows thatthe number of arcs is always equal to the number of crossings in the diagram ofa knot. We will use some formal notation to better describe the crossings of aknot diagram before we progress further.

Notice that the specified orientation of the knot in its diagram naturallyinduces an orientation and subsequent classification on its crossing points. Sup-pose the knot is a one-way street and cars are constrained to move only inthis one-way direction, which is determined by the knot’s orientation. Let theover-passing strand at a crossing point, be a bridge over another road whichlies underneath: this being the under-passing strand. If we stand on the bridge,facing in the direction in which the cars are coming from, then the cars on theroad below will either pass from right to left or from left to right. In the first casewe call the crossing right-handed, and in the second case we call it left-handed.

Definition 2.11. A crossing is left-handed if the underpass traffic goes fromleft to right, or it is right-handed if the underpass traffic goes from right to left.We denote a left-handed crossing with −1 (or LH) and a right-handed crossingwith +1 (or RH).

Definition 2.12. Whether left-handed or right-handed, each crossing is determ-ined (locally) by three arcs and we denote the over-passing arc with i, and theunder-passing arcs with j and k just as in the picture below:


3 THE REIDEMEISTER THEOREM

Figure 2.7:A left and right handed crossing, [Hodorog, 2010]

If we apply this notation to an example of the (left) trefoil, we get somethingthat looks like this:

Figure 2.8:Diagram of the Trefoil, with conventional notation for arcs and crossings.[Hodorog, 2010]

One piece of information we can derive from the crossing notation is thewrithe of the knot diagram. First, we associate each crossing of a knot or a linkwith a number ±1, this number is called the local writhe number. Then we takethe sum of these numbers at all vertices or crossing points.

Definition 2.13. The writhe of a knot diagram K is the sum of the local writhenumber of the crossing signs:

w(K) =

n∑i=1

ε(ci)

where each ci is a crossing of a diagram with n crossings and ε(ci) is the sign(±1) of the ci crossing.

This raises the question of whether the writhe changes as we change theknot, i.e. is the writhe a knot invariant? This is a question that we will returnto later once a series of knot manipulation techniques, called the Reidemeistermoves, and the notion of a knot invariant have been defined.

3 The Reidemeister TheoremAs was alluded to in the introduction, the fundamental question in knot theoryis the knot recognition problem. Unfortunately there is, as of yet, no (efficient)


3 THE REIDEMEISTER THEOREM 3.1 Reidemeister Moves

algorithm for finding the answer to this question.Instead of tackling this rubric head on, we will try to simplify the problem

to this statement: given the projection of a knot, can we tell whether it is theunknot? For something like a simple loop, this is an easy question. As anexample, if we study the following projection, the reader will be able to seewithout too much effort, how it may be manipulated into the unknot:

Example 3.1. In this figure 3.1 we can clearly see that this is the unknot, withjust a loop or twist. We can easily untwist this loop out, and thus obtain theunknot.

Figure 3.1:A diagram, but not a minimal diagram, of the unknot.

Example 3.2. Now what about these knots 3.2, are they equivalent to theunknot, or are they entirely different knots? Indeed, these knots is equivalentto the unknot, although one may not think so upon first inspection. In order tofind out, we must define special techniques of planar isotopy or unknotting .

Figure 3.2:Two knots that are in fact equivalent to the unknot.

When one starts trying to untangle and manipulate knots in order to makethem look like each other, one embarks upon what can seem like a labyrinthineprocess. However, in a very natural and elementary manner, exemplified by themanner of deduction from knots above, progress has been made on methods todifferentiate one knot diagram from another by way of certain techniques.

3.1 Reidemeister MovesOne such technique was devised by Kurt Reidemeister (1893-1971) in his paper:Elementare Begründung der Knotentheorie [Reidemeister, 1926]. Reidemeisterwas able to prove that two diagrams representing the same knot are related bya sequence of ‘moves’ called the Reidemeister Moves. There are three types ofmoves which may be made on a knot locally in the neighbourhood of a crossing



point. These moves do not change the knot type locally, and thus over allcrossings, they do not change the nature of the knot overall.

Definition 3.1. (Reidemeister Moves) A Reidemeister move is one of threeways to change the diagram of a knot by changing the relation between its cross-ings.

1. The First Reidemeister move 1: allows us to put in or take out a twist inthe knot.

Figure 3.3:The First Reidemeister move, which we shall denote R1.

2. The Second Reidemeister move 2 allows us to either add two crossings orremove two crossings.

Figure 3.4:The Second Reidemeister Move (R2) before (left) and after (right).

3. The Third Reidemeister move 3 allows us to slide a strand of the knotfrom one side of a crossing to the other side of the crossing. This strandmay lie above, below or in between the two other strands it slides over.

Figure 3.5:The Third Reidemeister move (R3) before (left) and after (right).

Together, these moves produce the following Theorem:



Theorem 3.1. Two knots in space can be deformed into each other by anambient isotopy if and only if their diagrams can be transformed into each otherby planar isotopy and Reidemeister moves.

In other words, two links or knots are equivalent if and only if one can betransformed into the diagram of the other by a sequence of Reidemeister moves.

Proof. We shall provide a sketch proof, combining ideas from [Manturov, 2012],[Roberts, 2012] and [Stokman, 2012]. For a complete proof see [Murasugi, 1996].

The beauty of this Theorem is that it allows us to think of a knot as beingan equivalence class of diagrams under the application of Reidemeister moves.This simplifies technical elements that we have already covered, and is in factthe approach taken in [Gilbert & Porter, 1994]. However, without the previoustechnical theory, this may seem somewhat artificial.

Note that the “if” part in the statement of the Theorem is rather trivial. It isclear that the application of Reidemeister moves does not change the equivalenceclass of the knot, represented by its diagram. Therefore if is there is a sequenceof moves that can be used to deform one diagram of a knot into another, thenboth diagrams represent the same knot. Hence we shall consider the “only if”part.

Suppose thatK andK ′ with diagrams D and D′ respectively, are equivalent.Then there exists a sequence of ∆-moves such that we can arrive at K ′ from K.We shall borrow from the description of ∆-moves in [Portnoy & Mattman, 2005].Here we shall use our knot definition from the introductory sections: a knot asa polygonal path.

Note that we can approximate knots as accurately as we like in this manner.We can change such a polygonal path with what is called a ∆-move. We takea triangle (in a plane) whose interior does not intersect the knot. The ∆-moveinvolves replacing one edge of the triangle with the other two (or vice versa).That is, if one edge is part of a polygonal path defining a knot, then replacingthe edge by the other two edges of the triangle is called a ∆-move.

Figure 3.6:A ∆-move

We can project the triangle over crossings in the diagram D. If there is acomplicated crossing, i.e. a part of D that contains a lot of detail and several


3 THE REIDEMEISTER THEOREM3.2 Effects of the Reidemeister Moves

twists, crossings or arcs, then we can partition the large triangle into smallertriangles. This corresponds to viewing the ∆-move as a composition of smaller∆-moves. We can do this for all the Reidemeister moves. Here is a picture ofthe R2 move as a ∆-move:

Figure 3.7:The R2 move as a ∆-move

This Theorem makes it possible for us to find invariants of knots based ondiagrams, because one can check the invariance of a particular characteristic ofa knot by showing that it does not change when any of the Reidemeister movesare applied to the diagram it describes. This means that the Reidemeister movesprove sufficient to describe ambient isotopy.

3.2 Effects of the Reidemeister MovesThe writhe, w(K), of a knot K was explained earlier on, and now that theReidemeister moves have been defined, we will examine their effect upon thewrithe.

1. ForR1, we see the loop transforms from w = ±1, depending on orientation,to w = 0. Therefore it is not preserved under this move.

Figure 3.8:In this figure we can see that the writhe of a knot is not invariant under theapplication of R1.

2. For R2, w = 0 before the move and remains zero after the move, regardlessof the link’s orientation.


4 BASIC INVARIANTS

Figure 3.9:Here we see how the initial two crossings give +1 and −1, or −1 and +1,depending on the oriented crossings respectively. This implies that w = 0 beforethe R2 move. After the move we can see that the crossings are removed sow = 0.

3. Finally with R3 there is an over-crossing at the start, implying w = 1 andthere is and over-crossing after the move, so w = 1 is preserved.

Figure 3.10:Observe that we did not alter the sign of the crossing of the two bottom strands aswe moved the third strand, so this obviously does not effect the writhe, regardlessof the orientation of this crossing.

Hence, we can deduce that the writhe of a knot does indeed change under theapplication of R1, i.e it is not a knot invariant (described in detail in the nextsection) and thus, it cannot be used to differentiate equivalent knots.

The principle conclusion from this example is that the Reidemeister movesprovide a practical analytical measurement tool for directly determining if twoknots are equivalent to one another. This can be easily done in the case of un-complicated knots with few crossings and a low complexity. With more intricateknots, however, simply trying moves in a heuristic manner to compute the pos-sibilities of deforming one knot diagram into another becomes very inefficientand laborious. What is needed is an algorithmic and more elegant means at ourdisposal, which can be used to prove, in a finite number of steps, that two knotsare either of the same knot type or of different knot types. This is the reasonwe direct our focus towards knot invariants.

4 Basic InvariantsAn invariant with respect to a transformation, is a knot or link attribute thatis not altered by the application of the transformation. Similarly, in the case ofknots, an invariant is a property that is not altered under ambient isotopy or, interms of knot diagrams, a property does not change after the application of any


4 BASIC INVARIANTS 4.1 Unknotting and Crossing Number

of the Reidemeister moves. Thus, any two equivalent knots will have the samevalue for a particular invariant. This is the usefulness of invariants for solvingthe knot recognition problem.

As in the case of knots, link invariants serve to show whether two links areequivalent. Again, two equivalent link diagrams can always be transformed intoone another using the Reidemeister moves, and so these can be used to verifypotential invariants [Long, 2005].

Definition 4.1. A link invariant is a function from the set of links to someother set whose value depends only on the equivalence class of the link.

Any representative from the class can be chosen to calculate the invariant.There is no restriction on the kind of objects in the target space (i.e. integers,polynomials, matrices or groups). [Hodorog, 2010]

Some invariants of knots/links include :

• numeric invariants: unknotting number and crossing number. These arethe most instinctive knot invariants to study. They remain easy to definebut their computation is difficult, which results in some of the most fun-damental questions that arise from them remaining unanswered. Theunknotting number remains unknown for many knots [Hodorog, 2010].

• (tri-)colourability: this is the simplest knot invariant that can distinguishthe trefoil knot from the unknot [Przytycki, 1995]. It involves ‘colouring’the arcs of a knot according to a definition. This method can also begeneralised for p-colourings.

• polynomial invariants (Alexander, Jones, HOMFLY polynomials). Thisassociates a polynomial to a knot class, which should in theory be thesame for each knot in that class. Later, we will focus our investigationspecifically on the Alexander polynomial.

The invariants above are common to both knots and links. Along with theseinvariants, we also have two other classes of invariants which characterise onlylinks:

• the linking number for oriented knots

• the number of components in the link.

4.1 Unknotting and Crossing NumberThe unknotting number of a knot is a minimal number of required crossingchanges in order to change any diagram of the knot into that of the unknot.Formally, we can define it in conjunction with the crossing number like this:

Definition 4.2. The unknotting number n of a knot K, denoted u(K) = n,exists if there is a diagram of K such that changing n crossings in the diagramresults in the diagram changing from the original knot K to that of the unknot.

In addition to this, there must be no other diagram such that the diagramof K could be changed into the diagram of the unknot by changing less than ncrossings.


4 BASIC INVARIANTS 4.1 Unknotting and Crossing Number

Definition 4.3. The crossing number of a knot K, denoted c(K), is the leastnumber of crossings that occur in any diagram of the knot.

Lemma 4.1. [Roberts, 2012] (Pg. 21) Any knot may be changed to a diagramof the unknot by altering some crossing points.

Proof. Let K be a knot and D its diagram, obtained by projecting K onto aplane. Let us choose a point p on D such that p is not a crossing point, and letus also choose an orientation for the diagram. Following the path from p of thediagram in the direction of its orientation, we will come to a crossing for thefirst time. At this point we will switch the diagram of the crossing if it is anunder-crossing, to that of an over-crossing. Upon arrival at the crossing if it isalready an over-crossing, we will do nothing and continue on along the path ofthe knot diagram. We employ this procedure for every crossing along the pathuntil we return to p.Claim. Here we claim that this procedure results in the changing of the diagramof the knot to that of the unknot.

In order to prove this claim we design an unknot that has the projectionof the altered diagram on a plane in R3 that is created by the aforementionedprocedure.

At our start point p let t = 0 and when we return to p let t = T . Letting(x, y) be the coordinates in a plane and using t 7→ (x(t), y(t), z(t)) for 0 ≤ t ≤ Tas a parametrisation of the knot, consider the curve C obtained by gluing t 7→(x(t), y(t), t) to a vertical line segment parallel to the t axis that connects itsendpoints (p, 0) and (p, T ).

Figure 4.1:

Our Tangent Line

Diagram

Gluing t 7→ (x(t), y(t), t) to a vertical line segment

This gives us a closed curve K ′, of K. If we view K ′ from the negativet axis, we notice that because an over-crossing always has a smaller value of tthan an under-crossing, this projection is the diagram described by our previousprocedure. We can observe how K ′ is unknotted by projecting it onto any planecontaining the t axis. We know that no two points of the curve C have the samet-coordinate because as we said at the start, p is not a crossing point. Thusthe projection is a simple closed curve with no crossings. Hence K ′ has thesame projection as the unknot as required by our claim and thus the Lemma isproven.

Theorem 4.1. For any knot K, we have

u(K) ≤ 1

2c(K)


4 BASIC INVARIANTS 4.2 Linking number invariant

Proof. Apply the procedure from Lemma 4.1 to a diagram with c(K) crossings.In this way, we make a maximum of c(K) changes to the crossings. If u(K) ≥12c(K), change K to the unknot K ′ with z-coordinate 0 ≤ t ≤ 1, i.e change thecrossings that we did not change the first time. Thus we have:

u(K) ≤ 1

2c(K)

Example 4.1. Here we see the unknotting procedure, or deformation, of thetrefoil into the unknot. The trefoil has a crossing number of three. We need onlyapply two Reidemeister moves, R2 and R1 in sequence, in order to deform thetrefoil into the minimal diagram unknot. Notice that after the application of R2

a crossing on the trefoil, we are left with a diagram that is ambient isotopic tothe unknot: it is simply an unknot with a twist. This illustrates that the trefoilhas an unknotting number of one. Optionally, we can apply R1 to this diagramof the twisted unknot, which yields the minimal diagram of the unknot.

Figure 4.2:c(trefoil)=3, u(trefoil)=1

4.2 Linking number invariantThe linking number was introduced by Carl Gauss, and has many applicationsnot only in knot theory, but also in algebraic topology, differential geometryand DNA super-coiling.

Definition 4.4. Consider an oriented link diagram of a link D. Choose twocomponents α, β of D. Let α u β denote the set of crossings of α with β (notincluding self-crossings) of which c is one of the crossings of α and β. Setc = +1 if the crossing on the link is right-handed, or c = −1 if the crossing onthe link is left handed. Then the linking number, ( lk), of α and β is:

lk(α, β) =1

2

∑c∈αuβ

sgn(c)

This describes the linking of two closed curves in three-dimensional space. AsKauffmann explains, the linking number is one-half of the sum of crossing signsof one curve with another [Kauffman, 1988].

Example 4.2. A torus link:



Figure 4.3:The two curves of this (2, 4) torus link have linking number four.

Example 4.3. Here are some links, and their corresponding linking numbers:

Figure 4.4:Different Links and their linking numbers: from left to right: Linking Numbers,lk= −2,−1, 0, 1, 2. In each pair the orientation of the leftmost link is anti-clockwise, whereas the orientation of the rightmost link is clockwise.

To prove that the linking number is an invariant, we shall check that itremains preserved after the application of the Reidemeister moves.

Theorem 4.2. The linking number of a knot is an invariant.

Proof. Examining each move individually, we find:

1. Under Reidemeister move one, one can see that the original linking numberis lk = 0, because (locally) there is only a single component link involvedin the move. Hence it is not a crossing between links and does not effectthe linking number.

2. Under Reidemeister move two, there are two cases like this:



Figure 4.5:The two cases in the proof of R2. Source: http://personal. kenyon.edu/holdenerj/ Math108Spring2007/ knottheory /invariant5.html

Here one can see that the initial linking number is zero regardless of ori-entation, since the two components are unlinked before the move. Afterthe move notice that the two new crossings created c1 and c2, have op-posite signs, thus they cancel each other out, leaving a linking number ofzero after the R2 move.

3. Here we refer the reader to [Murasugi, 1965] for a proof regarding thethird move.

Example 4.4. One particularly interesting example of a link is called theWhitehead link, discovered by J. H. C. Whitehead in 1934. It is used to constructthe Whitehead manifold. One can compute its linking number as follows:

1

2(1 + 1− 1− 1) = 0

This illustrates a somewhat counter-intuitive result: the linking number is zero,but the Whitehead link is clearly linked!

Figure 4.6:Two different diagrams of the same Whitehead link.


4 BASIC INVARIANTS 4.3 Colourability

Remark 4.1. However, if we compare the linking number of the Whiteheadlink to that of the Hopf link from earlier, their respective linking numbers aredifferent, hence we know that we cannot apply a sequence of Reidemeister movesto transform one diagram into the other. Indeed, the links cannot be equivalent.

4.3 ColourabilityAnother method of describing simple invariants of knots is by colourability.It involves colouring the arcs of knots in a specially defined way, in order todetermine which knots are colourable and which are not. This yields a basicinvariant, by way of an equivalence relation: we can then group together allthe colourable knots, distinct from the non-colourable knots. This method waspioneered by Ralph Fox, and used to make knot theory easily accessible toeveryone in a lecture he gave at Haverford College in 1956.

4.3.1 Tri-colourability

Definition 4.5. A diagram of a knot is tri-colourable if and only if:

1. each arc can be drawn using one of three colours such that at each crossingeither the 3 different colours meet or the same colour comes together, and

2. at least 2 colours can be used to colour the knot.

One can may interpret the colours as elements of the field F3 of order three.Let us define,

T (D) = {(x1, ..., xk) ∈ F3 : xi + xj + xk ≡ 0 mod 3

at each crossing of the arcs Ai, Aj , Ak}

as the solutions set to l homogeneous linear equations with k unknowns over thefield F3. Thus T (D) is the set of colourings for a particular diagram D. Fromthis, one can establish the following Theorem:

Theorem 4.3. T (D) is an F3 vector space, therefore

ξ(D) = 3dimT (D)

is a power of 3, where ξ(D) be the number of 3 colourings from T (D).

Proof. The solutions of the equation

xi + xj + xk = 0 mod 3

form a vector space with 3dimT (D) elements.

This tells us that the number of three colourings for any knot diagram is apower of three.

Example 4.5. The colourability of the trefoil knot:



Figure 4.7:The trefoil knot is colourable. [Hodorog, 2010]

Example 4.6. Now compare this with 4.6 below:

Figure 4.8:As a counter example, note how the figure of eight knot is not colourable: thereare crossings where only two different colours meet, which violates part one ofour definition of colourability 4.5. [Hodorog, 2010]

Remark 4.2. Also, the unknot is not colourable because we can only use onecolour to colour it, which is classed as the trivial colouring. We need to use atleast two according to the definition.

From this method of colouring we can make the following deductions:

• that the trefoil is not equivalent to the unknot or the figure-of-eight knot,hence they are different knots,

• that any colourable knot is non-trivial.

Remark 4.3. The elementary properties of colouring are as a direct result of arelationship with the Jones polynomial, which is discussed in the appendix.

One must concede that colourability is not a complete invariant for knots perse. Consider the figure-of-eight and the unknot from the previous examples,which we both found to be uncolourable. This means that colourability cannotbe used to show that the figure of eight knot is different from the unknot, andthereby non-trivial, but the figure of eight knot is clearly non-trivial.

In order to build upon the concept of colourability we will investigate casesof higher order colourings known as p-colourings.

4.3.2 P-colourings

Once three-colour colouring has been established, one may be naturally inclinedto investigate other ‘higher-order’ types of colouring. It can be interesting toexperiment with higher numbers of colours, although this does not seem to yieldinvariants of knots directly, nor easily.



In order to simplify these difficulties, one can turn to the approach of givingknots a different type of labelling; that of mod p labellings, where p is a positiveinteger. If p is a prime, the colours 0, 1, . . . , p− 1 form a field. Hence Tp(D),the set of p-colourings of a diagram of a knot D, is a vector space over Fp.

If p were not prime, Tp(D) would only form a module over the ring Fp. In de-scribing this notion, we shall use the definition from page 11 of [Przytycki, 1995].

Definition 4.6. We say that a link diagram D is p-coloured if every arc iscoloured by one of the numbers 0, 1, ..., p − 1 in such a way that at each crossingthe sum of the colours of the under-crossings is equal to twice the colour of theover-crossing modulo p. For a number p, Tp(D) is the set of colourings of thearcs of a diagram D. Formally, this is defined as

Tp(D) = {(x1, ..., xk) ⊆ Fp : xj + xk ≡ 2xi mod p,at the crossing with arcs Ai passing over Aj , Ak for 1 ≤ i, j ≤ k}

Thus, Tp(D) is the vector space of solutions of linear homogeneous equationsand ξp(D) = pdimTp(D).Remark 4.4. Furthermore, finding the mod p labellings of a knot is equivalentto solving a system of linear equations mod p for the particular knot. Thedimension of the solution space for such a system is called the mod p rank ofthe knot. Armed with both of these pieces of information, we can apply thefollowing Theorem:

Theorem 4.4. If a diagram of a knot is colourable then all of its diagrams arecolourable, which implies the number of three colourings is a link invariant. If aReidemeister move is performed on a colourable diagram, then it easily followsthat the resulting diagram is colourable:

Proof. Here we shall use modulo arithmetic and some diagrams of the Re-idemeister moves:

1. R1 involves just a single strand, so we can always use a single colour toyield the trivial colouring:

Figure 4.9:Colourability of R1. Source:[Wikipedia, 2012]

2. R2: Before the move we can simply use just two trivial colourings of theunlink: x1 and x2. In our colouring after the move, if x2 is the colour ofthe strand on top, x1 is the colour of the strand below, we call 2x2−x1 thecolour of the part of the x1 coloured strand enclosed by the x2 colouredstrand. At each of the two crossings 2x2 − x1 satisfies the congruenceequation:

x1 + 2x2 − x1 ≡ 2x2 mod p⇒ 2x2 ≡ 2x2 mod p



which satisfies our condition for colourability at a crossing. Thus the linkis colourable before and after the move.


3. R3: Colour the strand at the back (set ‘deepest’ into the page, so to speak)x1, then colour the strand passing over the two component strands of thecrossing x2, and finally colour the remaining middle strand x3. We followthe x1 strand from the bottom of the left-hand diagram (before the move)to the top, which we can colour using:

2x2 − x1 mod p

at the first crossing, and we can use the colour

2x3 − 2x2 + x1 mod p (1)

at the second crossing. Thus at the end of the x1 strand we have thecolouring

2x3 − 2x2 + x1 mod p

For the x2 coloured strand we can maintain this colouring throughout thestrand as it crossed over the others. Then for the x3 coloured strand wecan maintain this colouring until it reaches the second crossing, where itpasses under strand x2. Here we can colour x3 with:

2x2 − x3 mod p (2)

Now let us examine the after effect of the R3 move. Beginning in the sameway with the x1 coloured strand and following it from the bottom of theright-hand figure, we can use the colouring:

4x2 − 2x3 − x1 mod p

after the first crossing, and

−2x2 + 2x3 + x1 mod p

after the second crossing. Note that this is the same colouring as the endresult of our colouring for the x1 strand before the move, i.e. equation(1). Thus this colouring remains invariant. Once again we can maintainthe colour x2 throughout the x2 strand after the move. For the strandinitially coloured x3 after R3, we use the colour:

2x2 − x3 mod p

after its crossing. This is also the same as the colouring in (2) before R3.Thus this strand’s colour remains invariant under R3.


4 BASIC INVARIANTS 4.4 Polynomial Invariants


As these congruence equations are mod p and not just mod 3, p-colouring in-variance is proved for all Reidemeister moves.

4.4 Polynomial InvariantsCurrently a collection of three polynomials, called the polynomial invariants,contain the most successful method of identifying one knot from another.

• The Alexander Polynomial (1928): distinguishes all knots of eight cross-ings or fewer. Later we will discuss the approach taken in [Alexander, 1928],the paper which introduced this polynomial invariant.

• Jones Polynomial (1984): this was the first major development of moderntimes in polynomial invariants since Alexander’s discovery. Summarisingfrom [Jones, 2005], It distinguishes all knots of ten crossings or fewer, anda knot from its mirror image, but it does not differentiate mutant knots.A mutant knot produces the same polynomial form as another knot, butit remains structurally different.

• HOMFLY Polynomial (1985/87): this was a collaborative breakthrough[Freyd et al., 1985] which generalises the Alexander and Jones Polynomi-als, but again does not distinguish mutant knots.

We will primarily discuss the Alexander polynomial, but the reader will find acondensed treatment of the Jones and HOMFLY polynomials in the appendix.However, before we meet this polynomial invariant in the next part, we will firststudy some features of algebraic topology that we will use later in our treatmentof this polynomial.


5 ALGEBRAIC METHODS

Part III

Further Invariants“It was then that I began to look into the seams of your doctrine. I

wanted only to pick at a single knot; but when I had got that undone,the whole thing ravelled out. And then I understood that it was allmachine-sewn.”

Henrik Ibsen (1828–1906),

quoted from Mrs. Alving, in ‘Ghosts’, act 2.

5 Algebraic MethodsIn this chapter we will examine knots by employing some algebraic methods.Our goal is to construct the fundamental group of the complement of a knotin R3. This is essentially the fundamental group of the space remaining in R3

once we remove points belonging to the knot, but this will be discussed in depthlater.

Our motivation behind this construction is that it imparts information abouta given knot. Starting with the notions of paths, loops, fundamental groups andpresentations, we will prove the Van Kampen Theorem. This is a method forobtaining the fundamental groups of spaces. This method presents the funda-mental group in the form of a presentation consisting of words, which we willconsider after we have discussed these other components first. We now establishsome definitions from algebraic topology.

5.1 IntroductionWe know that K ⊂ R3, is a knot if and only if there exists a homeomorphismof the unit circle S1 into R3 whose image is K, as the knot is an embeddingin R3. Our problem is how does one define the fundamental group of the knotcomplement? First we will examine the structure of the fundamental group anddevelop our theory from there.

Definition 5.1. A path in a topological space X is a continuous map of someclosed interval into X. The images of the end points of the interval are the endpoints of the path, with the path being a joining of its two endpoints [Alexander, 1928].It is represented as such,

f : [a, b]→ X

Furthermore, a path is said to be closed or is called a loop, if the initial and theterminal points are the same. The loop is said to be based on the common endpoint.

Finally, utilising the above definition and the definition of a homotopy fromdefinition 1.2, we can explain the notion of the fundamental group of a topologicalspace.


5 ALGEBRAIC METHODS 5.1 Introduction

Definition 5.2. [Wilkins, 2012] Let X be a topological space, and let x0 ∈ Xbe any point of X. We can define an equivalence relation on the set of allcontinuous loops9 based on the common point x0 of X, where two such loops γ0

and γ1, γ0, γ1 : [0, 1]→ X, are equivalent if and only if

γ0 ' γ1

We can denote this equivalence class of a loop γ : [0, 1]→ X based at x0, as [γ].This is referred to as the based homotopy class of the loop γ. Moreover, the set ofequivalence classes of loops based at x0 is denoted by π1(X,x0). Therefore, twoloops γ0 and γ1 represent the same element of π1(X,x0) if and only if γ0 ' γ1.We shall callπ1(X,x0) the fundamental group of the Topological space X. Theidentity element of the fundamental group is the constant loop: εx.

All that remains to be shown is that π1(X,x0) is a group, which is illustratedin the proof of the following Theorem.

Theorem 5.1. [Wilkins, 2012] (3.1) Let X be a topological space, let x0 be somechosen point of X, and let π1(X,x0) be the set of all based homotopy classes ofloops based at the point x0. Then π1(X,x0) is a group, the group multiplicationon π1(X,x0) being defined according to the rule [γ1][γ2] = [γ1 · γ2] for all loopsγ1 and γ2 based at x0.

Proof. See [Wilkins, 2012] for a full proof, however a similar but more refinedstatement relating to the fundamental group of a knot will be proven in thenext subsection.

Next we will examine a method of describing the fundamental group, itselements and the operation which acts upon them. This method of descriptionis called a presentation.

5.1.1 Presentations

Definition 5.3. If S is a set of symbols a, b, c, . . ., let S denote the set of symbolsa, b, c, .. We define the set of words in S, denoted W (S), to be the set of all finitestrings of symbols from S ∪ S, including the empty word: ∅.

If w1 and w2 are two words, we can concatenate them in the obvious wayto make a new word: w1w2. Furthermore, any word can be written backwards,with all bars and unbars exchanged, giving an operation w 7→ w. [Roberts, 2012](Pg. 55)

Definition 5.4. [Roberts, 2012] (Pg. 55) In a continuation from the previousdefinition, S is a set of generators, and R ⊆ W (S) a set of relators, thus wecan define a group π1, called the fundamental group, as follows:

As a group, π1 = (W (S),∼), where ∼ is an equivalence relation defined byw ∼ w′if and only if there exists a finite sequence of words w = w0, w1, ..., wn =w′ such that each word differs from its predecessor by one of the following twooperations:

1. Cancellation: w1aaw2 ↔ w1w2 ↔ w1aaw2, where w1, w2 are words anda is any generator in S. Hence, this allows the insertion or deletion of abar-unbar pair of generators at any point in a word.

9Formally, a path in a topological space X, say γ : [0, 1] → X, at a point x0 such that:γ(0) = γ(1) = x0.



2. Relations: w1rw2 ↔ w1w2, for w1,w2 any words and r any element of R.This means that an element of R can be inserted or deleted from any pointof a word.

We shall write [w] to denote the equivalence class represented by a wordw. Notice that [w] ∈ π1. If we pay attention to group characteristics, themultiplication operation is induced by concatenation of equivalence classes ofwords: [w1][w2] = [w1w2]. The identity element is [∅], and the inverse of [w] is[w].

In this manner, we say that π1 has a presentation 〈S : R〉. For the purposesof this section we shall only consider the cases when S and R are finite sets,which means that π1 is finitely-presented. The reader will notice that we havethe elements of a group structure. It will now be proved rigorously by thefollowing Lemma.

Lemma 5.1. The preceding procedure defines a group structure for π1(W (S),∼).

Proof. We need to prove that the operation of multiplication is actually well-defined (since it’s expressed using representatives of equivalence classes), thatit is associative, and that the identity and inverse work properly.

• First, note that for any words, u ∼ v implies both

uw ∼ vw

andwu ∼ wv

by concatenating w at the start or end of all words in a sequence relatingu and v. Therefore, if w1, w′1 are representatives for [w1] and similarly ifw2, w′2 are representatives for [w2], then

w1w2 ∼ w1w′2 ∼ w′1w′2

and so[w1][w2] = [w′1][w′2]

as required.

• Associativity is obvious, because concatenation of words is associative.

• Concatenation with the empty word obviously leaves everything unchanged.

• Inversion is well-defined because any sequence of cancellations and rela-tions also works when the barred words are used. In particular, note that

r ∼ ∅ ⇒ rr ∼ r ⇒ ∅ ∼ r

Hence, inverses of relators can also be considered as relators.

• And finally, for any word w we have:

ww ∼ ∅ ∼ ww

by repeated cancellation of opposite pairs from the middle of those words,therefore [w][w] = 1. (Proof from pg. 56 of [Roberts, 2012])



Now we must construct a way of describing homomorphisms from groups,deriving from presentations of different groups. Suppose π1 = 〈S : R〉 is a groupdetermined by a presentation, and let G be a different group. The followingLemma says:

Lemma 5.2. [Roberts, 2012] (Pg. 56) There is a bijective correspondencebetween functions f : S → G and functions f : W (S)→ G which satisfy

f(w1w2) = f(w1)f(w2)

for all words w1, w2 ∈W (S).

Proof. We shall follow the exposition on pg. 56 of [Roberts, 2012]. Any fdefined on W (S) also defines an f on S by restricting it to the words of length1, which include single symbols of S.

Conversely, given an f on S, we can extend it to S by setting f(a) = f−1(a)

(the inverse is the inverse in G), and then define f on a word w by reducing theword into its constituent generators in S∪S. We then apply f , and multiply theresulting elements of G together. Such an f obviously satisfies the multiplicativeproperty, note that this identity also implies that:

f(w) = f−1(w) and f(∅) = 1G

Hence, these two operations f and f are mutually inverse, and thus bijective.

Lemma 5.3. Let π1 = 〈S : R〉 be a group given by a presentation, and G besome other group. Then there is a bijective correspondence between homomorph-isms θ : π1 → G and functions f : S → G whose associated f functions satisfyf(r1) = f(r2) for any relation r1 = r2 in R.

Proof. Any homomorphism θ : π1 → G determines a function f : S → Gby setting f(a) = θ([a]), for any generator a ∈ S. Clearly the associatedf : W (S) → G in this case is given by f(w) = θ(w), if one carries out theabove construction and uses the fact that θ is a homomorphism. Therefore itsatisfies f(r1) = f(r2) for any relation, because [r1] = [r2] in π1.

Conversely, any function f : S → G determines an f : W (S) → G by theprevious lemma. This function satisfies f(w1aaw2) = f(w1w2) = f(w1aaw2)

automatically, because of the way f . If the f satisfies the extra hypothesis in thestatement of the lemma then it also satisfies f(w1r1w2) = f(w1r2w2) for eachrelation. Therefore f induces a function θ : π1 → G. Because of the definitionof f , this is a homomorphism. Again, the two operations are mutually inverse,giving a bijection. (Pg. 56)[Roberts, 2012]

Remark 5.1. It is here that we see an equivalent formulation between the twogroup structures of the fundamental group we have just described. This meansthat words can be used to describe loops, the equivalence class of loops corres-pond to homotopy classes and equivalence between two equivalence classes ofwords may be interpreted as equivalence of the respective homotopy classes ofcorresponding loops. Thus the two groups are in fact merely different means ofdescribing the same group structure on the fundamental group.



Example 5.1. [Roberts, 2012] (Pg. 59)

1. The fundamental group of Rn is trivial, because all maps into Rn arealways homotopic using a linear homotopy

ft(x) = (1− t)f0(x) + tf1(x)

which indeed works rel{0, 1}, which means f0|A = f1|A for some subset Aof the domain of f0 and f1.

2. The fundamental group of Sn with n ≥ 2 is also trivial, by a Lebesguecovering lemma10 argument, which ensures that any loop is homotopic toone missing the north pole, and therefore homotopic to one into Rn, whichis homotopic to the constant loop.

3. For n = 1 this argument fails, and indeed π1(S1) ∼= Z for any base point.To prove this one uses the covering map11:

p : R → S1

x 7→ e2πix

Any map I → S1 can be lifted into a unique map to R, given a lift of itsstarting point, and the lift of any loop will end at a value n more than itsstarting point, where n ∈ Z. This integer is the winding number of theloop, and defines the isomorphism to Z.

4. If X is a path-connected space then π1(X,x0) ∼= π1(X,x1) ∼= ... for allxi ∈ X, i.e. the isomorphism class of group is independent of the basepoint. The isomorphism is defined by picking a connecting path γ : x0 →x1 in X, and then sending any loop α at x0 to the loop γ · α · γ−1, whichgoes back along γ from x1to x0 , around α, then forwards along γ from x0

to x1 again. Indeed this is reversible up to homotopy. Thus, we tend toignore the base point when referring to the fundamental group of a path-connected space, however it should not be completely forgotten about.

The fundamental group contains some crucial functorial attributes. Themost important pertains to enabling us to discover that mappings from onespace to another provide mappings from the fundamental group of one space, tothe fundamental group of the other space. These are described in the Lemmabelow.

Lemma 5.4. [Roberts, 2012] (7.2.4)

1. If f : X → Y takes x0 to y0 then composing it with loops in X induces ahomomorphism f∗ : π1(X,x0) → π1(Y, y0). The identity map ι : X → Xinduces the identity homomorphisms, and if g : Y → Z, y0 7→ z0, theng∗f∗ = (gf)∗.

10Throughout this section we will make frequent use of this Lemma. See [Fulton, 1995] (pg.371) for a proof of this Lemma.

11A covering map is a continuous mapping p : X → X such that each point x ∈ Xis in an open neighbourhood U such that p−1(U) is a disjoint union of open sets, each ofwhich is mapped homeomorphically by p onto U . Also X is called the covering space of X.[Fulton, 1995]



2. If f, g : X → Y both take x0 7→ y0 and are homotopic rel {x0} then f = g.

3. If f, g : X → Y have f(x0) = y0 and g(x0) = y1 where y0 is not necessarilyequal to y1, and f, g are homotopic, then one has to let γ be the patht 7→ F (x0, t) around which the image of x0 moves during the homotopyF , where F : f ' g. Then g∗(x) = γf∗(x)γ (compare (4) in the previousexample).

Proof. Proofs of these properties can be seen in [Wilkins, 2012](pg. 42).

Definition 5.5. [Roberts, 2012] (Pg. 59) We say that two spaces X and Y arehomotopy-equivalent if there exist maps f : X → Y and g : Y → X such that

fg ' ιY and similarly gf ' ιX

A space with such a property is called contractible.

Lemma 5.5. If X and Y are both homotopic-equivalent and path-connected,then their fundamental groups are isomorphic to each other.

Proof. Firstly, we know from the previous definition that gf ' ιX . Using thisin:

g∗f∗(x) = (gf)∗(x) = γ(ιX)∗(x)γ−1 = γxγ−1

shows that g∗f∗is an isomorphism from π1(X,x0) to π1(X, gf(x0)) throughπ1(Y, f(x0)). In a similar manner, we can show that f∗g∗ is also an isomorphismfrom π1(Y, f(x0)) to π1(X, fgf(x0)) through π1(X, gf(x0)). Both g∗s are thesame12: the first being surjective and the second being injective, thus this is anisomorphism.

5.1.2 The Van Kampen Theorem

In this sub-section we will examine a useful theorem which describes the fun-damental group of a union of two spaces in terms of the fundamental group oftheir intersection, under certain conditions.

A free product with amalgamation is a push-out in the category of groupsand homomorphisms:

G = G1 ~G0G2

In this context we have homomorphisms:

i : G0 → G1 , j : G0 → G2

k : G1 → G , l : G2 → G

and k ◦ i = l ◦ j. So we have a commutative square:

G2l−→ G

↑ j ↑ k

G0i−→ G1

However if G is the free product with amalgamation then more is true.12Note that the f∗s are actually different because different base points are involved.



We have a universal property given any group H with homomorphisms:

p : G1 → H , q : G2 → H

such that p ◦ i = q ◦ j, there exists a unique homomorphism r : G → H suchthat p = r ◦ k and q = r ◦ l. Thus we have

−→ −→ Hq

↗ ↗∃!r

↑

G2l−→ G ↑

j ↑ ↑ k ↗p

G0 −→i

G1

A corresponding universal property is satisfied in the category of topologicalspace and continuous maps where G0, G1, G2 and G are replaced by the fun-damental groups of U ∩ V, U , V , and U ∪ V respectively, where U , V are opensubsets of X with U ∪ V = X and where i, j, k and l are the inclusion map-pings. In order for the theorem to hold, we also require U , V , and U ∩ V tobe path-connected, and all of X, U , V , and U ∩ V to have universal coveringspaces.Remark 5.2. Note that this description of the universal property matches upwith the presentation of the free product with amalgamation, and that the VanKampen Theorem states that π1(X,x) is the universal group with this property.

Theorem 5.2. Making the substitutions above for G0, G1, G2 and G, we maystate the Van Kampen Theorem as follows:

for any homomorphisms

p : π1(U, x)→ H and q : π1(V, x)→ H

such that q ◦ j = p ◦ i, there is a unique homomorphism

r : π1(X,x)→ H

such that r ◦ l = q and r ◦ k = p.

Proof. A sketch proof will be provided later: see the sketch proof of the similartheorem in terms of presentations, Theorem 5.1.2.

Now we will investigate particular cases.

Example 5.2. Let G0 be the trivial group. Then

G ∼= G1 ~G2

where this is the free product of G1 and G2.

Example 5.3. In this case let G2 be the trivial group. Then

G ∼= G1/N

where N is the normal subgroup generated by i(G0). In particular, if G1is givenin terms of a presentation and if G0

∼= Z then this just adds an extra relation:

i(generator of G0) = identity

to the presentation.



Utilising the description of the fundamental group in terms of presentations,we may also state the Van Kampen Theorem in a different manner, paraphrasedfrom [Roberts, 2012].

Theorem 5.3. [Roberts, 2012](Pg. 60) Suppose X be a topological space withsubsets U,W and V , as in our previous statement including our assumptions,but such that W = U ∩ V . Letting x0 be a base-point in W and supposing thatthe fundamental groups of U, V and W are given by presentations:

π1(U, x0) = 〈SU : RU 〉 , π1(V, x0) = 〈SV : RV 〉 , π1(W,x0) = 〈SW : RW 〉

we examine the inclusion mappings:

i : W ↪→ U, j : W ↪→ V

and their corresponding induced mappings of fundamental groups: i∗ and j∗. Foreach g ∈ SW , we can choose a word wU (g) ∈W (SU ) that represents the elementi∗(g). Similarly, we can do the same for a word wV (g) ∈ W (SV ) representingthe element j∗(g). Then it follows that π1(X,x0) has a presentation:

〈SU ∪ SV : RU ∪RV ∪ {wU (g) = wV (g) : ∀g ∈ SW }〉

What this formulation of the Theorem says is that we begin with a union ofboth presentations of π1(U) and π1(V ). The problem is that a loop in W maybe described by a word in the SU generators but also by a different word inthe SV generators. These words represent distinct elements because there areno relations that shuffle the two generators. In π1(X) they should describe thesame element, hence we must include some new relations identifying that thesetwo generators are equivalent so that we can prevent duplication. Althoughit proves sufficient to add such a new relation for each generator of π1(W ),instead of adding one for each loop. Therefore, provided that π1(W ) is finitelygenerated, we need only add finitely many of these new relations.

Proof. A sketch proof shall be provided. For a full proof, see Knots & Surfacesby [Gilbert & Porter, 1994]. We begin by constructing a homomorphism fromthe free group13 〈SU ∪ SV 〉 to π1(X). Indeed this can be done in a simplemanner: the generators in SU and SV represent loops in U and V . We canmap a word in these generators to a concatenation of their respective loops.The surjectivity of this mapping follows from the Lebesgue covering Lemma14

and the path-connectedness15 of X. Indeed every relation RU , RV and thesuperfluous ones used in the Theorem are satisfied by this surjective mapping.From this we may induce the epimorphism:

〈SU ∪ SV : RU ∪RV ∪ {jU (g) = jV (g) : ∀g ∈ SW }〉 → π1(X,x0)

13A free group is a group with a subset such that any element of the group may be writtenas a product of finitely many elements of the subset, including the inverses of said elements,but neglecting trivial variations of the combinations.

14Dissect any path in X based at x0 into a finite number of smaller paths, each one lyingcompletely inside at least one of U and V . This is proven in the appendix.

15At each point of dissection, which lies in X, we insert an extra loop, contained within X,to the base point and then reverse along it before continuing along the next small segment.Now the path is a visible composite of loops, each contained inside at least one of U or V ,which is represented by some word in the generators.


5 ALGEBRAIC METHODS 5.2 Seifert Surfaces

The remainder of the proof proves injectivity. We assume that some word inW (SU ∪SV ) is mapped to a null-homotopic16 loop in X and splits up this null-homotopy into small parts17, each of which correspond to a homotopy in U orin V , which may already be known. It is the additional relations which cause achange of coordinates between the sets U and V , which can only occur in theintersection and nowhere else.

Next we shall see the application of the Van Kampen Theorem in our con-struction of the knot complement, but first we will begin by describing theconcept of a knot as being the boundary of a surface.

5.2 Seifert Surfaces18

A surface in the mathematical sense, is a two dimensional manifold which may beorientable or non-orientable, and with or without a boundary [Long, 2005](pg.35). Although a surface has no thickness, we think of orientable surfaces asthose that have two distinct sides which we could ‘colour’ two different coloursdetermined by the direction of a normal vector on a particular side of the surface.

Definition 5.6. A Seifert surface for an oriented link L is a closed and compactorientable surface in R3, whose boundary is the link L. [Manturov, 2012] (pg.18)

The orientation of this link L is determined by the orientation of the Seifertsurface.

Example 5.4. A sphere is an example of an oriented surface without boundary.A cylinder is another oriented example, but with a boundary consisting of twocircles. A Möbius band (a strip connected end-to-end with one half-twist) has aboundary and is non-orientable. The boundary has one component which lookslike a circle with a twist [Long, 2005].

In 1930, Frankl and Pontrjagin proved that for all knots there exists a connec-ted orientable surface with the knot as its boundary [Frankl & Pontrjagin, 1930].The German mathematician Herbert Seifert improved on this result in [Seifert, 1934]by giving a separate proof which also included an algorithm for creating such asurface. It is this surface that we are interested in.

Remark 5.3. For each knot there is more than one Seifert surface. This isbecause the form of the Seifert surface depends on which diagram of the knotis used.

5.2.1 Construction of the Seifert Surface

Construction of the Seifert surface begins at the crossing points of the knot orlink. We follow the orientation of the knot from any point until we arrive at acrossing point. Here we shall use an analogy of a motorway. If the reader canimagine a positive (right-handed) crossing point as a crossing where one roadpasses over another, with the traffic travelling from left to right. This is the

16A function f is null-homotopic if it is homotopic to a constant function.17Again by using a similar Lebesgue lemma idea.18I would like to acknowledge the assistance of my supervisor, in these forthcoming sections

leading up to the Alexander Polynomial.



over-crossing, and similarly, the road lying perpendicular below is the under-crossing, with its traffic travelling from bottom to top. We imagine that wewould like to get from the higher road to the lower road. In order to do this weneed to construct a slip road from the left half of the over-crossing that bendsto the left and declines down to the upper half of the lower road. Similarly, ifwe want to travel from the lower road to the higher road, we need to construct aslip road from the bottom half of the under-crossing to the right half of the over-crossing so that it inclines and bends to the right. For a negative (left-handed)crossing this process is inverted.

We preform this procedure at each crossing on the diagram, changing thespecified crossing in the appropriate manner. Once this is done, we find thatall the crossings have been eliminated to obtain a collection of disjoint unknots,which are called Seifert circles. We then attach a disk to each of these in thelower half-space such that the disks or bowls do not intersect.

Then say each Seifert circle is represented by a perfect circle in the planez = 0 so that we have a disjoint collection of Seifert circles. Take each circle asthe equator of a hemispherical bowl in the lower half-sphere. The bowls do notintersect, thus we have the respective portion of the Seifert surfaces.

Example 5.5.

Figure 5.1:A link diagram of 52 and its Seifert circles. Source:[1]

We also specify the orientation on the smooth portion of an oriented surfaceby specifying the normal direction. Think of using this unit normal to picturea plane of ‘red’ and ‘blue’ sides. If we consider an oriented path through thesurface determining a corresponding orientation at a point of the path set by apositively oriented orthogonal or orthonormal triad t

∼, l∼and n

∼. Here t

∼points

along the direction of travel, l∼points into the surface to the left and n

∼points

towards your head.Now think of what happens as you go around the boundary of a bowl. In

the anti-clockwise direction: t∼

points anti-clockwise, l∼

points downwards, n∼

points inwards, and at the very bottom of the bowl n∼points upwards. With a

clockwise circulation around the boundary: l∼points upwards out of the surface,

n∼points outwards, and at the very bottom of the bowl n

∼points downwards19.

Thus we see that replacing crossings with ‘slip-roads’ and constructing ‘bowls’under each resulting Seifert circle yields a collection of bowls with a defined ori-entation on each bowl.

19This is the standard orientation convention i.e. for Stokes’ Theorem.



Figure 5.2:An oriented diagram of the 52 knot (left), and our construction of Seifert circles(right). Note that in the right-hand diagram, Seifert circles that have the sameorientation lie in different planes, and those which lie in the same plane haveopposite orientations.Source: http://www.sgwater.org/math/knots/

The boundary will consist of the knot away from the crossings together withthe slip-roads at the crossings. We complete the construction of the surface byputting a strip with a half-twist at each crossing. Note that the strip twistsanti-clockwise to your left as you go through a right-handed crossing on thesurface in the direction of travel indicated by the directions of the knot arcs.

Figure 5.3:The twisting of a right-handed crossing into a Seifert surface, illustrated with achecker-board colouring.Source:[2]

Example 5.6. The Seifert Surface of the trefoil. Following the procedure forconstructing the Seifert surface for 31, we draw a circle bounding the edge ofthe entire knot, and then we draw second circle inside the first, which boundsthe interior arcs. Then we make the circles discs by filling them in, we elevatethe smaller inside disc to a higher level and connect the two discs together bythree bands (one for each crossing point). Each disc has a boundary which isclockwise oriented, and each band has one right-handed half-twist.


5 ALGEBRAIC METHODS 5.3 The Knot Complement

Figure 5.4:Here we see the Seifert Surface of the trefoil. Note the two circles on differentlevels and the ‘half-twist’ bands connecting both levels.Source:[3]

Theorem 5.4. Every knot is the boundary of an orientable surface.

Proof. We shall use a proof based upon the interpretation of an orientable sur-face as having two faces that we can paint red and blue depending on theorientation of the normal vector on a side of the surface. Our Seifert construc-tion yields a surface with the original knot as its boundary. Then we colourthe Seifert circles with two colours. Viewing the pile of discs from above, forall of the Seifert circles at each level, if the orientation of a disc’s boundary isclockwise then let us colour the ‘top side’ of the disc red. If the orientation isanticlockwise then we colour it blue.

Next, colour the ‘bottom side’ of the discs with the opposite colour to the topsides. Thus, we have systematically coloured whole surface with two colours. Ifwe begin on the red side of a disc and travel on the same level to another disc,then the half-twist ‘ramp’ takes us to the bottom of the next disc, which is alsored. If we go up a level, then the half-twist ensures we remain on the top of thenext disc, which is red. Now we have that the red faces of the Seifert circlescomprise one entire side of the Seifert surface. In a similar manner, the bluesides make up the other face and we have an orientable surface as required.

Now we will use the Seifert surface to obtain an infinite cyclic cover of theknot complement by slicing the knot complement along the Seifert surface ofthe knot, and gluing together infinitely many copies of the resulting manifoldwith boundary in a cyclic manner [Wikipedia, 2012].

5.3 The Knot Complement5.3.1 Infinite Cyclic Cover of the Knot Complement

Our motivating 2-dimensional example is the helicoidal covering of the punc-tured plane. Equivalently, take a 2-sphere and choose two points joined by aline segment or arc, and then construct a covering space so that you go up astorey on crossing the arc in the positive direction. The Helicoidal covering isthe special case of this when one of the endpoints is the ‘point at infinity’.



Figure 5.5:The Helicoid. Source: [Wikipedia, 2012]

To realise this take a continuous function f(x), which can be smooth, suchthat it is zero on the positive side of the arc, and one on the negative side of thearc. Then take X = 2-sphere minus the endpoints of the arc, and X = {(x, t) ∈X ×R : t− f(x) ∈ Z}. Then f defines the height on a ramp, as in a multi-storycar-park.

5.3.2 The Knot Complement

Definition 5.7. The knot complement X, of a knot K is the space in S3 ex-cluding the points of K i.e. X = S3\K

Remark 5.4. If K is a knot in R3, then let X be the complement space of theknot (X\K). This is a path-connected non-compact 3-manifold.

The definition of the knot complement may differ in two ways. The first maybe the idea of knots as being situated in S3, which is R3 plus the union of apoint at infinity. This does not effect knot theory, since knots and sequences ofdeformations can always be assumed not to hit infinity. Secondly, a small openε-neighbourhood20 of a knot is homeomorphic to an open torus. The removalof this neighbourhood creates a 3-manifold X ′ with a torus as its boundary.These different compliments have the same fundamental group, therefore it isirrelevant which we use, so long as we are consistent in our use throughout.

We can conduct an analogous procedure to that of our construction of thehelicoid, albeit in one higher dimension, to get our infinite cyclic cover of theknot group. If we have an oriented knotK in S3 which bounds the Seifert surfaceΣ, an oriented surface without self intersections. Then construct a continuousor smooth function f : S3\Σ→ [0, 1] so that f = 1 along the negative side of Σand f = 0 along the positive side of Σ. Extend f to have the value 0 on Σ thentake infinite cyclic cover Y as:

Y = {(x, t) ∈ S3\K : t− f(x) ∈ Z

Now let [π, π] be the (normal) subgroup21 of π generated by αβα−1β−1 forall α, β ∈ π. If π = π1(X,x0) for any topological space X then:

π

[π, π]∼= H1(X)

20An ε-neighbourhood of a set U ⊆ X, where (X, d) is a metric space, is the set of pointsin X that are within a distance ε > 0 of some u ∈ U .

21The subgroup generated by these things is normal because γ−1(αβα−1β−1)γ =(γ−1αγ)(γ−1βγ)(γ−1αγ)−1(γ−1βγ)−1 ∈ [π, π].



where H1(X) is the first homology group22 of X.So we get the infinite cyclic cover of the knot group:

p : Y → S3\K

Let π = π1(S3\K,x0) where x0 is a base-point in the knot group. Then

π1(Y, y0) ∼= p#(π1(Y, y0)) ⊂ π1(S3\K,x0)

From the Wirtinger presentation (to be discussed later) we can show that:

π1(Y, y0) ∼= [π, π]

the commutator subgroup of π. For what it is worth:

H1(Y ) ∼=[π, π]

[[π, π], [π, π]]

Now we have a homeomorphism θ : Y → Y where θ((x, t)) = (x, t + 1), witht− f(x) ∈ Z of course. Now θ induces an isomorphism θ∗ : H1(Y )→ H1(Y ).

We know that H1(Y ) is an Abelian group, so if we can have a Laurentpolynomial23, this acts on H1(Y ) with:

(∑p

nptp)(γ) =

∑p

np(θ∗)p(γ) , np ∈ Z

Thus H1(Y ) becomes a module over the ring R of Laurent polynomials withinteger coefficients, where R = Z[t, t−1]. The Alexander polynomial, featured inthe next section, is an algebraic invariant ofH1(Y ) considered as an R-module24.In recipes to calculate this, one represents H1(Y ) as an exact sequence25 of R-modules:

Rq′→ Rq → H1(Y )→ 0

and then determines the Alexander polynomial from the determinants of theminors of the matrix representing Rq

′ → Rq as a generator of the ‘fitting ideal ’associated with this matrix. Before we continue, we will first discuss fittingideals and elaborate further upon the above representation in the next part.

Remark 5.5. The homology of the knot complement S3\K is not particularlyinteresting:

H0(S3\K) ∼= Z , H1(S3\K) ∼= Z

H2(S3\K) = 0 , H3(S3\K) = 0

22For a definition of this, we direct the reader to [Wilkins, 2012].23In Z[t, t−1].24A (left) R-module M say over a unital ring R, consists of an Abelian group (M,+) and

an operation R ×M → M with r(x + y) = rx + ry, (r + s)x = rx + sx, (rs)x = r(sx) and1Rx = x, where x, y ∈M and r, s, 1R ∈ R. [Wilkins, 2012](pg. 5).

25This is a sequence where Image(incoming)=Ker(outgoing).



5.3.3 Fitting Ideals

If R is an integral domain, namely R = Z[t, t−1] and M is a finitely-presentedR-module, this means that there is an exact sequence:

Rs → Rq →M → 0

of R-modules.In this situation:

RsA→ Rq

ϕ→M → 0

where A is represented by a q×smatrix with coefficients in R. Here ϕ : Rq →Mis surjective and Ker ϕ = Im(A), so

M ∼= Rq/A(Rs)

Thus if

g1 = ϕ(1R, 0, ..., 0)

g2 = ϕ(0, 1R, 0..., 0)

......

...gq = ϕ(0, ..., 0, 1R)

then g1, g2, ..., gq are generators of M and the columns of A specify relationsbetween those generators. We let the jth fitting ideal of M , call this Fj(M), bethe ideal of R generated by the determinants of (q − j) × (q − j) minors of Afor i < q, Fj(M) = R for j ≥ q. Now: F0(M) ⊂ F1(M) ⊂ F2(M) ⊂ ... Here wemake the following claim:Claim 5.1. The ideals Fj(M), are algebraic invariants of the R-module M thatare independent under choice of presentation:

RsA→ Rq

ϕ→M → 0

Proof. First considerRs

A→ Rqϕ→M → 0

andRs′′ A′′→ Rq

ϕ→M → 0

two presentations with the same homomorphism ϕ : Rq →M .Now A(Rs) = A′′(Rs

′′) = Ker ϕ, therefore we can find an R-module homo-

morphism B, such that A = A′′ ◦B

Rs

|A

↘B Rq

↓ ↗A′′

Rs′′

For each generator in the obvious free basis of Rs pick an element of Rs′′with

the same image in Ker ϕ and extend by linearity.



It follows that because A = A′′ ◦B, the determinants of the (q− j)× (q− j)minors of A are linear combinations of those of A′′. Similarly those of A′′ arelinear combinations of those of A, with coefficients in R. Thus the ideal of Rgenerated by the (q− j)× (q− j) minors of A coincides with that generated bythe (q − j)× (q − j) minors of A′′. This completes the first step.

Now suppose we have presentations:

RsA→ Rq

ϕ→M → 0

andRs′ A′→ Rq

′ ϕ′→M → 0

with Rq+q′ ∼= Rq⊕Rq′ . Define θ : Rq⊕Rq′ →M such that θ(x, y) = ϕ(x)+ϕ′(y)

and let K = Ker θ. Now we have an R-module homomorphism C : Rq → Rq′

such that ϕ = ϕ′ ◦ C. Then θ(x,−Cx) = ϕ(x)− ϕ′(x) = 0, thus (x,−Cx) ∈ Kfor all x ∈ Rq.

Thus if (x, y) ∈ K then (0, y + Cx) ∈ K so y + Cx = A′z for some z ∈ Rs′ .So K is the image of:

ψ : Rq ⊕Rs′→ Rq ⊕Rq

′

where ψ(x, z) = (x,−Cx + A′z). Hence the matrix for ψ has block form asfollows:

. . . · · · |. . . · · ·

... Iq×q

... |... 0

q×s′

...

· · ·. . . | · · ·

. . .

|−Cq′×q

| A′q′×s′

|

Now any non-zero determinant of a minor must involve entries from the top-leftand bottom-right. A non-zero n × n minor of the block matrix must equal ann′×n′ minor of A′ where n− q ≤ n′ ≤ n. If n ≥ q we can get an n− q minor ofA′ by taking the whole of the q×q leading I block. Thus the ideal generated bythe n×n minors of a matrix of ψ is the ideal generated by the (n− q)× (n− q)minors of A′.

Therefore the ideal generated by the ((q+ q′)− j)× ((q+ q′)− j) minors ofthe matrix of ψ is that generated by the (q′ − j)× (q′ − j) minors of A′. So wehave that an ideal of R generated by determinants of (q− j)× (q− j) minors ofA, is equal to an ideal generated by determinants of (q+ q′− j)× (q+ q′− j) ofthe block matrix of ψ, which is equal to an ideal of R generated by determinantsof (q′ − j)× (q′ − j) minors of A′.

Then this ideal Fj(M) is independent of presentation, and is therefore analgebraic invariant of M . If it is a principal ideal, then its generator is analgebraic invariant of M , determined up to multiplication by a unit26 of R.Hence, our claim is proven.

With fitting ideals firmly established, we may progress on to the constructionof the knot complement.

26An invertible element of R.



5.3.4 Building up the Knot Complement

Here we resume our discussion of the knot complement. In the various ap-proaches to calculating the Alexander polynomial (to follow later) one obtainsthe finite presentation:

Rq′→ Rq → H1(Y )→ 0

of the first homology group H1(Y ) of the infinite cyclic cover Y of the knotcomplement. This is considered as a module over the ring Z[t, t−1], and where tacts as a deck transformation27 of Y generating the infinite cyclic group of decktransformations of this covering space. The Alexander polynomial generates thefitting ideal F0(H1(Y )) of Y .

With this established we can begin to construct the knot complement. Ima-gine the knot diagram is a tunnel in a coal mine with level crossings, like in thefigure below:

Figure 5.6:The trefoil as a coal mine, with the black sections representing ‘pillars’ of coalsay, and the white regions representing ‘tunnels’ around the pillars.

There is coal above and below, and a level tunnel separating out ‘pillars’of coal connecting the ceiling to the floor. The hollowed space is not (yet)homeomorphic to a tube around the knot because of the crossings.

However, we can put in a 2-dimensional ‘cell’ as a bridge at the crossings tocompletely separate the crossing portions of the tunnel.

Figure 5.7:The ‘bridge’ at each crossing.

27A deck transformation f : X → X, is a homeomorphism which acts in conjunction witha covering map p : X → X such that p ◦ f = p.



Suppose the crossings are at right-angles and we have a hollowed out cube atthe crossings. It follows that we convert the space X1, that is the complement ofthe intersecting level tunnels to the space that is the knot complement by addinga 2-dimensional cell at each knot crossing whose boundary is contained in X1.The resulting space is a strong deformation retract28 of the knot complement,and shares the same fundamental group and homology groups.

We then suppose that the level tunnels have a square cross-section with floorat height z1 and ceiling at height z2. Let

X0 = {(x, y, z) ∈ R3 : z ≥ z2} ∪ {(x, y, z) ∈ R3 : z ≤ z1}

and note that X0 consists of two contractible connected components. To gofrom X0 to X1 we have to add in the ‘pillars’ and the space exterior to thetunnels. A level tunnel represents a graph where each vertex is of degree 4.Thus let nV , nE , nR represent the number of vertices29, ‘edges’ and regionsregions respectively, including the external region. From Euler, we have

nV − nE + nR = 2

and also

2nE = 4nV

⇒ nE = 2nV

thus nR = 2 + nV . We will now describe this information in terms of presenta-tions of the fundamental group.

5.3.5 Fundamental Group Presentation

Let Xbase = X0 together with the external part of X1 i.e. not the pillars,including the ‘point at infinity’; S3 = R3∪{∞}. So Xbase is homeomorphic to aclosed ball30. We get from Xbase to X1 by adding in the c(K) + 1 pillars. Eachpillar adds in a free generator of the fundamental group represented by a loopγi going down through pillar number i and back through the external regionfrom bottom to top, with linking number −pi31 with the knot itself.

Using the Van Kampen Theorem, we see that π1(X1, x0) is a free group onc(K) + 1 generators: [γ1], [γ2], ..., [γc(K)+1]. We also use γ0 for a null-homotopicloop going down through the external region (‘pillar 0’) and back up again.Thus γ0 represents the identity element of π1(X1, x0).

Then at each crossing α, we add on a 2-dimensional cell along its boundarycircle. This introduces a relator into the presentation of the fundamental group:

28A continuous map f : X × [0, 1] → X is a deformation retraction of a space X ontoa subspace U if ∀x ∈ X and u ∈ U we have: f(x, 0) = x, f(x, 1) ∈ U, and f(u, 1) = U .This is a homotopy between a retraction and the identity map, with U called the deformationretract of X. Moreover, a deformation retraction is called strong if it satisfies the requirement:f(u, t) = u ∀t ∈ [0, 1]. This fixes the points in U throughout the homotopy.

29These are knot crossings in this instance.30We can obtain Xbase by hollowing out a ball within S3.31Here pi is equal to the minimum number of times we cross the Seifert surface on our way

down with negative intersections.



Figure 5.8:

il

k j

A positive crossing with pillars:

pi = p− 1

pj = p

pk = p+ 1

pl = p

At the boundary of a ‘bridge’ we have:

Figure 5.9:

j

l i

k

At a bridge, going down through i, up through j etc. gives:

[γi][γj ]−1[γk][γl]

−1 = e

Thus, we obtain a presentation of π1(S3\K,x0), which will discuss in detail


5 ALGEBRAIC METHODS 5.4 The Knot Group

shortly.For the first homolgy group of the knot complement we get a similar present-

ation that involves free Abelian groups. Using the set-up we have discussed, wefind that H1(X) is generated by the homology classes of cycles z1, z2, ..., zc(K)+1

which correspond to the loops γ1, γ2, ..., γc(K)+1 with relators [zi]− [zj ] + [zk]−[zl] = 0 at the crossings.

What we discover is that circumnavigating the knot once always producesthe same homolgy class i.e.

[zi]− [zj ] = [zl]− [zk]

[zi]− [zl] = [zj ]− [zk]

Thus because this class is preserved across crossings, and because the knot isconnected, we find that:

H1(X) ∼= Z

and is generated by loops going down a pillar on one side of the knot and comingup on the other.

Now that we have studied the knot complement, we are ready to turn ourattention to its fundamental group. This is the focus of the next section.

5.4 The Knot GroupDefinition 5.8. The knot group of a knot K, denoted π1(K) = π1(R3 − K),is the fundamental group of all the points in R3 that are not in K i.e. it is thefundamental group of the knot complement.

Next we shall investigate a number of statements about the knot group takenfrom pg. 63 of [Roberts, 2012]:

1. “The knot determines the complement”. This implies that equivalent knotsinduce complements which are homeomorphic to each other: consider theeffect of a ∆-move, it follows quite easily that the homeomorphism typeof the knot’s complement remains invariant after this move.

2. The knot group is an invariant of knots: this is because of statement one,which implies that equivalent knots also have isomorphic fundamentalgroups32.

3. While the previous two statements may be unsurprising, the following oneis counter-intuitive: knots are determined by their complements. This wasproven in [Gordon & Luecke, 1989], though it had been a conjecture whichwas assumed true for a very long time. It states, more precisely, that iftwo knots have homeomorphic complements then they are equivalent33.An analogous theorem for links is immediately false.

32It is the isomorphism class of the group which is really considered as the invariant in thiscase.

33This may possibly only be up to mirror-imaging. However, such ambiguity can be removedif one requires an orientation-preserving homeomorphism between the complements.



4. Another surprise statement is that the knot group determines the knot.[Whitten, 1987] was able to show that two prime34 knots which have iso-morphic groups, also have knot complements that are homeomorphic toeach other. It may then be deduced by the Gordon-Luecke Theorem men-tioned in the previous statement, that they are indeed equivalent.

The first part of this statement fails for composite knots35: compare thereef knot and the square knot. They are isomorphic, but their knot groupsare not homeomorphic, demonstrating that composite knots are not givenby their groups.

We will now introduce the Wirtinger presentation. This provides a means ofcalculating the knot group.

5.4.1 The Wirtinger Presentation of the Knot Group

Theorem 5.5. (The Wirtinger presentation) [Roberts, 2012] (Pg. 63) Apresentation of π1(K) may be obtained as follows. Let D be a diagram of K andlet us orient it, labelling the arcs a1, . . . ., ak, whilst letting S = {a1, a2, ..., ak}.At each signed crossing there are three incident labels x; y; z as shown below:

Figure 5.10:

Z

Y

ZY

X X

A right- and left-handed, and hence signed, crossing with the labellings x, y, z.

Associating the relation xy = z to each positive crossing, and xy = z to eachnegative crossing, we obtain a set of relations R. Thus, π1(K) ∼= 〈S : R〉 ⇔y−1xy = z.

Proof. We have a tame knot ‘nicely configured’, so that a vertical projectiononto the plane z = 0 gives a good knot diagram. Furthermore we require thatthe knot be confined to the region 0 ≤ z ≤ 1, and that at each knot crossing thelower portion is straight and horizontal at height z = 0 and the upper portionis also straight and horizontal at height z = 1.

We pick r > 0 such that everything in the knot or knot diagram that shouldbe separated, is separated by a distance of at least r. In particular if two edges ofthe knot or knot diagram do not meet at a vertex or cross on the diagram, then

34This is “a non-trivial knot which cannot be written as the knot sum of two non-trivialknots”. [Wikipedia, 2012]

35A knot which may be represented as the sum of two prime knots.



the edges are a distance apart of at least r. We also assume that the crossingsare straight within a horizontal distance of at least r of a knot crossing. Weenumerate the segments, the arcs and the crossings.

Take the subset X0 of R3 to consist of everything that does not lie on orunder the knot. Imagine the knot casts a shadow downwards, and X0 consistsof everything illuminated from above. Given any point of X0, the vertical linesegment joining the point to the corresponding point of the plane z = 2 lies inX0. Thus X0 is contractible and π1(X0, x0) is trivial for all points of X0.

Moreover we have a canonical or standard isomorphism:

π1(X0, x0) ∼= π1(X0, x1) x0, x1 ∈ X0

So we can move the base-point around X0 as necessary. Now define open setsUSh for each segment Sh and Vcα andWcα for each crossing cα. Here USh consistsof all points of R3\K whose projections lie on the interior of the segment Shor in the interior of one of the two regions of the knot diagram adjoining thatsegment.

Also we have

(X0 ∪ (∪h−1h′=0USh′ )) ∩ USh = X0 ∩ USh

The only thing that USh′ adds to X0 is points below Sh′ , which do not comewithin VSh . Hence we have:

π1(X0 ∪ (∪hh′=0USh′ )) = π1(X0 ∪ (∪hh′=0USh′ )) ~ Z

So then:X1 = X0 ∪all segments h USh

is a free product of N infinite cyclic groups, with one infinite cyclic group foreach segment:

X1 = 〈e1, e2, ..., eN 〉

Hence X1 is everything in R3\K that is not on or below a knot crossing.Now for a knot crossing cα = (xα, yα), define:

Vα = {(x, y, z) : |(x, y)− (xα, yα)| < r and 0 < z < 1}Wα = {(x, y, z) : |(x, y)− (xα, yα)| < r and z < 0}

It is not difficult to see that Vα ∩ X1 is topologically an ‘open millstone’ partof an open cylinder bounded top and bottom, with a hole drilled through fromtop to bottom. Whereas Vα itself is the relevant portion of the cylinder, thus itis contractible. This means that π1(Vα, x0) is trivial and:

π1(Vα ∩X1, x0) ∼= Z

Similarly,π1(Wα ∩X1, x0) ∼= Z

We see that each Vα and each Wα adds a relator to the presentation of thefundamental group. We first go under e3 with a negative crossing, then undere1 with a positive crossing. This is topologically equivalent to:



Figure 5.11:

e1

e4

e3

e2

Going under e3 by way of a negative crossing, and then under e1 by way of apositive crossing.

This introduces a relator:

e−13 e1 = identity⇒ e1 = e3

So if segments belong to the same arc then the generators are equal, hence wehave one generator per arc.

Let X2 = X1 ∪ (∪α∈CVα) where C is the set of all the crossings on the knot.Then π1(X2, x0) is a free group on M generators, with one generator per arc.Now R3\K = X2 ∪ (∪α∈CWα), so for the arcs z, and y:



Figure 5.12:

y

z

Considering the following:

Figure 5.13:

z

x

y

This implies that

zy−1x−1y = identity⇔ y−1xy = z

which is the relation in the Wirtinger presentation.



Example 5.7. We can use the Wirtinger presentation to find a presentation ofthe knot group of the trefoil, to be as follows:

〈x, y : |xyx = yxy〉

Figure 5.14:

Relations in the Wirtinger presentation for 31

Now that we have explored some of the fundamental ideas behind the basictheory, we are ready to tackle the advanced theory behind some of a morepowerful invariant. We will see how the ideas explained in this section provideus with a route to the computation of the Alexander polynomial in the firstsection of the next part.


6 ALEXANDER POLYNOMIAL

6 Alexander PolynomialThe Alexander polynomial can be computed in a number of different ways.

1. Alexander’s combinatoric method (1928): uses the diagram of the knot inconjunction with the Reidemeister moves.

2. Fox’s method (1963): uses a statement of the knot group.

3. Conway’s skein relation (1969): this utilises specialised equations that linkthe crossings of different knot diagrams. It was mentioned in Alexander’soriginal paper [Alexander, 1928], but Conway’s demonstration is clearerand instigated the discovery for the Jones polynomial 15 years later.

The method we will study (in the next subsection) is that of Fox’s method, asit employs our study of section five.

6.1 Deriving the Alexander polynomialUtilizing what we have established in the previous sections we will now computethe Alexander polynomial from the Seifert matrix.

Let K be a knot, X = S3\K (also called the strong deformation retract)and X be and infinite cyclic cover36 of X. Define the deck transformation

θ : X → X

which generates the group Deck(X) of deck transformations of the coveringp : X → X where:

p(x1) = p(x2)

⇔ x2 = θn(x1)

for some n ∈ Z.From the construction given earlier, there exists a continuous function h :

X → R such that:h(θn(x)) = h(x) + 1

for all x ∈ X. We choose a base-point x0 ∈ X, then choose x ∈ X such thatp(x0) = x0 and we define xn = θn(x0) for all n ∈ Z. We have a homomorphism:

λ : π1(X,x0)→ Z

now for any loop γ : [0, 1]→ X based at x0, there exists a unique lift

γ : [0, 1]→ X

with γ(0) = x0. Then it follows that γ(1) = xn = θn(x0) for some n ∈ Z. Wealso define λ([γ]) = n so that γ(1) = θλ([γ])(x0).

It can be shown without too much effort that Ker λ = p#(π1(X0, x0)) where

p# : π1(X, x0)→ π1(X,x0)

36The reader should remember that we constructed this by forming a Seifert surface, cuttingalong the Seifert surface and then gluing an infinite number of copies.


6 ALEXANDER POLYNOMIAL 6.1 Deriving the Alexander polynomial

is a covering map between fundamental groups. In fact, letting π = π1(X,x0)for short, we have that

Ker λ = p#(π1(X0, x0)) = [π, π]

where [π, π] is the commutator subgroup of π1(X,x0) generated by uvu−1v−1

for all u, v ∈ π1(X,x0). The Wirtinger presentation, by using relators of theform y−1xy = z illustrates one way in which this may be seen. It follows thatx and y have the same image under the quotient homomorphism π → π

[π, π].

Hence the image of the loop around the knot is preserved across knot crossings.Denoting the regions of our knot diagram as shown below,

Figure 6.1:

Region 2

Region 3

Region 4

Region 1

Region 0

The labeling of regions

we choose our base-point x0 outside of the regions enclosed by the knot, inregion 0. For each region i, with i > 0, we have a loop denoted αi, which goesfrom our base-point x0, above the knot, down through region i and then returnsto x0 under the knot. Note that α0 will be the constant loop at x0. Thenλ(αi) = −pi where pi is associate to the region in such a way as to increase inincrements when we cross under the knot in a right-handed crossing37.

Let αi : [0, 1]→ X be the lift38 of αi. Then:

αi(0) = x0, αi(0) = x−pi = θ−pi(x0)

If we focus in on a crossing, it looks something like this:37Observe that λ(αi) is equal to the linking number of αi with the knot K.38Let p : Z → Z be a covering map. A continuous map g : Y → Z is called a lift of another

continuous map g : Y → Z if and only if p ◦ g = g. [Wilkins, 2012] (pg. 41).



Figure 6.2:

+1

We consider the loop γ below:

Figure 6.3:

+1γ

This loop is the based homotopy class of αi · α−1j · αk · α

−1l = γ. Next we

consider what the lift of γ looks like.

αi(0) = αj(0) = αk(0) = αl(0) = x0

αi(1) = x−p+1, αj(1) = x−p, αk(1) = x−p+1, αl(1) = x−p



So the lift of γ is:αi · (θ ◦ αj)−1 · (θ ◦ αk) · α−1

l

This loop is null-homotopic in X.Now let us digress somewhat in order to map to the homology group. We

may use the Hurewicz homomorphism39:

νX : π1(X, x0)→ H1(X)

with Ker νX = [π1(X, x0), π1(X, x0)]. Also let A = {xn : n ∈ Z}, so A is asubspace of X and A = p−1(x0).There is a relative homology group H1(X, A)and natural homomorphisms:

H1(X)→ H1(X, A) and ∂1 : H1(X, A)→ H0(A)

Where H0(A) is a free Abelian group on the set of path components of A, whichis equivalent to being an Abelian group on the set A because A is a discretetopological space40. Furthermore, H1(A) = 0 because it is discrete. We havean exact sequence:

H1(A)→ H1(X)→ H1(X, A)→ H0(A)→ H0(X)

We can represent an element of H0(A) by a Laurent polynomial with integercoefficients. A 0-chain

∑n∈Z sn[xn]A is represented by:∑

n∈Zsnt

n ∈ Z[t, t−1]

An element of H0(X) is of the form s[x0]X because X is path-connected. Thenatural homomorphism H0(A) → H0(X) determined by the inclusion A ↪→ Xsends

∑n∈Z sn[xn]A to

(∑n∈Z sn

)[x0]. Thus if p(t) is the representing Laurent

polynomial, then the homomorphism maps p(t) 7→ p(1). Thus

Ker(H0(A)→ H0(X)

)∼= {p(t) ∈ Z[t, t−1] : p(1) = 0}

Recall H1(A) = 0 because A is discrete. Therefore, we have a short exactsequence:

0→ H1(X)→ H1(X, A)∂1→ H0(A)→ 0

where H0(A) = Ker(H0(A)→ H0(X)

)∼= {p(t) ∈ Z[t, t−1] : p(1) = 0}. Then

H1(X) ∼= Ker ∂1, thus we can consider H1(X) a subgroup of the Abelian groupH1(X, A).

A pathγ : [0, 1]→ X

in X from x0 to some point xn in A, determines a corresponding homology class[γ]X,A. Furthermore, if γ1 = γ2 rel {0, 1} then

[γ1]X,A = [γ2]X,A

39For any space X and n ≥ 0 there exists a group homomorphism: h∗ : πn(X,x0) →Hn(X) called the Hurewicz homomorphism, which is equivalent to the abelianisation map:h∗ : π1(X,x0)→ π1

[π1,π1]when n = 1. Thus H1(X) ∼= π1

[π1,π1].

40Every point of A is its own path component.



. Thus, given two loops γ1 and γ2 in X with base-point x0, there exist uniquelifts γ1 and γ2 which are paths in X beginning at x0, so γ1(0) = γ2(0). If[γ1] = [γ2] in π1(X,x0) then γ1 ' γ2 rel {0, 1} and therefore γ1(1) = γ2(1) and

[γ1]X,A = [γ2]X,A

in H1(X,A). Thus we obtain a function from π1(X,x0) to H1(X,A).Now if α and β are paths in X with α(0), α(1), β(0), β(1) ∈ A and if α(1) =

β(0), then [α · β

]X,A

= [α]X,A +[β]X,A

Let [γ] = [γ1] [γ2] in π1(X,x0) and let γ1, γ2 be the lifts of γ1 and γ2 to Xstarting at x0 respectively. Then γ1(1) = xλ([γ1]). The unique lift of γ2 startingat xλ([γ1]) = θλ([γ1]) ◦ γ2. It follows that:

[γ]X,A = [γ1]X,A +[θλ([γ1]) ◦ γ2

]X,A

= [γ1]X,A + θλ([γ1])∗ [γ2]X,A

Thus if we let D ([γ]) = [γ]X,A for all γ, where γ is the lift with γ(0) = x0, then:

D ([γ1] [γ2]) = D ([γ1]) + θλ([γ1])∗ [γ2]

Now make H1(X) and H1(X, A) into R-modules where R = Z[t, t−1] and(∑n∈Z snt

n)

(µ) =∑n∈Z snθ

n∗ (µ). Then

D (uv) = D (u) + tλ(u)D (v)

So we now have reached the stage where we have a function D : π1(X,x0) →H1(X, A) with the property above.

Now ∂1 : H1(X, A) → H0(A) acts on α : [0, 1] → X where α(0), α(1) ∈ Aby

∂1 (α) = [α(1)]A − [α(0)]

Thus if γ : [0, 1] → X lifts γ : [0, 1] → X with γ(0) = x0 and γ(1) = xn withn ∈ λ ([γ]), then:

∂1D ([γ]) = ∂1γ = [γ(1)]A − [γ(0)] = [xn]− [x0] = tn − 1 = tλ([γ]) − 1

Hence ∂1D (u) = tλ(u)−1. Note that if p(t) = tλ(u)−1 then p(1) = 0. Moreover,D(u) ∈ H1(X) if and only if ∂1D (u) = 0, which is if and only if λ(u) = 0. Thishappens if and only if

u ∈ p#

(π1(X0, x0)

)= Ker λ = [π, π]

Thus we have set-up the function D : π1(X,x0)→ H1(X, A) with

D (u) ∈ H1(X)

⇔ u ∈ Ker λ⇔ u ∈ p#

(π1(X0, x0)

)and D (uv) = D(u) + u ·D(v), where

u · z = tλ(u)z

= θλ(u)∗ z



for all u ∈ π1(X,x0) and z ∈ H1(X, A).Notice that D|Ker λ is a homomorphism from Ker λ to H1(X), with

Ker λ = p#

(π1(X0, x0)

)∼= π1(X0, x0)

This homomorphism D|Ker λ, in fact corresponds to the Hurewicz homomor-phism from π1(X0, x0) to H1(X). However, the intriguing property of this isthat this is the restriction of a function D : π1(X,x0) → H1(X, A) defined onπ1(X,x0). This function is a derivation that is defined by Fox, and can beexpressed in terms of Fox derivatives, which we will discuss shortly.

6.1.1 Fox’s Method

The American mathematician R. H. Fox pioneered a method in which the Al-exander polynomial can be recovered from a presentation of a knot group usinga form of calculus called Fox derivatives. Fox’s method also uses the study offundamental groups of the complementary spaces of knots.

Returning back to our diagram

Figure 6.4:

+1

Our original diagram from 6.1 with the labeled regions.

We let ui = [αi] etc. in π1(X,x0), so we have a relation:

ui · u−1j · uk · u

−1l = e

So we should have

D(ui · u−1

j · uk · u−1l

)= 0

⇔ Dui − uiu−1j ·Duj + uiu

−1j ·Duk − uiu

−1j uku

−1l ·Dul = 0

⇔ Dui − tλ(uiu−1j ) ·Duj + tλ(uiu

−1j ) ·Duk − tλ(uiu

−1j uku

−1l ) ·Dul = 0

⇔ Dui − t−pi+pj ·Duj + t−pi+pj ·Duk − t−pi+pj−pk+pl ·Dul = 0

⇔ Dui − t ·Duj + t ·Duk −Dul = 0



which looks like this on our diagram:

Figure 6.5:

𝑝𝑗 = 𝑝

𝑝𝑖 = 𝑝 − 1

𝑝𝑙 = 𝑝

𝑝𝑘 = 𝑝+1 𝑡

-𝑡

+1

−1

Remark 6.1. Some Remarks on [Alexander, 1928]: We may regard Alexander’swork on his polynomial as an inversion of our conventions. This is evident inhis placing of the coefficient t to the left of the under-crossing route, althoughcrossings what he denoted to be right-handed are in fact left-handed under ourconventions.

Fox Derivatives Fox derivatives are a method of defining a partial derivativefor a monomial in non-commuting symbols, such as the words which form therelations in our group presentation. It should be noted that the derivativeformed in this way cannot be treated as a word in the same sense as before,but will be a formal sum of words in the same symbols [Long, 2005]. Alsoobserve that Di is a free derivative. This is a map from a free group F to thecorresponding group ring Z [F ].

Definition 6.1. Suppose r1and r2 are relations in symbols x1, ..., xi, xj , ...xnand their inverses. Then the Fox derivatives of the words with respect to thesymbol xi, denoted ∂

∂xi= Di, are obtained using the following rules:

1. Di(xi) = 1

2. Di(xj) = δij

3. Di(1) = 0

4. Di(x−1j ) = −δijx−1

i

5. Di(r1 · r2) = Di(r1) + r1 ·Di(r2)

The procedure is to differentiate the leftmost term at each stage and use (5) tocreate a sum of terms until the end of the word is reached.

Suppose we have a presentation of π1(K), then we will have n words thatare the relations of π1(K) in n variables. By using Fox’s calculus we can obtainthe derivative of each word with respect to each generator and describe their



values in an n × n matrix, called the Jacobian of π1(K). This is how we willbegin.

Letting the knot group of a knot K be the group π1(K) = G, there is agroup homomorphism:

α : F → G

(x1, . . . ., xn) 7→ 〈x1, . . . ., xn|r1, . . . .rn〉

which can the be extended to: α∗ : Z [F ] → Z [G]. Now Kerα∗ is generated byr1, ..., rn. We then define

β : Z [G] → Z[t, t−1

]xi 7→ t

for each 1 ≤ i ≤ n. Observe that β is an abeliansation of Z [G], and β(G) is aninfinite cyclic group. Now we assign a n×n matrix J called the Jacobian whichhas entries

βα (Dj(ri))

or

J = Dj(ri) =

(∂ri∂xj

)=

∂r1∂x1

∂r1∂x2

· · · ∂r1∂xn

......

. . ....

∂rn∂x1

∂r1∂x2

· · · ∂rn∂xn

to any presentation 〈x1, . . . ., xn|r1, . . . .rn〉 of G. Where each relation ri is acombination of the generators xj : ri = ai1x1 + ... + ainxn. We call the n × nmatrix with aij entries for 1 ≤ i, j ≤ n the presentation matrix [Yu, 2004].

The Alexander matrix is a presentation matrix of H1(X) as a Z[t, t−1

]-

module. An Alexander matrix of size n×n yields a fitting ideal for a particularknot that is generated by n × n minors of the Alexander matrix. This is aprincipal ideal because the Alexander matrix of a tame knot is square, andfurthermore any generator of this principal ideal is the Alexander polynomial(∆). This means that for ∆ ∈ Z

[t, t−1

]we have:

H1(X) =Z[t, t−1

]∆

We remove one row and one column from the matrix, obtaining a (n− 1)×(n − 1) matrix, and substitute t in for all variables. This means our fittingideal is the zero’th fitting ideal (also called an Alexander ideal) as introducedearlier, is then generated by the (n − 1) × (n − 1) minors of J . Finally wetake the determinant of this remaining matrix and this produces the Alexanderpolynomial, which is the greatest common divisor of the determinant of the(n− 1)× (n− 1) matrix.

Moreover, we can prove that the Alexander ideal is a knot invariant, andalso that it is not dependent upon the presentation of the knot in question. Thisis done using the Tietze Theorem from [Crowell & Fox, 1963] and [Fox, 1962].

Example 6.1. We will compute the Alexander polynomial of the Hopf linkwith positive and negative orientations:



Figure 6.6:H+ (top) and H− (bottom). Source:[Wikipedia, 2012]

We use our derivation technique on theWirtinger presentation of π1(S3\H±).In this case the Wirtinger presentation yields:

π1(S3\H±) =⟨a, b | aba−1b−1

⟩Using the Fox derivatives of the relator with respect to each word gives:

∂(aba−1b−1)

∂a= 1− aba−1 and

∂(aba−1b−1)

∂b= a− aba−1b−1

This produces the row matrix M :

M =(1− aba−1 a− aba−1b−1

)We then reduce M to the matrix Mt in Z[t, t−1] by subbing in for a, b = t, andworking out 1− aba−1 = 1− t, and a− aba−1b−1 = t− 1 we have:

Mt =(1− t t− 1

)Now we remove the first column: this leaves us with the Alexander matrix A.Then the determinant of this new matrix A:

A =(t− 1

)is of course t−1. Since this is a polynomial of degree one, we multiply by t−

12 to

obtain:∆H±(t) = t

12 − t− 1

2

Remark 6.2. Two Alexander polynomials are equivalent if the differ by a factorof ±tk.Remark 6.3. Note how the Alexander polynomial is the same regardless of theorientation of the Hopf link. This means the Alexander polynomial cannot beused to distinguish between H+ and H− links. This aspect will be discussed inthe next section.

Note that in choosing which group presentation to use there is a negativecorrelation between the algebra and the calculus, whereby a trade-off must bemade. If one uses a presentation with a large number of generators then therelations are relatively easy to differentiate, but one is left with a large squarematrix, for which it requires much effort to calculate the determinant. Onthe other hand, if one begins with a presentation of minimal generators, whichrequire no calculation of determinants at all, but the small number of generatorsincreases words to a cumbersome length and thus complicates Fox derivativecalculation.



Part IV

Conclusion“To cut the Gordian Knot is not to untie it.Children and lunatics cut that, which the poet spends his life pa-

tiently trying to untie.”

Jean Cocteau (1889-1963)41

This project has provided just a brief foray into the realm of equivalencein knot theory. There are many more continuously burgeoning developmentsbeing made in this dynamic field regarding invariants, like those of the Vassilivinvariant, the Khovanov and Conway polynomials. However a gap which yetremains in this incursion into knot theory, is a brief discussion of some of itsapplications.

ApplicationsOne area which is developing almost equally as rapidly as the research into in-variants, is that of the applications of knot theory. Knot theory is a particularlydynamic branch of topology in this regard. Topologists working with biologistshave discovered concrete applications of knot theory in the study of enzymesacting on DNA strands. The long and complicated strands that are DNA, aretangled up in the same structures that we observe with mathematical knots.They are long, thin strands contained in the nucleus of a cell. If one can ima-gine the nucleus as a football, and inside it there is thin wire 200 km in lengthtightly packed inside; this represents DNA. DNA must be topologically manip-ulated in order for vital processes to take place. It must first ‘untangle’ itselfso that it can interact with enzymes. Tightly packed DNA in the genes mustalso quickly unknot itself in order for replication or transcription to occur. Bythinking of DNA as a knot, we can use knot theory to estimate how hard DNAis to untangle.

Principles of topology give cell biologists a quantitative, powerful, and invari-ant way to measure properties of DNA. In this manner, the ideas of knot theoryhave been able to illuminate the process of how enzymes untangle DNA. Meas-uring changes in crossing number have also been instrumental in understandingthe termination of DNA replication and the role of enzymes in recombination[Beals et al., 1999].

One interesting topological feature that can be observed in a protein is thatof a distinct knot. These knotted structures represent some of the smallest“self-tying” knots observed in nature [Mallam et al., 2008] and add an extracomplication to the protein-folding problem.

41In the original French: “Couper le nœud Gordien n’est pas dénouer le nœud Gordien.Enfants et fous coupent ce que le poète met toute une vie de patience à dénouer.” (Bern:Peter Lang AG, International Academic Publishers, 2007.



Figure 6.7:The Knots are hard to identify from the protein’s structure (top row). But ifthe trend path is plotted (bottom row), the presence or absence of a knot can beeasily discerned. The colours in this instance indicate different structuralmoieties of the protein chain. Source: http://www.nature.com/nmat/journal/v10/n2/fig_tab/nmat2953_F1.html

It has been found that the α/β-knot protein contains a deep trefoil knot at itscarboxy terminus42. In recent research there has been a growing number of these‘naturally’ occurring knots in increasingly more complex forms: in addition totrefoil knots, a highly intricate figure-of-eight knot and a knotted structure withfive projected crossings have been observed [Mallam et al., 2008]. The role ofinvariants in such circumstances, is to provide a tool for distinguishing differentknot proteins and to aid in the study of their particular constructions.

ProblemsAs in most areas of Mathematics, there remain many open problems in knottheory. However, a difference may be that a significant number of these unsolvedproblems seem unassuming: often being articulated in a trivial way, yet theycan cunningly evade solution. Therefore I have chosen problems with relativelyeasy formulations, but which have great significance in modern knot theory.

Here are some cited in [Manturov, 2012] (appendix D):

Problem 6.1. Does the following statement hold:

c(K1#K2) = c(K1) + c(K2)

where # is the operation that adds or ‘combines’ one knot to another.

Problem 6.2. Does the connected sum of n knots (not unknots) have unknot-ting number at least n?43 The positive answer to this question would followfrom the positive answer to the following question:

Does this statement relating to the unknotting number hold:

u(K1#K2) = u(K1) + u(K2)

42The carboxy- or ‘C’-terminus is the end of an amino acid chain (protein or polypeptide).A protein’s sequence is created from left to right, from the N-terminus to the C-terminus.

43Formulated by de Souza.



LimitationsIf we return to the main invariant discussed in this document, the Alexan-der polynomial, even this invariant has some short-comings and is thus not acomplete knot invariant. For example, it is unable to differentiate the Kinoshita-Terasaka knot, which has a trivial Alexander polynomial (∆KT (t) = 1), fromthe unknot, which also has ∆unknot(t) = 1:

Figure 6.8:The Kinoshita-Terasaka knot which is clearly not equivalent to the trivial knot,despite its trivial Alexander polynomial.

Example 6.2. The Alexander polynomial also cannot identify a knot from itsmirror image i.e. amphicheiral knots.

One famous example of the trickiness of establishing knot equivalence isexemplified by the famous Perko pair: 10161 and 10162. These equivalent knotsconsidered to be different for 75 years, but the equivalence of the two was finallyproven in [Perko, 1973].

Figure 6.9:The Perko Knots. [Perko, 1973]

Areas of Further Research & Project SummaryIn terms of future endeavours, the four-dimensional aspect of knot theory hasbeen left relatively undeveloped. Another area that is possibly playing catch-upto mainstream knot theory is the area of (computer) visualisation, as highlightedin the thesis [Scharein, 1998].Summary. In a brief synopsis, we have investigated equivalence via invariantsfrom the ground up. We began by forming the abstraction of a knot in three-dimensional space through using several definitions to shape the concept. Thenwe turned to special manoeuvres applicable to knots; by way of Reidemeister’sTheorem, which lead us to the question of equivalence of knots.



The difficulty of determining equivalent knots, caused us to investigate in-variant properties between them, leading to some elementary knot invariantslike p-colourings. We found a useful tools in the elements of algebraic topo-logy, which provided a path to the Alexander polynomial. Finally, some of theburgeoning applications of knot theory we alluded to, in tandem with possiblefuture areas of research.

Conclusion. Intriguingly, this final employment of knot theory in the futuredevelopmental research into DNA proteins, serves the same purpose, and sharesthe same motivation, as that encompassed by the early fascination with theuntying of the Gordian Knot. Establishing whether one link is equivalent toanother, is intrinsic to the endeavour of untangling and unknotting complicatedlinks.

It is the ability to differentiate such intricate structures, which can construc-ted by even simple components, that invariants possess. This apparatus foridentification of equivalence between knots and links in mathematics and fur-ther afield in the natural world, contains a hidden beauty, which we have seen inthe build-up to the different invariants. Thus, the construction of invariants inorder to determine equivalence is as applicable to microscopic protein structuresas it is to embeddings in space, or indeed the knots in our shoelaces.


REFERENCES

Part V

BibliographyReferences[Alexander, 1928] Alexander, J. W., Topological invariants of knots

and links, Transactions of the American Mathem-atical Society, Volume 30, 1928. Pg. 275–306.

[Anstee et. al., 2004] Anstee, R. P., Przytcki, J. H., Rolfson, D., Knotpolynomials and generalized mutation, 2004.

[Ashraf, 2005] Ashraf, Rehana, Recurrence Relations for HOM-FLY Polynomial and Rational Specializations,Ph.D Thesis, GC University Lahore, Pakistan,2005. Pg. 18.

[Beals et al., 1999] Beals, M., Gross, L., Harrell, S., Ac-cessed on 6th December 2012 from website:http://www.tiem.utk.edu/ ~gross/bioed/ web-modules/ DNAknot.html

[Colberg, 2011] Colberg, Erin, A Brief History of knottheory. Accessed on 20th November2012 from:http://www.math.ucla.edu/~radko/191.1.05w/erin.pdf

[Collins, 2007] Collins, J., The Alexander Polynomial, 2007.Accessed from http://www.maths.ed.ac.uk/~s0681349/research.html on 9th February 2013.Pg. 2-12.

[Crowell & Fox, 1963] Crowell, R. H. & Fox, R. H., Introduction to KnotTheory, Ginn and Company, 1963. Pg. 43-46.

[Fox, 1962] Fox, R. H., A quick trip through knot theory,from Topology of 3-Manifolds and related topics.Prentice-Hall, Inc., 1962. Chapter 3, Pg. 120-167.

[Frankl & Pontrjagin, 1930] Frankl, P.; Pontrjagin, L., Ein Knotensatz mitAnwendung auf die Dimensionstheorie, Mathemat-ische Annalen Volume 102, Issue 1, 1930. Pg. 785-789.

[Freyd et al., 1985] Freyd P., Yetter D., Hoste J. , Lickorish, W.B.R.,Millet, K. and Ocneanu, A., A new polynomial in-variant of knots and links, Bull. Amer. Math. Soc.12(1985). Pg. 239-246.

[Fulton, 1995] Fulton, W., Algebraic Topology; A First Course,Springer Graduate Texts in Mathematics, 1995.Pg. 371-2.

lxxi

REFERENCES

[Gay, 2007] Gay, D., Explorations in Topology: Map Coloring,Surfaces, and Knots, Academic Press, 2007. Pg.300.

[Gilbert & Porter, 1994] Gilbert, N. D. & Porter, T., Knots and Surfaces,Oxford University Press, 1994. Pg. 1-11, 111-150,197-250.

[Gordon & Luecke, 1989] Gordon, C. McA. & Luecke, J., Bull., Knots are de-termined by their Complements, Amer. Math. Soc.(N.S.) Volume 20, Number 1 (1989). Pg. 83-87.

[Hagge, 2006] Hagge, Tobias J., Every Reidemeister Move IsNeeded for Each Knot Type, Proceedings of theAmerican Mathematical Society, Vol. 134, No. 1(Jan., 2006). Pg. 295-301.

[Hodorog, 2010] Hodorog, Madalina, Basic knot theory, JohannRadon Institute for Computational and AppliedMathematics, Research Institute for SymbolicComputation, Johannes Kepler University Linz,Austria, 2 February, 2010. Hodorog, M., knot the-ory, http:// people.ricam.oeaw.ac.at/m.hodorog/software/knot/knottheory.html#auto-4.

[Jones, 2005] Jones, Vaughan F. V, The Jones Polynomial, fromhttp://math.berkeley.edu/~vfr/jones.pdf, 2005.Accessed 20th November 2012.

[Kauffman, 1987] Kauffman, L. H., State models and the Jones poly-nomial, Transactions of the American Mathemat-ical Society, Vol. 318, no. 2, Topology 26 (1987),no. 3. Pg. 395-407.

[Kauffman, 1988] Kauffmann, L. H., New Invariants in the Theory ofKnots, American Mathematical Monthly, Vol. 95,Issue 3, March 1988. Pg. 195-242.

[Kauffman, 1990] Kauffmann, L. H., An Invariant of Regular Isotopy,American Mathematical Monthly, Vol. 318, No. 2.,April 1990. Pg. 417-471.

[Kirby, 1995] Kirby, R., Problems in Low Dimensional Topology,1995.

[Lickorish & Millet, 1988] Lickorish, W. B. R. & Millet, K. C., The New Poly-nomial Invariants of Knots and Links, Mathemat-ics Magazine, Vol. 61, No. 1. (Feb., 1988). Pg. 3-23.

[Lickorish, 1997] Lickorish, W. B. R., An Introduction to knot the-ory, Graduate Texts in Mathematics 175, Springer-Verlag New York, 1997. Pg. 110-122, 179-192.

lxxii

REFERENCES

[Lomonaco, 1996] Lomonaco, S. J. Jr, The modern legacies of Thom-son’s atomic vortex theory in classical electro-dynamics, from Louis H. Kauffman, editor, TheInterface of Knots and Physics, Volume 51 ofProceedings of Symposia in Applied Mathemat-ics, pages 145-166, Providence, Rhode Island, 1996.American Mathematical Society.

[Long, 2005] Long, E., Topological invariants of knots: threeroutes to the Alexander Polynomial, 2005. Accessedfrom:http://www.ucl.ac.uk/ ~ucbpeal/ alexander-mac.pdf. Retrieved on 24th October 2012. Pg. 19-44.

[Mallam et al., 2008] Mallam, Anna L., Morris , Elizabeth R., Sophie E.Jackson, Exploring Knotting Mechanisms in Pro-tein Folding, Proceedings of the National Academyof Sciences of the United States of America,Vol.105, No. 48 (Dec. 2, 2008), Pg. 18740-18745.

[Manturov, 2012] Manturov V. O., The Knot Book. Ac-cessed on 12th November 2012 from:http://www.varf.ru/rudn/manturov/book.pdf.

[Massey, 1991] Massey, W.S., A Basic Course in Algebraic Topo-logy, Springer, 1991. Pg. 36.

[Massuyeau, 2008] Massuyeau, G., A Short Introduction to the Alex-ander Polynomial, 2008. Pg. 1-6.

[Murasugi, 1996] Murasugi, K., Knot Theory and Its Applications,Birkhäuser Boston, 1996.

[Murasugi, 1965] Murasugi, K., On a certain numerical invariant oflink types, Trans. Amer. Math. Soc.,117, 1965. Pg.387-422.

[Nardo & Payne, 2000] Nardo, J. & Payne, B., A Brief Historyof knot theory. Accessed on 2nd December2012 from: http://www.oglethorpe.edu /faculty/~j_nardo/knots/index.htm.

[Nash, 1997] Nash, C., Topology and Physics – a historical es-say, National University of Ireland, 1997, from AHistory of Topology, edited by Ioan James and pub-lished by Elsevier-North Holland, 1998.

[Perko, 1973] Perko, K.A., On the classification of knots, NoticesAmer. Math. Soc., 20, 1973. Pg. 453-454.

[Portnoy & Mattman, 2005] Portnoy & Mattman, Knot Module Lecture Notes,2005. Accessed on 5th December 2012 from:http://www.csuchico.edu/math/ mattman/NSF/Lecture notes.pdf.

lxxiii

REFERENCES

[Przytycki, 1992] Przytycki, J. H., Knot theory from Vandermonde toJones, Aportaciones Matematicas ComunicacionesII, 1992. Pg 173-185.

[Przytycki, 1995] Przytycki, J. H., 3-coloring and other elementaryinvariants of knots, An extended version of twotalks given at the Mini-semester on knot theoryat the Stefan Banach International MathematicalCenter in Warsaw, July 17th- August 18th, 1995.Pg. 3-16.

[Radko, 2011] Radko, Olga, Lecture Notes from A ba-sic introduction to knot theory, 2011-12.Accessed on 23rd October 2012 from:http://www.math.ucla.edu/~radko/191.1.05w/index.html.

[Reidemeister, 1926] Reidemeister, Kurt, Elementare Begründung derKnotentheorie, Abh. Math. Sem. Univ. Hamburg,5, 1926. Pg. 24-32.

[Roberts, 2012] Roberts, J., Knots Knotes, Univer-sity of Edinburgh. Accessed from:http://math.ucsd.edu/~justin/Papers/knotes.pdf,on 20th November 2012.

[Scharein, 1998] Scharein, R. G., Ph.D Thesis, Interactive Topolo-gical Drawing, Department of Computer Science,The University of British Columbia, 1998.

[Seifert, 1934] Seifert, H., Über das Geschlecht von Knoten. Math.Annalen 110 (1), 1934. Pg. 571–592.

[Sossinsky, 2002] Sossinsky, A., Knots: Mathematics with a twist,Harvard University Press, 2002. Pg. 1-119.

[Stokman, 2012] Stokman, J., Quantum Groups and knot the-ory: lecture notes week 37. Accessed from:http://staff.science.uva.nl/~jstokman/week37.pdf.Accessed on 12th November 2012. Pg. 4-5.

[Thistlewaithe, 1985] Thistlethwaite, M., Knot tabulations and relatedtopics. In I. M. James and E. H. Kronheimer,editors, Aspects of Topology in Memory of HughDowker, Volume 93 of London Mathematical So-ciety Lecture Note Series, pages 1-76. CambridgeUniversity Press, 1985.

[Trace, 1983] Trace, Bruce, On the Reidemeister Moves of aClassical Knot, Proceedings of the American Math-ematical Society, Vol. 89, No. 4 (Dec., 1983). Pg.722- 724.

lxxiv

REFERENCES

[White, 1969] White, J H., Self-Linking and the Gauss Integralin Higher Dimensions, American Journal of Math-ematics, Vol. 91, No. 3 (Jul., 1969). Pg. 693-728.

[Whitten, 1987] Whitten, W., Knot complements and groups, To-pology 26, 1987. Pg. 41-44.

[Wikipedia, 2012] Wikipedia, knot theory page. Ac-cessed on 5th December at:http://en.wikipedia.org/wiki/Knot_theory.

[Wilkins, 2012] Wilkins, D. R., MA3427-8 Algebraic Topology Lec-ture Notes 2012. Michaelmas Term, pg. 41-55, Hil-ary Term pg. 5.

[Erickson’s Proof] Erickson, Proof of Wirtinger Present-ation: http://faculty.bucks.edu/ erick-son/algebraic_topology/ wirtinger.pdf. Accessed16th December 2012.

[Yu, 2004] Yu, J., Alexander polynomialof knots, 2004, accessed from:http://math.berkeley.edu/~hutching/teach/215b-2004/yu.pdf on the 16th January. Pg. 1-9.

[1] Image accessed from:http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method= retrieve&_eid=1-s2.0-S0040938306000553&_image=1-s2.0-S 0040938306000 553-gr1.jpg&_ba=&_fmt=full&_ orig=na&_i ssn=00409383&_pii=S0040938306000553&_acct=C000228 598&_version=1&_urlVersion=0&_userid=10&m d5=ee541826611b0300c08d78914aa93daa, on 14th January2013.

[2] Image accessed from:http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_eid= 1-s2.0-S0040938399000580 &_im-age= 1-s2.0-S0040938399000580-gr4 .gif&_ba=&_fmt=full&_orig=na&_issn=00409383&_pii= S00409 38399000580&_acct=C000228598 &_version=1&_url Version=0&_userid= 10&md5 =424feb90de716a4af43 7fb666c65cdc7, on 12th January 2013.

[3] Image accessed from:http://cdnimg.visualizeus.com/thumbs/e6/aa/computer,generated,image,seifert,

lxxv

REFERENCES

surface,trefoil,knot-e6aa0c60a5d2cb9eb467aaf67ed6fd56_h.jpg, on 12th January 2013.

lxxvi

A OTHER POLYNOMIAL INVARIANTS

Part VI

AppendixIn this section there is some supplementary information on two other polynomialinvariants which were discovered as a result of the Alexander polynomial. I haveincluded the work I conducted regarding these polynomials in this appendixsection because I felt that they would would offer the reader a perspective ofhow influential Alexander’s discovery was.

This subsequent work also illustrates the relatively modern advancementsmade in the context of polynomial invariants. This also serves to offer the readera ‘scale’ of sorts for polynomial invariants: one can compare the characteristics,power and limitations of these polynomial invariants. Thus, the intention ofthis additional section is to ‘frame’ this research project within the context ofthe evolution of polynomial invariants.

A Other Polynomial invariantsAfter the construction of the first knot polynomial, we can use some theoryabout the intermediary ‘bracket polynomial44’ as our bridge to the Jones poly-nomial. The Jones polynomial serves as an extremely robust invariant. Noticingthe inter-connectivity and mutual dependence of these ‘strands’ of knot theory,we will attempt a formulation of the HOMFLY polynomial. It will be shownhow this is a powerful synergy of the Alexander and Jones polynomials.

A.1 Bracket PolynomialIn order to begin discussing the Jones polynomial we need to first define thebracket polynomial. The Bracket polynomial is an invariant of framed links45.We will establish that although the bracket polynomial is not an invariant,if we choose a suitable normalisation it can be tempered into an invariant.Furthermore, this yields the famous Jones polynomial.

A.1.1 States

If we examine a crossing, we notice that it can be split up or ‘spliced’ in twoways:

44Also known as the Kauffman Bracket polynomial, after Louis Kauffman’s discovery de-tailed in [Kauffman, 1987].

45This was first discussed in [Kauffman, 1987].

lxxvii


Figure A.1:

A

B

A

B

B

A

The ‘splitting’ of a crossing into ‘A’ and ‘B’ splittings

The ‘A Split’ joins the regions labelled A, and the ‘B Split’ joins the regionslabelled B. By splicing crossings, one can obtain a family of diagrams with theoriginal link diagram as the mutual ancestor:

Figure A.2:Variations of the trefoil constructed using splittings [Radko, 2011]

This can continue until we eventually obtain n copies of the unknot. A splitlabelled A or B can be reconstructed to form the original crossing:

lxxviii


Figure A.3:The ‘reconstruction’ of crossings from splittings. [Radko, 2011]

We can then reconstruct the splices to form the original knot.

Figure A.4:The states of the trefoil knot. [Radko, 2011]

The last, or final ‘descendants’ of a crossing, which has been spliced, arecalled ‘states’. Any state with labels can be used to reconstruct a knot. Weshall construct invariants of knots and links by averaging over states.

Definition A.1. If K is a knot and δK is its state, let 〈K|δK〉denote the productof the labels attached to δ.

Example A.1.

Figure A.5:⟨Ktrefoil|δKtrefoil

⟩= A3. [Radko, 2011]

The labels are determined by the structure relative to K. And let ‖δ‖ =# of components− 1. For example the labels of the trefoil knot is equal to

‖δtrefoil‖ = 2− 1 = 1

Definition A.2. The bracket polynomial is a three-variable, A,B, d, polynomialof a knot K that commutes. It is defined as:

〈K〉 = 〈K〉 (A,B, d) =∑δ

〈K|δ〉 · d‖δ‖

Example A.2. We can compute the bracket polynomial from a state decom-position. If we compute the bracket polynomial for the trefoil we get:

〈Ktrefoil〉 = A2B +A3d+AB2d+A2B +AB2d+B3d2 +A2B +AB2d

= 3A2B + (A3 +AB2 +AB2 +AB2)d+B3d3

= 3A2B + (A2 + 3B2) ·Ad+B3d

lxxix


Lemma A.1. The bracket polynomial of a crossing, consisting of strands A andB, is equal to the sum of the bracket polynomials of the splices A and B.

Proof. Let K be a knot and KA be the knot obtained by an A splicing at somecrossing. Similarly let KB be the knot obtained by a B splicing of K at thesame crossing. Then the following holds:

States of K = (States of KA) ∪ (States of KB)

by definition of the bracket polynomial. Indeed, it follows that

〈K〉 = A · 〈KA〉+B · 〈KB〉

which is the statement of the Lemma.

Example A.3. We can use this recursion relation to compute the bracketpolynomial of the Hopf Link, H:

〈H〉 = A2d+AB +AB +B2d

= 2AB + (A2 +B2)d

This leads us to question the behaviour of the bracket polynomial under theapplication of the Reidemeister moves we defined earlier.

Lemma A.2. The bracket polynomial is invariant under the application of theReidemeister moves, and hence:

⟨R⟩

=

{(Ad+B) 〈δ0〉(A+Bd) 〈δ0〉

which is determined by the orientation of R, where R is the crossing before theRi Reidemeister move, and where δ0 is the trivial state.

⟨R2

⟩= AB 〈δB〉+AB 〈δR2

〉+ (A2 +B2) 〈δA〉

Proof. We shall only prove invariance for the second Reidemeister move. Forthe second move we have:

〈R2〉 = AB 〈δB〉+AB 〈δR2〉+ (A2 +B2) 〈δA〉

Let us impose conditions upon A,B, d so that they are preserved under R2:⟨R2

⟩=⟨R2

⟩where R2 is the crossing before the R2 move, and R2 is the crossing after theR2 move. The first is that AB = 1, and the second is d = −(A2 + A−2). Thisimplies that

〈δR2〉+ (A2 +B2) 〈δA〉 = 0

⇔ d 〈δA〉+ (A2 +A−2) 〈δA〉 = 0

lxxx


With these conditions and the equality of⟨R2

⟩=⟨R2

⟩, we have⟨

R1

⟩= (−A · (A2 +A−2) +A−1) 〈δ0〉

= −A3 〈δ0〉So by utilising these two conditions we can guarantee that the bracket polyno-mial remains invariant under R1and R2.

Furthermore, the same conditions ensure that the bracket polynomial isinvariant under R3, under the following Lemma:

Lemma A.3. If B = A−1 and d = −(A2 +A−2), then⟨R3

⟩=⟨R3

⟩Proof. We shall prove this diagrammatically from [Radko, 2011], as it is clearerand easier to follow.

Figure A.6:[Radko, 2011]

The crossing is A and B split respectively, then R2 is applied, and finallythe original crossing is obtained again from the splittings. Hence, this showsinvariance under R3.

A.1.2 Normalised Bracket Polynomial

Definition A.3. The normalised bracket polynomial of a knot K, denoted LK ,is:

LK = (−A3)−w(K) · 〈K〉This special case of the bracket polynomial has an interesting property pertain-ing to the next Theorem.

Theorem A.1. The normalised bracket polynomial is invariant under ambientisotopy.

Proof. We already know that both w(K) and 〈K〉 are invariant under R2 andR3, thus it suffices to check that LK is invariant under R1.

LK(R1) = (−A3)−w(R1) ·⟨δR1

⟩= (−A3)−(1+w(R1)) · (−A3) · 〈δ0〉= (−A3)−w(R1) · 〈δ0〉= LK(R1)

lxxxi


Having studied the invariance of the normalised bracket polynomial, we shallnow move on to investigating the bracket polynomials of the mirror images ofknots.

Bracket Polynomials of Mirror Images If K is a knot, let K∗ be themirror image of the knot K. Then,

〈K∗〉 (A) = 〈K〉 (A−1)

LK∗(A) = LK(A−1)

Thus one can deduce that if LK∗(A) 6= LK(A−1), then K is not ambient isotopicto K∗.

Example A.4. If we examine the trefoil knot, recalling that w(T ) = 3.

〈T 〉 = 3A2B + (A2 + 3B2) ·A · d+B3d2

= 3A+ (A2 + 3A−2) ·A · (−1) · (A2 +A−2) +A−3 · (A2 +A−2)2

= −A5 −A−3 +A−7

Using the expression for the normalised bracket polynomial of the trefoil,

LT = (−A3)−3 〈T 〉= −A−9(−A5 −A−3 +A−7)

= A−4 +A−12 −A−16

Comparing this with the expression for the normalised bracket polynomial ofthe mirror image of the trefoil T∗,

LT∗ = A4 +A12 −A16

∴ LT 6= LT∗

Therefore the trefoil knot is not equivalent to its mirror image.In the next sub-section we shall see an application of the bracket polynomial.

It will be used to create a robust polynomial invariant that was worthy of a FieldsMedal.

lxxxii


A.2 Jones PolynomialThe Jones polynomial remains a rather poorly-understood thing, even manyyears after its discovery [Roberts, 2012]. The polynomial VL(t) is a Laurentpolynomial in the variable

√t, which is defined for every oriented link L but

depends on that link only up to orientation preserving diffeomorphism, or equi-valently isotopy, of R3 [Jones, 2005].

Definition A.4. The Jones polynomial VK(t), assigned to an orientated link Lso that the following properties are satisfied:

1. VL(t) is invariant under ambient isotopy,

2. Vunknot(t) = 1

3. t−1 · Vleft crossing − t · Vright crossing = (√t− 1√

t) · Vstate

The three properties above can be verified by the following theorem.

Theorem A.2. Let VL(t) = LL(t−14 ), where LL is the normalised bracket poly-

nomial of a link L. Then VL(t) as defined above, satisfies properties one tothree.

Proof. Properties one and two follow from the corresponding properties of LL,so it remains only to prove property three. If CR is a right-hand crossing andCL a left-hand one, then we have:

〈CL〉 = A · 〈δA〉+B · 〈δB〉

and similarly,〈CR〉 = B · 〈δA〉+A · 〈δB〉

which implies,

B−1 · 〈CL〉 −A−1 · 〈CR〉 = (A

B− B

A) · 〈δA〉

⇒ A · 〈CL〉 −A−1 · 〈CR〉 = (A2 −A−2) · 〈δA〉 (†)

Let w = w(δA), then

w(CL) = w + 1 and w(CR) = w − 1

Let α = −A3, then multiplying our equation † by α−w we get:

A · 〈CL〉 · α−w −A−1 · 〈CR〉 · α−w = (A2 −A−2) · 〈δA〉 · α−w

⇒ A · α 〈CL〉α−(w+1) −A−1 · α−1 · 〈CR〉 · α−(w+1) = (A2 −A−2) · 〈δA〉 · α−w

and using the definition of the normalised bracket polynomial we arrive at thepolynomial,

−A4 · LL +A−4 · LR = (A2 −A−2) · LδAFinally, by putting A = t

14 ,we obtain property three.

The next theorem demonstrates an interesting property of the Jones poly-nomial, namely the reversing property.

lxxxiii


Theorem A.3. Let L be a link, K1, ...,Kn its components and L′ the linkobtained from L by reversing the direction of one component. Letting λ =lk(K1, L − K1) be the total linking number of with the rest of L, the follow-ing holds true:

VL′(t) = t−3λ · VL(t)

Proof. Firstly note the effect of reversing the direction of K1 on the writhe ofthe link:

Figure A.7:The effect of a change in orientation of K1 upon the writhe.[Radko, 2011]

Computing the linking number, we have

lk(K1, L−K1) =1

2

∑c∈K1∩L−K1

sgn(c)

where K1 ∩ L − K1 are the crossings of the two diagrams. Computing w(K)and w(K ′) yields,

w(L) = w(L−K1) + w(K1) +∑

c∈K1∩L−K1

sgn(c)

andw(L′) = w(L−K1) + w(K1)−

∑c∈K1∩L−K1

sgn(c)

Subbing in for the w(K1) term into the equation for w(L′) gives,

w(L′) = w(L)− 2∑

c∈K1∩L−K1

sgn(c) = w(L)− 4lk(K1, L−K1)

Thus,w(L′) = w(L)− 4λ

which we shall use later. Focusing on LK′(A), we have

LL′(A) = (−A3)−w(L′) · 〈L′〉 = (−A3)−w(L′) · 〈L〉

= (−A3)−w(K)+4λ · 〈L〉by our equation above,

= (−A3)4λ · LL(A)

thus,VL′(t) = LL′(t) = t−3λ · LL(t−

14 ) = t−3λ · VL(t)

as required.

lxxxiv


A.2.1 The Jones Polynomial and Alternating links

We shall now discuss an interesting conjecture pertaining to the Jones poly-nomial. This is Tait’s Conjecture. This claims that any knot described by analternating diagram cannot be represented by any other diagram with fewercrossings [Roberts, 2012]. It remained unproven for around a century, until thediscovery of the Jones polynomial. First, we shall begin with some preliminarydefinitions.

Definition A.5. We say that a diagram is connected if the projection of thediagram is a connected subset of the plane.

From this one can easily see that any knot diagram is connected. A diagramdivides the plane into several regions. In fact, if the diagram is connected, allregions are homeomorphic to the unit disc and the number of regions is equalto the number of crossings plus two.

Definition A.6. An isthmus is a crossing that contains less than four distinctregions.

Figure A.8:An Isthmus. [Radko, 2011]

An isthmus is analogous to a bridge between two separate diagrams. Onecan remove an isthmus by rotating half of the diagram as shown below.

Example A.5.

Figure A.9:The removal of an isthmus [Radko, 2011]

Definition A.7. We say that a diagram is reduced if there are no isthmi. Anydiagram can be reduced simply by rotating half of it several times.

Breadth of the Jones Polynomial We shall now investigate what we caninfer from the bracket and Jones polynomials, by the means of examining thebreadth of the Jones polynomial.

Definition A.8. The breadth of a Laurent polynomial p(t), denoted Br(p), isthe difference of the maximal exponent of t in p, and the minimal exponent of tin p.

Theorem A.4. The breadth of the bracket polynomial of a reduced alternatingknot diagram with c crossings is exactly 4c.

lxxxv


For now, let us suppose this theorem holds, and we shall return to it with aformal proof later. Part of the proof will require Tait’s Conjecture.

Corollary A.1. (of Tait’s Conjecture) Any reduced alternating diagram isminimal.

Proof. Let D be a reduced alternating diagram of a knot K with c crossings.Then the breadth of the bracket polynomial of D is

Br 〈D〉 = 4c

and the breadth is a knot invariant46. Thus by Theorem A.4 we cannot obtainany diagrams of K with less than c crossings.

Corollary A.2. Note that the A.1 is equivalent to stating that:

1. any non-trivial reduced alternating diagram depicts a knot which is non-trivial, and

2. all reduced alternating diagrams of a knot have the same number of cross-ings over the entire knot class of that knot.

We shall now return to the proof of Theorem A.4.

Proof. Recall that,〈K〉 =

∑δ

〈K|δ〉

where δ is a state of K. If δA is a state with only A-resolutions, and δB is astate with only B = A−1 resolutions, then δA contributes the highest positivepower of A, and similarly δB contributes the lowest (negative) power of A. Let|δ|, the number of ‘loops’ in δ, be such that

|δ| − 1

Lemma A.4. For a reduced alternating diagram D with δA,δB as defined above,one has,

|δA| > |δB |for any state δ1 which has exactly one B-splitting.

Proof. Colour the regions at each crossing in the diagram like this:


46Recall how we proved that the bracket polynomial remains invariant under the applicationof Reidemeister moves, thus the breadth of the bracket polynomial is also invariant.

lxxxvi


Split D to δA. Then the loops of δA are boundaries of the unshaded regions.For δ1 one crossing will be different:


Since the diagram was reduced, this vertex is not an isthmus. Thus twoshaded regions in the left-hand diagram are different, whereas in the right-handdiagram they are connected. Thus,

|δ1| = |δA| − 1

as required.

Next we shall employ the following Lemma.

Lemma A.5. Suppose D is a diagram with c crossings, and δK is any state withk amount of B-splittings and thus, c − k A-splittings. Let δA = δ0,δ1, ...,δk−1,δkbe a chain of states, obtained from each other, and having i B-splittings for δi.Then the maximum power in:

〈D|δi+1〉 ≤ 〈D|δi〉 ∀i : 0 ≤ i ≤ k

Proof. We prove this as follows

〈D|δi〉 = (−1)Ac−2j · (A2 +A−2)|δi|

〈D|δi+1〉 = (−1)Ac−2(j+1)(A2 +A−2)|δi+1|

The power of (A2 +A−2) increases at most by 1, but∑δ decreases by 2.

Claim A.1. It was stated earlier in Theorem A.4 that the breadth of the bracketpolynomial of any reduced alternating knot diagram with c crossings is equal to4c. We will now prove this statement

Proof. Let δA be the state consisting of all A-splittings. The highest power in〈K|δA〉 is

c+ 2 · ‖δA‖

where ‖δA‖ is the number of components in δA − 1. Let δ1 be a state withsplicings of one type, A, only except for one which is of type B. Then thehighest power in 〈K|δ1〉 is

(c− 2) + 2(‖δ1‖ − 1).

lxxxvii


Using Lemma A.4 we have,

(c− 2) + 2(‖δ1‖ − 1) < c+ 2 ‖δA‖

and by the second Lemma A.5, no state other than δ1can contribute a degreehigher than that of δ1. Thus the maximum degree is equal to

c+ 2 ‖δA‖ .

Reversing the roles of the A and B splicings, we find that the minimum degreeis equal to

−(c+ 2 ‖δB‖).

The breadth isBr(K) = 2c+ 2(‖δA‖+ ‖δB‖).

However, the components of δAand δB are boundaries of unshaded and shadedregions respectively. Thus,

‖δA‖+ ‖δB‖ = #δA + #δB − 2 = c

∴ Br(K) = 2c+ 2c = 4c

Thus our claim is proven.

Now we will resume the proof of A.4

Proof. We need to see what changes occur if we consider an arbitrary diagram,one that is not necessary reduced alternating. According to Lemma A.4 , thereis no strict reduction in degree from δA to δ1. Combining this with the statementfrom Lemma A.5, we can deduce that the maximum power that may, or maynot occur is still:

c+ 2 ‖δA‖

Hence:Br(K) ≤ 2c+ 2(‖δA‖+ ‖δB‖).

However, we still need an estimate of ‖δA‖ + ‖δB‖ for an arbitrary diagram.Here we shall use the following Lemma.

Lemma A.6. For a state δ, let δ∗be the dual state obtained by exchanging allA and B splittings. Then,

‖δ‖+ ‖δ∗‖ ≤ c

where c is the number of crossings in the original diagram.

Proof. To prove this lemma we shall use induction on c. For c = 1, we have{‖δ‖ = 1

‖δ∗‖ = 0

which look like this:

lxxxviii


Figure A.12:Here we see δ and δ∗ for c = 1. [Radko, 2011]

Assume that the lemma holds for all c ≤ c0. Let D be a diagram with c0 + 1crossings. Let δ∗(D) be the dual state of δ with respect to D, and δ∗(D1) be thedual state of δ with respect to D1. Then δ∗(D1) differs from δ∗(D) only at theoriginal crossing. Thus, ∥∥∥δ∗(D1)

∥∥∥ =∥∥∥δ∗(D)

∥∥∥± 1

By the induction hypothesis we have,∥∥∥δ∗(D1)

∥∥∥+∥∥∥δ∗(D)

∥∥∥ ≤ cSubstituting in

∥∥∥δ∗(D1)

∥∥∥ =∥∥∥δ∗(D)

∥∥∥± 1, we obtain:

‖δ‖+ ‖δ∗‖ ± 1 ≤ c

Which implies‖δ‖+ ‖δ∗‖ ≤ c+ 1

as required by our Lemma.

This completes our proof of Theorem A.4.

A.3 HOMFLY Polynomial“His word is as good as a tied knot”-Russian Proverb

Beginning with a brief overview of this section, we will analyse some of thestatements from the HOMFLY47 original 1985 paper, and Lickorish’s textbook:An Introduction to Knot Theory. The HOMFLY polynomial is a famous linkinvariant48 that generalises the Jones and Alexander polynomials. First, wewill prove the existence of this invariant in two indeterminates and that it iswell-defined. Second, we will examine some simple properties of this polynomial.

In 1985 the editors of The Bulletin of the American Mathematical Societyreportedly received, virtually within a period of a few days in late Septemberand early October 1984, four research announcements, each describing the sameresult—the existence and properties of a new polynomial invariant for knotsand links [Freyd et al., 1985]. There were clear independent variations in the

47HOMFLY is an acronym for the first letters of the simultaneous but independent foundersof this new polynomial invariant; Hoste, Ocenau, Millet, Freyd, Lickorish and Yetter.

48We remember that a link invariant is, as Asharaf describes, “a function from the set ofall links to any other set such that the function is unchanged as the link is changed (up toisotopy)”[Ashraf, 2005].

lxxxix


approaches taken by the four groups and in their corollaries and elaboration.These were: A new invariant for knots and links by Peter Freyd and DavidYetter; A polynomial invariant of knots and links by Jim Hoste; Topologicalinvariants of knots and links, by W. B. R. Lickorish and Kenneth C. Millett, andA polynomial invariant for knots: A combinatorial and an algebraic approach,by A. Ocneanu.

For the purpose of this section we will concentrate on Lickorish and Millet’sapproach, as briefly outlined in their paper [Lickorish & Millet, 1988], but wewill supplement this with details from Lickorish’s 1997 text.

A.3.1 Existence

Lickorish & Millet utilise two ideas in their proof of the existence of the HOM-FLY polynomial. The first is that of an ascending link diagram, which can beinterpreted as follows.

We call a diagram L of an oriented link ordered if an ordering of the linkcomponents is chosen. We call it based if a base point is explicitly selected in Lon each link component. Hence if L is ordered and based, then the ascendingdiagram αL is constructed from L by changing the crossings so that when wefollow all the components in a specified order, being sure to always begin at thebase point of each component, each crossing is first encountered as an under-pass. What this means is that the link represented by αL can be though of aslying in R3 above the diagram, with each component lying below its subsequentcomponent in the given order as one moves around it away from its base point.This is somewhat like an ascending staircase. Thus, αL in actual fact, representsthe trivial link.

The second notion used by Lickorish is the fact that the Reidemeister movesdo not increase the number of crossings of a diagram above a certain bound. Inthe section on Reidemeister moves, we proved that they are an invariant of thecrossing number.

Both of these yield the following theorem:

Theorem A.5. If P is a function, P : D →Z[l±1,m±1], where D is an orientedlink diagram in S3, that satisfies the following conditions:

1. Normalisation: i.e. P (Kunknot) = 1

2. Skein relation: if the three oriented link diagrams, L+, L− and L0, shownbelow,

Figure A.13:The three oriented link diagrams.

Source: http://www.mathadore.nl/mathadore/knots/old/ConwayLs.png

xc


are the same, except in the neighbourhood of of one crossing, then:

lP (L+) + l−1P (L−) +mP (L0) = 0

P (L) is called the HOMFLY polynomial of the oriented link L.

Proof. Lickorish & Millet use induction on the number of crossings, whichdefines P on the link diagrams. This guarantees the validity of the skein relation:

lP (L+) + l−1P (L−) +mP (L0) = 0 (†)

for the diagrams in 2, and verifying invariance under Reidemeister moves. Notethat the equation (†) determines uniquely any one of P (L+), P (L−) and P (L0)from the other two. A solution is of the form:

(P (L+), P (L−), P (L0)) = (x, x, µx)

where x is arbitrary and µ = −m−1(l + l−1).Letting Dn be the set of all oriented link diagrams in the plane with at most

n crossings, suppose that P : Dn−1 → Z[l±1,m±1] has been defined such thaton Dn−1:

1. the relation (†) hold for any three diagrams in Dn−1;

2. P (L) is invariant under the Reidemeister moves on L that involve lessthan n− 1 crossings;

3. if L is any ascending diagram of a link in Dn−1 with #L components,then P (L) = µ#L−1.

Induction on L0, in which any diagram is ascending, leaves nothing toprove.

In his text Lickorish extends the definition of P over Dn in the followingway: if L is an n crossing diagram, choose an ordering of its components, selecta base point on each component and let αL be the resulting ascending diagram.Define P (αL) = µ#L−1where #L is the number of link components of L. It isclear that the crossings of αL can be changed to obtain L. If L+ and L− arethe diagrams before and after such a change, and L0 is the diagram with thecrossing removed, then the value of P on the diagram after the change can becomputed using (†) from P (L0) and the value on the diagram before the change.The value of P (L0) is known by induction. The value of P (L) is then definedto be the value computed from P (αD), obtained by changing the crossings ofαL to produce L. The sequence of these changes does not effect P (L), howeverwe must show that P (L) is independent of component order and choice of basepoints.

Maintaining the order of the components, we can observe that the basepoint b of a link component L is moved from before a crossing, to a point b′just after the crossing. Call βL the ascending diagram using b′ as its base pointinstead of b. If the other segment of the link involved in the crossing is partof a different component, then αL = βL. Otherwise βL can be constructedfrom αL by simply changing the crossing. Note that the diagram L0 is also anascending diagram, with #L+1 link components, and is indeed in Dn−1. Thus,

xci


by induction, we have P (L0) = µ#L and since P (αL)=µ#L−1, using these inour skein formula gives P (βL) = µ#L−1. This result implies that the order ofdefinition and calculation of P (αL) and P (βL) is irrelevant. This means we canchoose b, define P (αL) and calculate P (βL), or we can choose b′, define P (βL)and compute P (αL), both will produce the same result. Hence this implies thatthe definition of P (L) is independent of base point choice.

P is well-defined on n-crossing diagrams, and (†) is satisfied all such dia-grams, once L+ and L− have the same orderings. Reidemeister moves on anordered diagram L, with fewer than n-crossings will now be discussed. It is clearthat a Reidemeister move is invariant upon the ordering of L after the move.

Finally, it remains to be proved that supposing βL is an ascending diagramconstructed from L with reference to a different ordering, does P (L) remainthe same, i.e. is it independent of the ordering of its components? This wouldcomplete the induction, as βL is an arbitrary ascending n-crossing diagram.

In order to check this, observe that any component with no crossing thatbounds a disc whose interior is disjoint from the diagram nay be moved awayfrom the rest of the diagram49 using an ‘extension’ of the R2 move50. Turningour attention to the ‘innermost’ loop of the diagram51, using the R1 move wecan remove a crossing, This leaves P (L) unchanged, but our new diagram nowhas n − 1 crossings and is still ascending. Thus by the induction we haveP (βL) = µ#D−1.

If this is not the case, then the other arc of βL traverse the loop. Thesetransversals intersect each loop at two points: one such transversal and loopsegment bounds a 2-gon, which is most likely crossed by many transversals.Denoting the two arcs p and q, which meet the loop at A and B respectively,and bound the region R, any remaining transversals that enter R meet each of pand q. Within R the transversals meet each other in at most one point. This isthe situation described in Lemma 15.1 of Lickorish’s text52. Thus, there existsa 3-gon with an edge in p. Assuming that all the base points lie outside R, thefact that βL is ascending implies that the 3-gon has over-crossings at its threevertices in the scheme of an R3 type move. Applying this moves p across the 3-gon, changing R to a new region. Repeating this eliminates all the 3-gons in thenew region of R, and it follows that this region can be removed completely byan R2move. Therefore, βL can be changed by Reidemeister moves that involveno more than n crossings, to an ascending diagram with n− 2 crossings. ThusP (βL) = µ#L−1 by induction. Note that our choice of base points outside of Rwas valid because the choice of base points in an ascending diagram does noteffect the value of P .

This completes the induction hypothesis, and P is well-defined on Dn for alln, the skein formula (†) is always satisfied and since any sequence of Reidemeistermoves remains within Dn for some n, we found that P (L) is unchanged by allReidemeister moves. Thus P (L) defines a link invariant.

49Into the unbounded complementary region.50In the aftermath of the usual R2 move, we can extend the newly created unlink out into

space.51This is a sub-arc of the diagram, bounded by the same crossing at both of its ends.52This states that the graph, with vertices that are all intersections of ti ∩ tj (where ti, tj ∈

t1, ..., tn are traversals in R), and with edges (p ∪ q) ∪ni=1 ti, partitions R into a collectionof v-gons. Amongst these, there is a 3-gon with an edge in p and a 3-gon with an edge in q[Lickorish, 1997].

xcii


Example A.6. Lickorish and Millet calculated P (31) i.e. the HOMFLY poly-nomial of the trefoil knot, in their submission to the Bulletin to be:

−2l2 − l4 + l2m−2

It follows that the HOMFLY polynomial is indeed a specialisation of boththe Alexander and Jones polynomials, in the following manner:

1. l = ιt−1, m = ι(t12 − t− 1

2 ) produces the Jones polynomial VK(t) of theknot K, with links L consisting of a number of {L−, L0, L+}, and skeinrelation:

−tVL− − (t−12 − t− 1

2 )VL0+ t−1VL+

= 0

2. l = ι, m = ι(t12 − t−

12 ) define the Alexander polynomial ∆K(t) of the

knot K, with links L consisting of a number of link diagrams, and skeinrelation:

−∆L− − (t12 − t− 1

2 )∆L0+ ∆L+

= 0

One can obtain a formula for ∆L from P (L) through substitution of variables.When this is juxtaposed with the formula for VL, we can see the similaritybetween both formulae, which resulted in the discovery of the HOMFLY poly-nomial [Lickorish & Millet, 1988]:

∆L+−∆L− + (t

12 − t− 1

2 )∆L0= 0

t−1VL+− tVL− + (t−

12 − t 1

2 )VL0= 0

xciii

B THE ROLFSON KNOT TABLE

B The Rolfson Knot Table

[Scharein, 1998]

xciv

Index31 Knot, 102-component links, 13

A Split, 78Alexander polynomial, 58Alexander the Great, 6alternating projection, 15ambient isotopic, 10, 11amphicheiral knots, 69arc, 15ascending link diagram, 90

B Split, 78based, 90Borromean rings, 13Bracket Polynomial, 77

carboxy terminus, 68closed polygonal curve, 12colourability, 28colourable, 28complete knot invariant, 69constant loop, 34contractible, 38covering map, 37covering space, 44crossing number, 16

deck transformation, 49diagram, 15DNA, 67

equivalence, 6, 10equivalence class, 34, 35

figure of eight, 29finitely-presented, 35Fox derivatives, 63Fox-Artin Wild Knot, 12Freyd, 90fundamental group, 33, 34fundamental group of a Topological space,

33

generators, 34Gordian Knot, 6group structure, 35

homotopic, 10

homotopy-equivalent, 38Hopf link, 13

invariants, 6isomorphic, 38isotopic, 6, 11isotopy, 11

knot, 9, 11, 12knot isotopy classes, 6knot protein, 68knot recognition problem, 6

Lebesgue covering lemma, 37left-handed crossing, 16Lickorish, 90link, 12link invariant, 23, 89linking number, 25local flatness, 11loop, 33loops, 34

Millett, 90minimal diagram, 16mod p rank, 30multiplicity, 12mutant knots, 32

normal direction, 42normal vector, 41

Ocneanu, 90ordered, 90oriented, 13oriented alternating projection, 15oriented knot diagram, 15over-crossings, 15

p-colourings, 29placement problem, 6planar isotopy, 18planar representation, 15polygonal curve, 12presentation, 35Presentations, 34projection, 14

Ralph Fox, 28

95

INDEX

regular projection, 14Reidemeister Moves, 18relators, 34right-handed crossing, 16Rolfson Knot Table, 94

segment, 15Seifert circles, 42skein relation, 90smooth knot, 9States, 77

tame, 6, 12the empty word, 34the polynomial invariants, 32The Wirtinger presentation, 53Trefoil, 10trivial colouring, 29Trivial Knot, 9trivial link, 13

under-crossings, 15Unknot, 9unknot, 18unknotting, 18unknotting number, 23unlink, 13

Valknut, 10Van Kampen’s Theorem, 38Vassiliv invariant, 67

Whitehead link, 13Whitehead manifold, 27wild, 6wild knots, 11winding number, 37words, 34writhe, 17

Yetter, 90

xcvi

a blueprint for equivalence: the quest for a complete knot invariant · 2015-12-28 · a blueprint...

Documents