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ECOLE POLYTECHNIQUEPROMOTION X2003ARBEL Julyan

RAPPORT DE STAGE D’OPTION

Arc presentations of knots and links

NON CONFIDENTIEL

Option : Departement de MathematiquesChamp de l’option : Topologie, theorie des noeudsDirecteur de l’option : Nicole Berline

Directeur de stage : Hugh R. Morton

Dates du stage : 10 avril - 23 juin 2006Adresse de l’organisme : Dept. of Mathematical Sciences

University of LiverpoolPeach St. Liverpool L69 7ZL

Abstract

We present a survey of applications of arc presentation of knots and links. In a first chapter,

we start by defining several basic notions in knot theory. We focus then on an invariant polynomial

which yields results on alternating links. Eventually we describe arc presentations. The second part

deals with three straightforward applications of arc presentation. We first make use of the similarity

between grid form and front projection of a Legendrian knot in order to relate the Bennequin number

β to the arc index α. Then we show the analogy between arc presentations and closed braids to

provide an upper bound for the braid index s of the inverse parallel satellite of a knot. At last we

study a pragmatic notion, namely the ropelength Rop of a knot, for which the compact structure

of arc presentations gives good upper bounds. Also, appendices provide some of the simpler knots,

as well as a succinct inventory of conjectures in knot theory.

Resume

Nous decrivons certaines applications de la presentation par arcs des nœuds et entrelacs. Dans

un premier chapitre, nous commencons par definir quelques notions de base de theorie des nœuds.

Nous nous focalisons ensuite sur un invariant polynomial duquel nous tirons des resultats sur les

entrelacs alternes. Nous decrivons enfin les presentations par arcs. La seconde partie traite de trois

applications directes de la presentation par arcs. Nous tirons d’abord parti de la similarite entre

forme en grille et projection frontale d’un nœud Legendrien pour relier le nombre de Bennequin β

a l’index d’arc α. Nous utilisons ensuite l’analogie de la presentation par arcs avec la fermeture

de tresse pour degager une borne superieure de l’index de tresses s du satellite parallele inverse

d’un nœud. Nous traitons finalement d’une notion pragmatique, la longueur d’un nœud, Rop, pour

laquelle la structure compacte de la presentation par arcs donne de bons majorants. On trouvera

egalement en appendice une table des nœuds les plus simples, ainsi qu’un inventaire succinct de

conjectures en theorie des nœuds.

Foreword & Acknowledgements

For an interesting reading of this report, a basic knowledge of topology is sufficient; for a

delightful reading, some taste in 3–dimensional–flavour mathematics may be required. I drew

knots, links and most of diagrams of the report using XY-pic. I apologise for probable mistakes I

typeset. I am glad to express my warmest gratitude to Prof. Hugh R. Morton for selecting relevant

articles for me in a quite abundant literature, and making me accessible some of the notions in knot

theory with limpid explanations and transparent illustrations. I also thank Prof. Nicole Berline

for helping me to go to Liverpool, and all staff at the Department of Mathematical Sciences for

making my visit interesting in every respect.

Introduction

Knots have been considered for a long time ago, as we can see on Celtic engravingson stone for example. In a scientific approach, they appeared at first in the late 18thcentury in the works of C. -F. Gauss, F. Klein and M. Dehn. Their first tabulationsgo back to the end of the 19th century. At that time, Lord Kelvin thought thateach knot should correspond to a chemical element, but this appeared to be wrong.Later, the English physicist P. G. Tait formulated three conjectures. However themost significant results in this theory have been obtained in the last two decades andso, in parallel with, and certainly thanks to, the development of computers.

Basically, a simple question arises as soon as we begin to tackle knots:

How could knots be classified ?

This problem is still not solved. In this subject, lots of questions which occur areso simple to state that they could be explained to a child. The funny thing is thatthey often require ideas from the forefront of research to be solved. The amountof Fields medals which award works in knot theory is impressive: V. F. R. Jones,E. Witten and V. G. Drinfeld in 1990 and M. Kontsevitch in 1998.

Besides, arc presentation consists in placing a link in an open–book infrastructure.It appeared for the first time in the works of Hermann Brunn, when he provedthat any link has a (singular) diagram with only one multiple point (not necessarilydouble). In the 90’s, Joan S. Birman and William Menasco used arc presentationsof companion knots to study the braid index of their satellites. Lee Neuwirth [N⁀84]used them to give presentations of the fundamental group of the link complement.With the grid form of an arc presentation, Herbert Lyon [L⁀80] studied 3–manifoldconstructions, whereas Lee Rudolph [R⁀92] worked on quasipositivity.

Peter R. Cromwell and Ian Nutt developed the arc index invariant α, as well asbasic properties of α for split or factorised links [C⁀95]. Hugh R. Morton and Elisa-betta Beltrami produced a lower bound for α in terms of the breadth of the Kauffmanpolynomial in [MB98]. In 2000, Yong Ju Bae and Chang Young Park [BP00] pro-duced an upper bound for α in terms of the crossing number.

Contents

Introduction iii

Contents iv

Glossary vi

I Toolbox in Knot Theory 1

1 Elementary definitions 31.1 Knots and links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Equivalence of knots and links . . . . . . . . . . . . . . . . . . . . . . 41.3 Link diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Knots arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Polynomial invariants and alternating links 132.1 Bracket polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Jones polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Diagrams of alternating knots . . . . . . . . . . . . . . . . . . . . . . 17

3 Arc presentation 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Grid form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.2 Equivalence between square-bridge and arc presentations . . . 233.3.3 Existence of arc presentations . . . . . . . . . . . . . . . . . . 25

3.4 Some useful properties . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Lower bound for arc index . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Upper bound for arc index . . . . . . . . . . . . . . . . . . . . . . . . 28

CONTENTS v

3.6.1 Alternating case . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6.2 Non-alternating case . . . . . . . . . . . . . . . . . . . . . . . 32

II Applications of arc presentation 34

4 Legendrian knots and links 354.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Legendrian links . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.2 Front projection . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.3 Bennequin number . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Grid form and front projection . . . . . . . . . . . . . . . . . . . . . . 40

5 Braid index of satellites 425.1 Closed braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Braid index and arc index . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3.1 Closed braid of the companion . . . . . . . . . . . . . . . . . . 475.3.2 Closed braid of the satellite . . . . . . . . . . . . . . . . . . . 48

6 Ropelength of knots 516.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Definition of ropelength . . . . . . . . . . . . . . . . . . . . . . . . . 526.3 Upper bound for ropelength . . . . . . . . . . . . . . . . . . . . . . . 53

A Knot Table 57

B Conjectures 58

Bibliography 60

List of Figures 62

List of Tables 64

Index 65

Glossary

Notations

K knotL linkB braid

unknot

trefoilD diagramG graphc crossingv vertexA arc presentationS square-bridge presentationf framingC companionP patternSh satellite with homeomorphism hSf satellite with framing f

L mirror image of L−L reverse image of LK1 # K2 connected sum of K1 and K2

L1 ⊔ L2 disconnected sum of L1 and L2

sprx(P ) Laurent degree of P in the variable x

τ(c) thickness of a curve cλ(c) length of a curve cdcsd(c) doubly-critical self-distance of a curve c

CONTENTS vii

Link invariants

c(L) crossing index of Lp(L) polygon index of Ls(L) braid index of Lw(L) writhe of Lα(L) arc index of Lβ(K) Bennequin number of Kcusps(K) number of cusps of KRop(K) ropelength of Klk(K1, K2) linking number of K1 and K2

⟨L⟩(A) Kauffman polynomial of L

VL(t) Jones polynomial of L

French translations

Braid TresseKnot NœudLink EntrelacTrefoil TrefleSpoke Rayon

Part I

Toolbox in Knot Theory

Chapter 1

Elementary definitions

1.1 Knots and links

This part aims at giving elementary definitions about knots. These are intuitive.Further definitions will be given later, in the corresponding sections.

Firstly, one can say there are many ways to represent knots. Shoelaces mightbe the most common one: they call in tow pieces of rope and ties them together.More generally, one can think about a piece of rope knotted at its middle. Its towendpoints lie in different directions. By convenience, we will consider knots wherewe have joined the ends together to form a loop. Hence the following definition :

Definition 1.1.1 (knot) A knot is a smooth embedding of S1 in R3, or, alterna-

tively, its image.

Remark. This definition prevents the knot from having double points.

Remark. Smooth means (at least) C1. This definition is too restrictive, because itputs aside certain kinds of knots, for instance polygonal knots (see Section 1.4). Wewill extend it later. However it is pleasant because it avoids pathological cases thatarise with knots with some kind non-differentiable points.

A first example of a knot is a simple circle in R3. It is called the unknot , or trivial

knot . The second simplest knot is knotted just once. Because of its shape when itis closed, it is called the trefoil knot , or simply trefoil. The figure-eight knot is usedin climbing. These three knots are drawn in Figure 1.1.

Let us introduce a second objet, a bit more complex. Considering now severalknots, we feel that they can interfere in many different ways in R

3. Consider forexample two unknotted loops. Whether they are circling one around the other ornot is physically different. On the one hand they are tied, on the other hand we can

4 Elementary definitions

Figure 1.1: Trivial knot, trefoil and figure-eight knot

move one away from the other one as much as we want. The following definitiontakes account of this point:

Definition 1.1.2 (link) A link is a finite disjoint union of knots : L = K1 ∪ . . . ∪Kn.Each knot Ki is called a component of the link. The number of components, n, iscalled its multiplicity and is denoted by µ(L). A subset of the components embeddedin the same way is called a sublink.

A link with one component is a knot. A link L whose components are trivialknots which all bound disjoint discs is a trivial link, of multiplicity µ(L) = n, or then-unlinked.

The simplest way to form a non trivial link is to close tightly together two rings:we get the Hopf link (Figure 1.2).

Figure 1.2: Hopf link

1.2 Equivalence of knots and links 5

1.2 Equivalence of knots and links

Given a new objet, we want to classify it. Considering a physical knot, neither itsplace in the space matter, nor, once it is tied up and closed, its shape. That is tosay, we want to consider knots up to a certain equivalence.

So let us define a deformation between two knots. We could think first of homo-topy. However a quick inspection shows that every knot is homotopic to the trivialknot. Indeed homotopy allows curves to pass through themselves. We should refinethe class of deformations.

We can avoid the latter problem by imposing that the set of points during thedeformation remains a knot. That is, we impose that the map is injective at alltimes. Such a deformation is called an isotopy. There is still a significant problem;a subtle inspection shows that every knot is isotope to the trivial knot too. Indeedone can make vanish any tied part of a knot by isotopy by reducing it continuouslyuntil it remains nothing else but an arc !

Let us be more meticulous. In the latter situation, a whole neighbourhood of theknotted part is crushed on an arc. There is an obvious problem of dimension. Weavoid that by considering only ambient isotopies, i.e. which carry the space withthem:

Definition 1.2.1 (isotopy equivalence) Two knots K1 and K2 are ambient iso-topic if there is an isotopy ϕ : R

3 × [0, 1] −→ R3 such that ϕ(K1, 0) = K1 and

ϕ(K1, 1) = K2.

At all time t in [0, 1], ϕ(., t) is one-to-one. Initially equal to K1 in K1, it is equalto K2 at the end.

It is straightforward to check that ambient isotopy is an equivalence relation onknots: two knots are equivalent if they can be deformed into one another. Eachequivalence class is called a knot type. By the sake of simplicity we will forget thedistinction between a knot and its type. From now on, a knot means with someabuse of terminology its type.

To prove that two knots are isotopic, one should present a step-by-step isotopytransforming one into another. Later on we shall present the list of Reidemeistermoves. Conversely, to show that two knots are not isotopic, one usually finds aninvariant having different values on these two knots.


1.3 Link diagrams

Usually, knots and links are encoded as follows. Let L be a link and π : R3 → R

2

a projection map. A part of π(L) which is the image of an interval of S1 is calleda branch. A point x ∈ π(L) is regular if | π−1(x)| = 1, and singular otherwise. If|π−1(x)| = 2 then x is called a double point . A double point is called transverseif it is the intersection of two crossing branches in the projection plan (not tangentfor example). A projection is called regular if π(L) has a finite number of singularpoints which are all transverse double points.

Each vertex V of this graph can be endowed with the following structure: let a,b be two branches of a link, whose projections intersect at vertex V ∈ π(L); sincea and b do not intersect in R

3, the two preimages of V have different height. So,we can distinguish which branch (a or b) comes over, or forms an over-crossing ; theother one forms an under-crossing .

under-crossing over-crossing

Figure 1.3: Local structure of a crossing

Definition 1.3.1 (link diagram) A diagram is a regular projection of a link withthe additional height information on the double points, also called crossings of thediagram.

The quadrivalent projection graph without an over/under-crossing structure iscalled the shadow of the knot.

Definition 1.3.2 (crossing number) The crossing number c(D) of a diagram Dis its number of crossings (!). The crossing number, or crossing index c(L) of a link(type) L is the minimum of c(D) over all diagrams D which represent the link (type)L.

A diagram of a link L whose number of crossings equals c(L) is called a minimaldiagram, or minimal-crossing diagram to avoid any ambiguity.

1.3 Link diagrams 7

In order to measure the connectedness of a diagram we look at its shadow, whichhas a genuine graph structure. The diagram is connected if it has a connected shadow .One should mind that a non connected link can have a connected diagram!

Sometimes, we will need diagrams whose structure is reduced, in the followingmeaning:

Definition 1.3.3 (reduction) A diagram is reduced if it has no crossing that is asplitting point. A crossing is said to be a splitting point, Figure 1.4, if it correspondsto a cut vertex in the shadow, i.e. a vertex whose removal disconnects the graph.

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

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

.....

Figure 1.4: A splitting point

Given a link L, we can form its mirror image L, i.e. the link obtained from theinitial one by reflecting it in some plane. Typical diagrams of a link and its mirrorimage can be obtained by switching all crossing types (over-crossing replaces under-crossing and vice versa). A link is called amphicheiral if it is isotopic to its mirrorimage. Otherwise it is said to be chiral .

Remark. Chemists tend to use preferably the term achiral to denominate am-phicheiral molecules.

One can also speak of oriented links, i.e. smooth mappings (images) of orientedcircles in R

3. By an isotopy of oriented links is meant an isotopy of links preservingorientation. Given an oriented link L, we can form its reverse −L by reversing theorientations of all its components. It is said reversible if it has the same isotopy typeas its reverse.

Latter properties are summed up in Table 1.1.

Definition 1.3.4 (alternativity) A link diagram is called alternating if while mov-ing along each component, one passes over-crossings and under-crossings alternately.A link is called alternating if it has (at least) one alternating diagram.

Remark. This kind of definition is common in knot theory: families of links aredefined in terms of their diagrams. Similarly, a positive link is an oriented link whichhas a positive diagram, i.e. with only positive crossings (Figure 1.5). One shouldmind that such properties are asked to be true on one diagram of the link, but are


link mirror image reverse

L L −L

amphicheiral link reversible link

L = L −L = L

Table 1.1: Chirality and reversibility

not necessarily true (i) neither on all its diagrams (ii) nor on its minimal-crossingdiagram. However, we will see later that an alternating diagram with connectedshadow and without splitting point is de facto a minimal diagram of its link. This wasconjectured by Tait about a century ago; we will prove it later with the Kauffman-Murasugi theorem.

??

+1

??

−1

Figure 1.5: Positive and negative crossings

1.4 Reidemeister moves

We have shown in the previous section that knots and links can be encoded by theirregular planar projection. While deforming a link, its planar projection might passthrough some singular states. These singular states give motivation for providing alist of simple moves for planar diagrams.

A knot is a smooth embedding of S1 in R3 so it can be arbitrarily closely approxi-

mated by an embedding of closed broken lines in R3. One decrees the approximation

is ‘closely enough’ when the knot type is preserved.

Definition 1.4.1 (polygonal link) A polygonal link is an embedding of a disjointunion of closed broken lines in R

3. A polygonal knot is a polygonal link with onecomponent.

One define a polygonal diagram similarly to a diagram of a smooth link (in termsof regular projection, c.f. Section 1.3). The polygonal number p(D) of a polygonal

1.4 Reidemeister moves 9

diagram D is its number of sticks. The minimum of p(D) over all polygonal diagramD of a given link L is called its polygonal index , and denoted p(L).

We have p( ) = 3 and p( ) = 6.

Remark. One should mind that any 4−valent polygonal graph endowed with anover/under structure is not necessarily a diagram of a polygonal link. Indeed some‘supposedly projection’ of sticks in R

2 may not correspond to any link in R3. For

example, see a closed star with five arms, each crossed with its two neighbors exceptone arm over its two neighbors. Then such a knot cannot be closed in R

3, unless weadd a sixth stick ‘vertically’, which contradicts the assumption of regular projection.

Definition 1.4.2 (tame link) A link is called tame if it is isotopic to a polygonallink and wild otherwise.

It is shown in [CF63] that any smooth link is tame.

Let us define an equivalence on polygonal links. Similarly to ambient isotopy, wework in R

3 (not in the projection plane). Let L be a polygonal link, and let ∆ be atriangle in R

3 such that:

• L does not meet the interior of ∆

• L meets one or two sides of ∂∆

• the vertices of L in L ∪∆ are also vertices of ∆

• the vertices of ∆ in L ∪∆ are also vertices of L

Definition 1.4.3 (∆-move) A ∆-move on L is defined as follow: one erases theside(s) of ∆ shared with L and replace it (them) by the other(s) of ∆, see Figure 1.6.

//////////////////// • •

• •

∆

←→

//////////////////////////////////////// • •

• ••

∆

Figure 1.6: A ∆-move


Remark. One should mind that a ∆-move is defined in R3, as ambient isotopy does.

It is not defined in terms of planar projection, with which a triangle ∆ may havecrossings.

Definition 1.4.4 (combinatorial equivalence) Two polygonal links are combi-natorially equivalent if there exists a finite sequence of ∆-moves that transforms oneinto the other.

It is clear that ∆-moves preserve ambient isotopy. Reversely:

Theorem 1.4.1 If two polygonal links are ambient isotopic, then they are combina-torially equivalent.

Although this result is intuitive, its proof is technically complicated. It can befound in [CF63].

The point of all this is that we shall use either smooth or polygonal approach forrepresenting links.

Ω1←→

Ω2←→

Ω3←→

Figure 1.7: The three Reidemeister moves

1.4 Reidemeister moves 11

Clearly, Reidemeister moves preserve ambient isotopy. Reversely, they are suffi-cient; the following theorem is fundamental in knot theory, and will turn out to beof a crucial usefulness later:

Theorem 1.4.2 (Reidemeister) Two diagrams of a link are related by a finite se-quence of planar isotopy and Reidemeister moves Ω1, Ω2 and Ω3 shown in Figure 1.7.

• •

• •

• •

CCCCCCCCCCCC

• •

• •

•

oooooooooooooooooooo

• •

•

•

• •

First Second Third Fourth

Figure 1.8: Types of ∆-triangles

PROOF. We choose to work with polygonal links. From Theorem 1.4.1 we know thatany two polygonal embeddings of a link are related by a finite sequence of ∆-moves.To prove the theorem one only need to check what happens to a diagram when onemove is performed. Let D be a polygonal diagram of a link L, let us perform a∆-move, and denote D′ the diagram obtained. One can choose the projection suchthat both D and D′ are regular.

Without loss of generality, one can assume that the two edges coming out fromL ∩∆ do not intersect ∆. Otherwise, one can add or remove a curl by Ω1.

Let us tile ∆ into small triangles of four types (shown in Figure 1.8) in sucha way that edges of small triangles do not contain vertices of D. Each first-typetriangle contains only one crossing of D; here edges of D intersect two sides ofthe triangle. The second-type triangle contains only one vertex of D and parts ofoutgoing edges. The third-type triangle contains a part of one edge of D and novertex. Finally, the fourth-type triangle contains neither vertex nor edge. Such atriangulation of ∆ can be constructed as follows. First, we cut all vertices and allcrossings by triangles of respectively the first and second types. Then we tile theremaining part of ∆ and obtain triangles of the last two types. The plan of the proofis now the following. Instead of performing the elementary isotopy to ∆, we performstep-by-step elementary ∆-moves for small triangles, composing ∆. It is clear thatthese elementary moves can be represented as combinations of Reidemeister moves


and planar isotopies. More precisely, the first-type triangle generates a combinationof Ω2 and Ω3, the second and the third-type triangles generate Ω1 or planar isotopy.The fourth-type triangle generates planar isotopy.

So D′ is obtained from D by a sequence of Reidemeister moves and planar iso-topy.

The three Reidemeister moves are independent, i.e. for each of the three moves,there exist two diagrams of the same link which cannot be transformed one into theother without the given move. We will not explicit these examples.

1.5 Knots arithmetics

Let us now discuss the algebraic structure on the set of knot isotopy classes. Let K1

and K2 be two oriented knots.

Definition 1.5.1 (connected sum) By a connected sum, composition or concate-nation of two oriented knots K1 and K2 is meant the oriented knot obtained byattaching the knot K1 to the knot K2 with respect to the orientation of both knots.The knot obtained is denoted K1 # K2. We say that K1 and K2 divide K1 # K2, orare its factors.

−→

Figure 1.9: Connected sum

This operation is well defined for oriented knots. It does not depend on the twoplaces of attachment: to see it, we can shrink for instance K2, and make it ‘slide’along K1 from one position of attachment until we reach another one.

It is commutative and associative.

It is not well defined if orientations are ignored: there are two ways to attachthe knots together. The product is different in general. It gives the same result onlywhen at least one knot is reversible (c.f. Section 1.3).

Another problem arises when we want to compose two links, namely we have tochoose which components to connect.

Another way to consider tow links it to look at them separately, without connex-ion:

1.5 Knots arithmetics 13

Definition 1.5.2 (disconnected sum) If each of two links L1 and L2 lies in oneof the two different domains delimited by any 2-sphere S in the 3-space, then we callthe union of L1 and L2 their disconnected sum, and denote it L1 ⊔ L2.

Contrary to the connected sum, the disconnected one is well defined for any link.

Definition 1.5.3 (split link) Let L be a link. L is said to be a split link if for twoof its components: L = L1 ⊔ L2.

Remark. One should mind not to mix up the connectedness of a link defined abovewith the one of a diagram given in Section 1.3.

As a nice result on these definitions, let us give the following theorem:

Theorem 1.5.1 The trivial knot has no non-trivial factors.

Manturov gives three different proofs of this result in [M⁀04].

Definition 1.5.4 (prime knot) A knot K is said to be prime if for any knots K1

and K2 such that K = K1 # K2, one of the knots K1 or K2 is trivial. All otherknots are said to be complicated.

Thus, we have proved in Theorem 1.5.1 that given a non-trivial knot K1, for anyknot K2, the knot K1 # K2 is not trivial either; that is, all elements of the knotsemigroup except the unknot have no inverse elements. We have another propertyof this semigroup, analogous to the Gauss Lemma in number theory:

Lemma 1.5.2 (Gauss Lemma) Let K1 and K2 be knots, and let K3 be a primeknot dividing K1 # K2. Then either K3 divides K1 or K3 divides K2.

Chapter 2

Polynomial invariants andalternating links

In this part we shall introduce several new polynomial link invariants, discovered inthe mid 80’s. We will describe the bracket polynomial and then the Jones polynomial,as they are understood by Louis Kauffman in [K⁀88].

2.1 Bracket polynomial

Let us begin with defining a 3−variable polynomial on unoriented diagrams. We willsee later which conditions are required for it to be an invariant.

Let D be an unoriented link diagram. We want to define the so-called bracketpolynomial of D, which is denoted 〈L〉 ∈ Z [ A, B, d ]. This notation may be ambigu-ous, so one should mind that a choice of a particular diagram D for L is done. Wewill sometimes specify 〈D〉. We require the following axioms:

1.⟨ ⟩

= A⟨ ⟩

+ B⟨ ⟩

⟨ ⟩= B

⟨ ⟩+ A

⟨ ⟩

2.⟨L

⋃ ⟩= d

⟨L

⟩

3.⟨ ⟩

= 1

Remark. Each of the elementary pieces , , and stands for a diagramwhich is modified only at a given crossing by the substitution of the correspondingpiece. In the second and third axioms, denotes the diagram of the unknot withno crossing.

2.1 Bracket polynomial 15

Remark. Let . . . be the n−unlinked. Then :⟨

. . .⟩

= dn−1

Thus, the bracket polynomial is computed by smoothing each crossing of thediagram one by one. A meticulous look at a crossing shows that the four partsdelimited by its arcs are of two kinds. Two opposite parts are swept out by the over-crossing arc rotating counterclockwise: we label them A; the two others are labeledB.

The smoothing of a crossing which opens the A parts (resp. B parts) and joinsthem into a so-called A-channel (resp. B-channel) is called the A-smoothing(resp. B-smoothing ). The two diagrams obtained are denoted respectively DA

and DB. A state of a diagram is a set of disjoint closed curves obtained from adiagram by smoothing all each of its crossing in either the A or B-channel. For adiagram with n crossings, there are 2n ways to compute a state (among which lotsgive the same state). For a given state S of a diagram D, let a(S) and b(S) berespectively the number of A-smoothing and of B-smoothing used to construct it.Let |S| be its number of components.

Then we have the:

Lemma 2.1.1 (bracket polynomial) The bracket axioms imply the following ex-pression for the bracket polynomial:

⟨L

⟩=

∑

S

Aa(S)Bb(S)d|S|−1

PROOF. We can compute⟨L

⟩by constructing a rooted binary tree in which the

vertices are labelled by link diagrams obtained from L by smoothings, and the edgeswith monomials in A and B such that:

1. the root vertex is labeled with a diagram of L,

2. each terminal vertex is labeled with a state of L,

3. each triple (parent, left child, right child) is of the form (D,DA, DB) for acertain D,

4. each left edge (resp. right edge) is labelled with A (resp. B):

We conclude using the remark on trivial links.

Let us check how does the bracket polynomial behave under Reidemeister moveΩ2:

16 Polynomial invariants and alternating links

D

DA

A

DB

B

22222222222

Figure 2.1: Computing the bracket polynomial with a rooted binary tree

Lemma 2.1.2 (Reidemeister move Ω2)

⟨ ⟩= AB

⟨ ⟩+

(A2 + ABd + B2

) ⟨ ⟩

PROOF.

⟨ ⟩= A

⟨ ⟩+ B

⟨ ⟩

= A

(A

⟨ ⟩+ B

⟨ ⟩)+ B

(A

⟨ ⟩+ B

⟨ ⟩)

= AB⟨ ⟩

+(A2 + ABd + B2

) ⟨ ⟩

thanks to the third axiom.

Thus, the bracket polynomial verify Reidemeister move Ω2 if and only if weimpose:

B = A−1 and d = −A2 −A−2

With some abuse of terminology, we keep the same name for the new polynomial,which is a Laurent polynomial.

The following lemma tackles the behaviour of the (new) bracket polynomial underthe Reidemeister move Ω3:

Lemma 2.1.3 (Reidemeister move Ω3) Invariance under Ω3 follows from the pre-vious rules.

PROOF. Let us apply the first axiom to the lower crossing:

⟨ ⟩= A

⟨ ⟩+ B

⟨ ⟩= A

⟨ ⟩+ B

⟨ ⟩

2.2 Jones polynomial 17

according to the invariance under Reidemeister move Ω2. The symmetric form ofthe latter expression shows that:

⟨ ⟩=

⟨ ⟩which concludes the proof.

We call the equivalence generated by Ω2 and Ω3 regular isotopy. Thus the bracketpolynomial is a regular isotopy invariant. The following lemma shows that it is notinvariant under Ω1:

Lemma 2.1.4 (Reidemeister move Ω1) The following holds:

⟨ ⟩= (−A)3

⟨ ⟩

⟨ ⟩= (−A)−3

⟨ ⟩

PROOF. The calculation is straightforward:

⟨ ⟩= A

⟨ ⟩+ A−1

⟨ ⟩

= (−A(A2 + A−2) + A−1)⟨ ⟩

rearranging ends the proof of the first statement, the second one is similar.

2.2 Jones polynomial

It is time to introduce another regular isotopy invariant:

Definition 2.2.1 (writhe number) The sum of all crossing signs ±1 of a diagramof an oriented link is called its writhe number and is denoted by w(L).

It is obvious to see that w(L) is unchanged by moves Ω2 and Ω3, hence it is aregular isotopy invariant. It increases or decreases of 1 when we apply the move Ω1

in the direct or inverse sense. We can construct a new invariant:

Lemma 2.2.1 fL(A) = (−A)−w(L)⟨L⟩

is an ambient isotopy invariant on orientedlinks.


PROOF. It suffices to note that fL is regular isotopy invariant and is, by construc-tion, invariant under Ω1.

Remark. The link L in the expression⟨L⟩

merely forgets its particular orientation;indeed

⟨.⟩

is defined on unoriented links.

It turns out that fL is the Jones polynomial up to a certain variable change.

Let L be a link (diagram). Let us choice one crossing. It is either positive

??

or negative

??

. In both cases we denote by L+ the link where this given crossing ispositive (and the rest of the diagram is unchanged), L− where it is negative and L0

where it is smoothed in the way shown bellow:

Definition 2.2.2 (Jones polynomial) For a given oriented link L, we define theLaurent polynomial VL in the variable t with integer coefficients by the followingaxioms:

1. t−1 V

?? − t1 V

?? = (√

t− 1√t) V ??

2. V//

oo = 1

3. VL(t) is an invariant of ambient isotopy.

It is called the Jones polynomial of L.

Lemma 2.2.2 fL(t−1/4) = VL(t)

PROOF. This is a straightforward calculation.

2.3 Diagrams of alternating knots

We will show in this section that reduced alternating diagrams are crossing-minimal,which remained for almost a century a conjecture, known as the First Tait conjecture.Recall that reduced diagrams were introduced in Definition 1.3.3.

Theorem 2.3.1 Let D be a reduced alternating diagram. Then the highest andlowest degrees in

⟨D

⟩are given by:

maxdeg⟨D

⟩= c(D) + 2(|SA| − 1)

mindeg⟨D

⟩= −c(D)− 2(|SB| − 1)

where SA and SB are the states with respectively only A-channels and B-channels.

2.3 Diagrams of alternating knots 19

PROOF. Let us call a flip the transformation of a A-channel to a B-channel or viceversa.

Let SA be the a-state channel. Note that any other state S ′ may be obtainedfrom SA by flipping some subset of state channels. Recall that

⟨L

⟩=

∑

S

Aa(S)−b(S)(−A2 − A−2)|S|−1

.Let

⟨L|S ′ ⟩ = Aa(S′)−b(S′)(−A2 − A−2)|S

′|−1 so that⟨L

⟩=

∑

S′

⟨L|S ′ ⟩. We

will prove the two following facts:

1. If S ′ is obtained from S ′′ by flipping an A-channel to a B-channel, thenmaxdeg

⟨L|S ′ ⟩ ≤ maxdeg

⟨L|S ′′ ⟩. The inequality is strict exactly when S ′

has fewer components that S ′′. That is when |S ′| = |S ′′| − 1.

2. If S ′ is obtained from the A-channel state by one flip, then |S ′| = |S| − 1.

Let us look at the first assertion. If⟨L|S ′′ ⟩ = Ax(−A2 − A−2)|S

′′|−1, then⟨L|S ′ ⟩ = Ax−2(−A2 − A−2)|S

′|−1. Since S ′ is obtained from S ′′ by one flip, weknow that |S ′| = |S ′′| ± 1.

• If |S ′| = |S ′′|+1 then⟨L|S ′ ⟩ = Ax−2(−A2−A−2)|S

′′| hence maxdeg⟨L|S ′ ⟩ =

maxdeg⟨L|S ′′ ⟩

• If |S ′| = |S ′′|−1 then⟨L|S ′ ⟩ = Ax−2(−A2−A−2)|S

′′|−2 hence maxdeg⟨L|S ′ ⟩ =

maxdeg⟨L|S ′′ ⟩− 4

So 1. is true. Assertion 2. results from the assumption of reduction.

It follows that for all states S ′′ different of SA: maxdeg⟨L|S ′′ ⟩ < maxdeg

⟨L|SA

⟩.

Thusmaxdeg

⟨L

⟩= maxdeg

⟨L|SA

⟩= c(D) + 2(|SA| − 1)

The case of minimum degree is analogous.

We can deduce the following theorem, which was proved independently by Kauff-man, Murasugi and Thistlethwaite:

Theorem 2.3.2 (Kauffman, Murasugi, Thistlethwaite) The number of cross-ings in a reduced alternating diagram D of a link L is a topological invariant ofL.


PROOF. Let spr(⟨D

⟩) be the Laurent degree of

⟨D

⟩, i.e. the difference between

higher and lower degrees. Since fL(A) = (−A)−w(L)⟨L⟩

is an ambient isotopy in-variant of L, we conclude that spr(

⟨L

⟩) is an ambient isotopy invariant of L too.

Theorem 2.3.1 gives us:

spr(⟨D

⟩) = maxdeg

⟨D

⟩−mindeg

⟨D

⟩

= (c(D) + 2(|SA| − 1))− (−c(D)− 2(|SB| − 1))

= 2c(D) + 2(|SA|+ |SB| − 2)

But |SA| + |SB| equals the number of faces in the corresponding graph, whichequals to c(D) + 2. Hence spr(

⟨D

⟩) = 4c(D), i.e. the number of crossings is an

invariant.

Let S be a state of a diagram D. Then we denote by S the dual state of S, i.e.the state where all the channels have been flipped. Then:

Lemma 2.3.3 Let D be any diagram. Then:

|S|+∣∣∣S

∣∣∣ ≤ R

, where R is number of regions of D (the number of faces of the corresponding graph).If D is non-alternating then:

|S|+∣∣∣S

∣∣∣ ≤ R − 2

The proof of this lemma is based on the existence of two consecutive over- orunder-crossings in a non-alternating diagram.

We deduce the following Proposition:

Proposition 2.3.4 For any reduced non-split diagram D of a link of any type (notnecessarily alternating), we have:

spr(VD) ≤ c(D)

Equality holds only for alternating diagrams and connected sums of them.

PROOF. The same arguments than in the proof of Theorem 2.3.1 shows that:

maxdeg⟨D

⟩≤ c(D) + 2(|SA| − 1)

mindeg⟨D

⟩≥ −c(D)− 2(|SB| − 1)

2.3 Diagrams of alternating knots 21

Notice that SB = SA so:

spr(⟨D

⟩) ≤ 2c(D) + 2(|SA|+ |SB| − 2) ≤ 4c(D)

Recall that fD(t−1/4) = VD(t) which implies that spr(⟨D

⟩) = 4 spr(VD), and there-

fore spr(VD) ≤ c(D)Using Lemma 2.3.3 for a non-alternating diagram gives a strict inequality.

We can know deduce the proof of the first Tait conjecture:

Proposition 2.3.5 (first Tait conjecture) Reduced alternating diagrams and con-nected sums of them are crossing-minimal.

PROOF. Let us assume the contrary: there exist two diagrams D and D′ of the samelink, D is alternating and reduced (or connected sum of them) with n crossings andD′ has n′ crossings, with n′ < n. Then:

spr(VD′) ≤ n′ < n = spr(VD)

The contradiction is that spr(VD′) = spr(VD).

Chapter 3

Arc presentation

3.1 Introduction

We will focus now on a special way of representing knots and links, called arc pre-sentation. In such a structure, a link lies within the pages of an open–book.

We will first give here definitions relating to arc presentations. We will then lookat the grid form derived from an arc presentation, which will be useful later on forthe study of Legendrian knots and links. In the end, we will look at a series ofinteresting properties of arc presentation, which we will use for example to boundthe rope length of a knot.

3.2 Definitions

We start with a definition:

Definition 3.2.1 (arc presentation) An arc presentation A of a link L is an em-bedding of L in a finite collection of open half-planes arrayed around a common axis,or binding, so that the intersection of L with each half-plane is a single simple arc.

The following definition looks very similar to the definition of crossing index of alink L, which is the minimal crossing number over all the diagrams of the link L.

Definition 3.2.2 (arc index) The number of half-planes is called the arc numberof the arc presentation A, and denoted α(A). The minimal arc number over all arcpresentations of a link L is an invariant of the link type, called its arc index. It isdenoted by α(L).

We shall prove later that every link has an arc presentation (see Theorem 3.3.1).

3.2 Definitions 23

Let A be an arc presentation of a link L, with arc number α(A) = m. Let usconsider cylindrical coordinates, with z−axis equal to the binding axis. Let labeleach half-plane anticlockwise from 1 to m, and each arc ai. The polar angle θi of theith half-plane does not matter, provided that it remains between the (i−1)th and the(i + 1)th (with a circular convention on the labels taken in 1, . . . , m modulo m).We will display the ith half-plane at polar angle 2iπ

m; this banality turns out to be

very appropriate to minimize the ropelength of a link, as we will see in Chapter 6.

Each arc has its two ends on the binding axis, and each endpoint links two arcs.Thus there are m endpoints on the binding. We label them from 1 to m accordinglyto the z−coordinate. Each of them can be seen as a floor, or a level of the arcpresentation.

The Figure 3.1 shows an arc presentation for the trefoil.

= 0 = 25 = 45

= 65 = 8512345

Figure 3.1: Arc presentation for the trefoil

An arc presentation can be specified by combinatorial data: a collection of mtriples in the form (xi, yi, θi), where each triple denotes an arc from level xi to levelyi on the half-plane at angle θi around the axis.

Let us end up this section with a notion of total distance traveled vertically bythe arcs in an arc presentation:

Definition 3.2.3 The total skip of an arc presentation A, denoted by Skip(A), is

24 Arc presentation

defined by

Skip(A) =α∑

i=1

|xi − yi|

Total skip will be useful to bound the ropelength of knots in Chapter 6.

3.3 Grid form

3.3.1 Presentation

The grid form of a link is a kind of a diagram. As we will see, the piece of data givenby a grid form is equivalent to the one given by an arc presentation. Grid formshave been studied separately from arc presentations: Herbert Lyon used them in3−manifold constructions and called them square-bride presentations; Lee Rudolphcalled them fences when he studied quasipositivity. As for us, we give a presentationof the grid form here in order to see later on in Section 4.2 how it is related to theLegendrian knots and links.

The name square-bridge presentation used by Lyon is quite transparent, providedthat one knows what a bridge presentation is. In one sense, a bridge presentationmeasures how non planar a link is:

Definition 3.3.1 (bridge presentation) A bridge presentation B of a link L is anembedding of L in a plane P except for a finite number of transverse bridges thatlie above the plane. The projection of the bridges onto the plane must consists ofdisjoint straight lines.

Remark. For a diagram of a bridge presentation, it is very convenient to choose itsown plane P for the projection.

Now we would like a specific presentation of links such that:

1. Their diagrams lie in a square pattern plane (Figure 3.2).

2. The overcrossings in their diagrams are in one and the same direction.

Let us specify the latter definition:

Definition 3.3.2 (square-bridge presentation) A square-bridge presentation Sof a link L is a bridge presentation of L such that:

1. The part of L in its plane P lies in a square pattern.

3.3 Grid form 25

Figure 3.2: Piece of a square pattern

2. The bridges are in one and the same direction of the pattern, above it.

Remark. Following the latter Remark, the plane used for a diagram of a square-bridge presentation is P. Such a diagram is called a grid form.

The left of Figure 3.3 presents a diagram of the left-handed trefoil. A grid formcan be deduced easily (right of Figure 3.3) because all the crossings are oriented ‘inthe same way’. It suffices to turn the head clockwise of 45 when we look at the gridform to see the similarity.

This similarity may seem pointless in this section. When looking at Legendrianlinks, we will see it turns out very useful.

Figure 3.3: A grid form deduced from a diagram

26 Arc presentation

3.3.2 Equivalence between square-bridge and arc presenta-tions

Let us see now how we can deduce a square-bridge presentation from an arc pre-sentation. The main idea consists in projecting the knot on a cylinder around thebinding axis, and then developing the cylinder by opening it.

Let A be an arc presentation of a link L. Let C1 and C2 be two full cylinders ofrespective radius 1 and 2 of axis the binding. Let us apply the following successivetransformations which conserve the type of L:

1. Project each arc ai, which links xi to yi, on the bound of C2, i.e. replace it bythe open 3−polygon of (r, z)−coordinates:

(0, xi), (2, xi), (2, yi), (0, yi)

At this stage we still have an arc presentation which is moreover polygonal.

2. At each level from 1 to m, transform the wedge formed by the two half spokesof radius 1 into the arc of circle of radius 1 which does not pass through θm

and θ1. At this stage we do not have an arc presentation anymore. Now thelink lies in C2\C1.

3. The link does not meet the wedge part of the space between the half planesθm and θ1. Thus we can cut C2\C1 by any half plane in between, and developit uniformly so as to obtain a parallelepiped whose basis is the symmetrictrapezium of bases 2π and 4π and height 1.

4. Stretch it uniformly so as to obtain a right parallelepiped V of lengths 4π, 1and m− 1.

Now we get easily a grid presentation of L. Let us look at the front face of V ,which is the former bound of C2: it contains vertical branches. On the back faceof V , the former bound of C1, we see horizontal branches overcrossed by the latterones. Eventually, we choose to look perpendicularly to these faces, so that we do notsee the remaining branches which link the vertical and the horizontal ones.

Hence that defines a square-bridge presentation, whose plane P is given by theback face of V .

Conversely, these four steps are invertible, so that one can recover an arc presen-tation from a square-bridge one. Thus both presentations are actually equivalent.This will allow us to provide some quite interesting relations in terms of Legendrianknots.

3.3 Grid form 27

1

1

2

2

3

3

4

4

55

2

1

3

2

4

3

5

4

15

3

1

4

2

5

3

1

4

25

Figure 3.4: Three grid forms deduced from an arc presentation

Let us illustrate several grid forms we can obtain for the trefoil with this method.We use the notations of the definition above. Cutting the cylinder of the arc pre-sentation of the trefoil given in Figure 3.1 between half-planes 5 and 1 gives the leftpart of Figure 3.4. We have added the labels of half-planes (above) and of levels(left side). We recognize the grid form of Figure 3.3. Cutting the cylinder betweenhalf-planes 1 and 2 gives the middle part of Figure 3.4. We recognize the trefoil asit is usually represented, with a stretched lobe. Eventually, the right part of Fig-ure 3.4 shows that a grid form (cutting between 2 and 3) is not necessarily a minimalcrossing diagram of a link. Here we have indeed four crossings for the trefoil.

3.3.3 Existence of arc presentations

As a first application of the equivalence between arc and square-bridge presentation,we will show that every link has indeed an arc presentation. The argument consistsin using planar isotopy.

Theorem 3.3.1 (existence of arc presentation) Every link has an arc presen-tation.

PROOF. Let L be a link. It suffices to find a grid form of L, from which we deducea square-bridge presentation, and then an arc presentation. It is easy to form a gridform from any diagram D of L: by planar isotopy, let us rotate (and possibly stretch)

every crossings of D in order to make all them look like , i.e. the crossing isperpendicular and the overcrossing is vertical.

Now let us place every crossing of the later diagram on a square pattern in thethe plane.

28 Arc presentation

Now let us try to make all branches lie in this square pattern. We may haveplaced several crossings too close to each others in the pattern so that it is notpossible anymore to make an existing branch run between them. We remedy this bymoving such crossings in a way which keep a path clear between them. That givesus a grid form of L.

3.4 Some useful properties

The length of the knot ran vertically is given by Skip(A) times the height of levels.We can think that Skip(A) is bounded in general in terms of α(A). When α(A)increases, Skip(A) should grow like α(A)2 because both the number of vertical skipsand the total height increase like α(A).

The number of vertical levels in an arc presentation A is precisely α(A). We willshow with a combinatorial argument that:

Lemma 3.4.1 If A is an arc presentation, then

Skip(A) ≤

α(A)2−12

if α(A) is odd,α(A)2

2if α(A) is even.

This bound is sharp.

PROOF. Let A be an arc presentation of arc number α(A) = m.

We have to find an upper bound for Skip(A) =

m∑

i=1

|xi−yi|. We first observe that

the difference |xi− yi| is one unit larger than the number of levels skipped over. Forexample, jumping from level 3 to level 6, a difference of 3 levels, skips the fourth andfifth levels. Thus, we can rewrite the sum

Skip(A) = m +m∑

i=1

number of levels skipped by the arc (xi, yi, θi)

Notice that any level j contributes to the above sum exactly when it is skipped

3.4 Some useful properties 29

over. We can rewrite our sum in terms of j as

Skip(A) = m +m∑

j=1

number of times level j is skipped

= m +

⌊m/2⌋∑

j=1

number of times level j is skipped

+

m−⌊m/2⌋−1∑

j=0

number of times level m− j is skipped,

where in the final equality we have split the second half of the sum off and letj 7→ m− j.

Now we bound the number of times level j is skipped over. The only way to hopover j from a higher level is to land on a lower level. There are j−1 levels below thejth on which such a jump can land. Further, each of these levels can act as a launchpad for a jump back up which crosses the jth level again. This gives at most 2(j−1)skips over level j. Similarly, the number of times we can skip over the m− jth levelis twice the number of levels above it, or 2j.

For even m, these estimates are sharp (as we will see below). However, when levelm − j is the central level of an arc presentation with 2k + 1 levels (j = k = m−1

2),

the situation is slightly different. Here all of the j levels above the middle cannotbe initial and terminal levels of arcs which skip level m − j. For if so, then no arcsland on level m− j, and we could have eliminated level m− j from the original arcpresentation. Thus level m− j is skipped at most 2j − 1 = m− 2 times.

Inserting these bounds into the latter equation, we apply the sum formulae forarithmetic progressions. When m is odd, we get

Skip(A) ≤ m +

m−1

2∑

j=1

2(j − 1) +

m−3

2∑

j=0

2j + (m− 2) =m2 − 1

2

If m is even, the proof is similar.

For the second part of the Lemma, we construct arc presentations which show thatthese results are sharp. Consider the arc presentation with even arc-index m = 2kdescribed by the data

(m, m/2, θ1), (m/2, m−1, θ2), (m− 1, m/2− 1, θ3), (m/2− 1, m− 2, θ4),

. . . , (m/2 + 1, 1, θ2k−1), (1, m, θ2k).

If we add up the lengths of the jumps, we get

30 Arc presentation

Skip(A) = m2/2.

The same approach yields a realisation of A so that Skip(A) = m2−12

for odd m.

As an aside, let us mention the to following relation due to Peter Cromwell [C⁀95]:

Proposition 3.4.2 Let L1 and L2 be two links, then:

α(L1 ⊔ L2) = α(L1) + α(L2)

α(L1 # L2) = α(L1) + α(L2)− 2

where L1 and L2 are connected at any two of their components.

Remark. Actually, the ‘≤’ part follows straightforwardly from the most natural arcpresentation of L1 ⊔ L2 and L1 # L2 that we get from those of L1 and L2. The otherpart is more difficult to prove.

3.5 Lower bound for arc index

In these last two sections, we give bounds for the arc index in terms of the crossingnumber.

Let us denote sprx(P ) the spread , or breadth, of the Laurent polynomial P in thevariable x, i.e. its Laurent degree, or difference between higher and lower degrees.

Let L be a link. Recall that FL(a, d) is the Kauffman polynomial of L. HughMorton and Elisabetta Beltrami provided in [MB98] the following lower bound ofα (L):

Theorem 3.5.1 spra (FL (a, d)) + 2 ≤ α(L)

The arguments are based on stacked tangles .Now let us recall that the Kauffman-Murasugi theorem states that the Laurent

degree of the Kauffman polynomial of an alternating link L equals its crossing num-ber. This yields the following corollary:

Corollary 3.5.2 If L is an alternating link, then c(L) + 2 ≤ α(L)

The Kauffman-Murasugi theorem does not allow us to find a lower bound of thearc index for non alternating links.

3.6 Upper bound for arc index 31

3.6 Upper bound for arc index

Computations for small knots show that up to 10 crossings, α(L) ≤ c(L) + 2 (PeterCromwell [C⁀98]). It was conjectured that for any non-split link L, α(L) ≤ c(L) + 2.

In 2000, Yong Ju Bae and Chang Young Park [BP00] described an algorithmwhich transforms a diagram of a link with n crossings into an arc presentation withat most n + 2 half-planes. This proved eventually the expected upper bound on arcindex. We will focus now on this algorithm as it is described by Cromwell in [C⁀04].

Firstly, let us introduce a special diagram of an arc presentation, obtained bylooking at the link from above:

Definition 3.6.1 (spoked form) A spoked form of an arc presentation A is a pla-nar star with α(A) legs (or spokes). Each of them corresponds to a half-plane of Aand is labelled with the two levels linked by its arc. The spokes respect the order ofthe half-planes.

For example, Figure 3.5 gives the spoked form of the arc presentation of the trefoilgiven in Figure 3.1.

2, 51iiiiii

iiii

1, 3

2

2, 43UUUUUU

UUUU

3, 5

4

1, 4

5666666

6666

Figure 3.5: Spoked form for the trefoil

Given a spoked form of a link, one can uniquely recover its arc presentation, sothese two notions are equivalent. The following algorithm converts any diagram ofa knot into a spoked form, i.e. an arc presentation.

A graph is said to be n-connected if at least n vertices must be deleted to dis-connect it. A graph G is disconnected if and only if n = 0. Recall that a cut vertexappears when a diagram can be reduced. So a 2-connected graph has no cut vertex.Besides, we defined an innermost loop at a vertex v as an edge whose endpoints arein the same vertex v. Such a loop necessarily bounds a domain which either is emptyor contain other innermost loop(s).

We will need later on the two following lemmas:

32 Arc presentation

Lemma 3.6.1 Let G be a 2-connected graph embedded in a plane with no 2-gons.Let Ge be the graph obtained by collapsing the edge e. Given a vertex v there is anedge e incident to v such that Ge is still 2-connected.

We shall not prove this Lemma, whose proof can be found in [C⁀04]. We will useit to prove the following one:

Lemma 3.6.2 Let G be a 2-connected plane graph with more than two vertices. LetG0

e be the graph obtained by collapsing the edge e and removing any innermost loopsfrom the resulting graph. Let v be a vertex, of any valence, and assume that all theother vertices are 4-valent. Then there exists an edge e incident to v such that G0

e isstill 2-connected.

PROOF. If a face of a G is a 2−gon then delete one of its edges; repeat this processuntil all the 2−gons have been deleted. This preserves 2−connectedness so we canapply Lemma 3.6.1 to select edge e. Replace the deleted edges to regain G. Let usenumerate the four different cases accordingly with the multiplicity of the creatededge:

1. If e is not replaced by a multiple edge then G0e is still 2-connected.

2. If e is replaced by a double edge then we can nominate either one as e. When eis collapsed the other edge will become an innermost loop which will be deleted,this leaves a 2-connected graph.

3. If e is replaced by a triple edge then we must nominate the middle edge as e.Collapsing it creates two innermost loops.

4. Let us look at the case where e is replaced by a quadruple edge. This imposethe graph to have only two vertices, linked with the quadruple edge. But thisis excluded by the statement of the lemma.

This latter lemma will allow us to proceed by induction in the proof of thefollowing theorem.

Theorem 3.6.3 If L is a non split link then α(L) ≤ c(L) + 2

PROOF. Let D be a minimal diagram of L. The trivial case of the unknot is true,because α( ) = 2. Thus let us assume that c(L) > 0.

The algorithm will be performed in three dimensions. The idea is the following:


⊙v

v1

e1e3

e2

e

•

⊙v

e1e3

e2

Figure 3.6: Collapsing the edge e

To erect a line perpendicularly to the diagram (the future binding axis),and to pick up and attach to this line the over-crossings of all the crossingsof D. To label the attachment points in order to get a spoked form for L.

Instead of working on D itself, we will consider its graph G. Choose a vertex vof G, and let B be the straight line erected through v. We will look at the graphtransversally, such that we see the binding B as a single point, which we denote by⊙ on the figures.

The process of picking up a strand and attaching it on the axis is the following:Choose an edge e of G incident to v and let v1 be the vertex at the other end of e.Let us label the four edges which meet at v1 anti-clockwise e, e1, e2 and e3 (see leftof Figure 3.6. If e1e3 is an over-crossing (resp. under-crossing) of D, then let usattach it to the axis above (resp. bellow) all the existing points and label it with thesmallest (resp. highest) integer greater (resp. smaller) than those already used sofar. The right of Figure 3.6 shows that it make the edge e collapse.

At intermediate stages of the algorithm, we have a non-regular diagram (orgraph), called an hybrid presentation, in that sense that the formerly 4−valent ver-tex v is of become of higher valence. In three dimensions, we have to keep in mindthat v collects all the different endpoints of the arcs of the arc presentation. Stepby step, we will create innermost loops at the vertex v. Under certain condition, wewill be able to transform them into labelled spokes.

What we have to look at now is whether the previous process can always beapplied or not. Firstly, while G has other vertices than v, there exists at least anedge e so that we can apply a new step.

We should particularly avoid creating a loop based at v which is not an innermostloop. Such a loop would be labelled at both ends and so should correspond to a spoke,but it may be impossible to make it lie in a half-plane of the arc presentation withouthaving to change the link type.

34 Arc presentation

e

v⊙1

•

e

v⊙2

•

e

v⊙3

•

e

v⊙4

•

e

v⊙5

•

e

v⊙6

•

e

v⊙7

•

e

v⊙8

•

Figure 3.7: Configurations of connexions

Having chosen an edge e and defined v1, e1, e2 and e3 as in Figure 3.6, thereare eight different configurations for their connexions, whether they are connectedto themselves or to other edges. These are enumerated in Figure 3.7.

Let us check each case. Case 1 does not create any innermost loops: we get anew label but no new spoke. Cases 2 et 3 create one label and one innermost loop,i.e. one extra spoke. Case 4 creates one new label, and two loops, i.e. two extraspokes.

The three following cases are forbidden: cases 5 and 6 would not occur thanks toLemma 3.6.2: indeed we have a triple edge but in that case the proof of Lemma 3.6.2leads us to choose the middle edge. In other words it comes down to case 4 whichis dealt above. If we collapse e in case 7, then v becomes a cut vertex (Figure 3.8).Lemma 3.6.2 states we can avoid this possibility.

⊙

Figure 3.8: Creating a cut vertex

We have eventually case 8. Because this case has only two vertices, it can onlybe the penultimate stage of the algorithm. Lemma 3.6.2 does not apply (only twovertices). Collapsing e creates two innermost loops that we exchange for spokes. Atthe end one and only one loop and spokes remain in the hybrid diagram. We choose apoint at random on this loop and attach it to the axis above all the other endpoints,


and label it accordingly. We have now a genuine spoked form, i.e. equivalently anarc presentation.

Let us finish the proof by counting the number of spokes, with a flavor of combi-natorics in graphs.

A 4−valent graph in the plane with n vertices has n + 2 faces, including thenon-bounded one. This is proved by induction: a simple closed curve with no verteximplies two faces, which initializes the proof. In our case, we begin with a graph of Lwhich has c(L) crossings, i.e. c(L) + 2 faces. Collapsing one edge does not alter thenumber of faces. When we transform one innermost loop into one spoke, we decreasethe number of faces of 1, and increase the number of spokes of 1. Their total sum ispreserved.

Now at the penultimate step, we have only one loop so exactly two faces, whichmeans there are c(L) spokes. The last loop will be exchanged for two spokes, so weend up with c(L) + 2 spokes. We have an arc presentation with c(L) + 2 arcs, whichmeans that the arc index is at most c(L) + 2.

Let us see now what happens whether L is alternating or not.

3.6.1 Alternating case

Using Corollary 3.5.2, we get straightforwardly:

Corollary 3.6.4 (arc index of alternating links) If L is a non-split alternatinglink, then:

α(L) = c(L) + 2

3.6.2 Non-alternating case

Empirical evidence tends to show that for non-alternating links, α(L) < c(L) + 2.Elisabetta Beltrami tackled in [B⁀02] the non-alternating case.

Proposition 3.6.5 (arc index of non-alternating links) If L is a non-split non-alternating link, then:

α(L) < c(L) + 2

Sketch Of Proof. The algorithm used by Beltrami is quite similar to the oneused in the proof of Theorem 3.6.3, with in addition a trick described in the sequel.

Recall that a non-alternating link L has no alternating diagram. In particular,let D be a minimal diagram of L, then D has (at least) two successive over-crossings.

36 Arc presentation

In the present case, we choose to erect the straight line introduced in the proof ofTheorem 3.6.3 through a vertex v adjacent to one of these two crossings, as in theleft of Figure 3.9.

⊙v

e1

e2 en

en−1e3

F

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

.........

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

.......

−→

⊙v

e1

e2 enF

.......

.......

−→

⊙v

e2 en

.......

.......

Figure 3.9: Two steps in contracting edges

The meticulous part of the proof consists in showing it is always possible to getafter successive edge contractions a hybrid diagram as shown in the middle of Fig-ure 3.9. Now the trick consists in contracting simultaneously the edges e2 and en. Itmakes e1 vanish, as well as the face F (right of Figure 3.9). If resp. 0, 1 or 2 spokesare created, then resp. 1, 2 or 3 faces are deleted, so that the total sum of bothspokes and faces decreases of one. At the end, we get a spoked form with at mostc(L) + 1 spokes.

Remark. Once more, empirical evidence shows that α(L) ≤ c(L) as soon as L isnon-split and non-alternating. It is true indeed for all knots K with c(K) ≤ 10,c.f. [B⁀02]. This remains a conjecture in the general case (see Appendix B).

Part II

Applications of arc presentation

Chapter 4

Legendrian knots and links

We will show in this section how arc presentations may lead quite naturally to Leg-endrian knots and links. We will prove a relation between the Bennequin numbersof a link and its mirror image, and their arc index.

4.1 Definitions

4.1.1 Legendrian links

Legendrian manifolds can be defined in any odd dimension. We will restrict ourstudy to the standard case of R

3. Let us introduce the notion of contact structure:

Definition 4.1.1 (contact structure) A contact structure on R3 is a smooth 1-

form ω on R3 such that ω ∧ dω is non-degenerate.

A contact structure ω provides a plane τ(v) at each point v: the kernel of thelinear form ω(v) is indeed a hyperplane.

In the sequel, we shall deal only with the following form:

ω = ydx− dz

We have ω ∧ dω = (ydx− dz) ∧ (dy ∧ dx) = −dz ∧ dy ∧ dx = dx∧ dy ∧ dz whichis the volume form (thus non-degenerated).

At each point v = (x, y, z), the tangent plane τ(v) is generated by the vectors(0, 1, 0) and (1, 0, y). Now we define:

Definition 4.1.2 (Legendrian link) A Legendrian link is a piecewise smooth linkin R

3 such that at each of its regular point v, the link is tangent to τ(v).

4.1 Definitions 39

4.1.2 Front projection

Among the possible diagrams available for Legendrian links, the ones defined onz = 0 and y = 0 have particularly interesting properties. The only thing we willsay about the first one is that it is called the Lagrangian projection. We will focus onthe second one, called the front projection, and denoted F . Thus, in order to realizea front projection of L we will look at L in the way given in Figure 4.1.

⊗y x//

zOO

Figure 4.1: The plane of front projection

Let γ be the projection on y = 0 of one component of a link L. Let t be theparameter which describes γ. We have y dx

dt= dz

dtso that y = ∂z

∂x. Thus we have the

following property:

Claim 1 The coordinate y of a Legendrian link L equals the slope of the curve z interms of x in its front projection.

This means that dxdt

cannot vanish, or equivalently that a front projection hasno vertical tangent line. Thus dx

dtcan only change its sign at non-continuous points.

These points are cusps, as drawn in Figure 4.2.

Figure 4.2: Left and right cusps

The latter Claim is quite nice because it enables us not to specify the over/under-structure of the crossings on the front projection. It is given de facto by the com-parison between the slope of both branches of the crossing. The one with higherslope under-crosses the other one (keep in mind the direction of the y−axis in a frontprojection).

Moreover, if we smooth all the cusps, we recover a standard diagram from thefront projection. These transformations are given in Figure 4.3.

40 Legendrian knots and links

−→

−→

−→

Front projection Standard diagram

Figure 4.3: Recovering a standard diagram from a front projection

On the other hand, the moves allowed on front projections are not as flexible asusual Reidemeister moves on standard diagrams. Indeed, one of the configurations ofthe second Reidemeister move Ω2, shown in Figure 4.4, is impossible for a Legendrianknot. Abstracted from the vertical tangents, there is an incompatibility in terms ofthe y−coordinate of the two branches.

Figure 4.4: Forbidden configuration on a front diagram

Dan Rutherford defines the Legendrian isotopy in [R⁀05] as follows:

Definition 4.1.3 (Legendrian isotopy) Two Legendrian links are Legendrian iso-topic if their fronts can be transformed into one another through a finite sequence ofthe three Legendrian Reidemeister moves shown in Figure 4.5 and planar isotopieson front diagrams.

There is no equivalence between Legendrian and standard links, because we canproduce possibly many front projections from a standard diagram. Indeed given a

4.1 Definitions 41

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

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

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

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

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

I.II.III.

! ! !

Figure 4.5: Legendrian Reidemeister moves

front projection of a topological links type L, adding or removing zig-zags (Figure4.6) does not modify the topological type, but changes the Legendrian type.

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

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

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

Figure 4.6: Zig-zags in a front

Firstly we can indeed replace all the extrema in the x−coordinate by correspond-ing cusps. Secondly we have to look at the crossings of the diagram. They are of twotypes: either ‘good’ if the under-crossing has the greater slope, or ‘bad’ otherwise.If a branch of one crossing is vertical, then it can be rotated a little bit to obtainone of the two previous types. A crossing of ‘bad’ type can be removed into the firsttype as shown in Figure 4.7: one rotates it up to a point when the under-crossinghas the greater slope, which adds two cusps, then we get a ‘good’ crossing.


xy~z| −→ xy~z|

Figure 4.7: Constructing a ‘good’ crossing from a ‘bad’ crossing

Thus we have proved that each link can be represented by (at least) a front, sowe have the following theorem:

Theorem 4.1.1 (existence of Legendrian representative) For each link isotopyclass, there exists a Legendrian link L representing this class.

Let us illustrate this existence with the example of the right trefoil shown inFigure 4.8.

Figure 4.8: Standard and Legendrian right-trefoil

4.1.3 Bennequin number

We have shown that a Legendrian links isotopy class have a unique standard rep-resentative up to standard isotopy. Since Legendrian isotopy is stronger than thestandard ambient isotopy, all standard invariants are Legendrian invariants.

4.2 Grid form and front projection 43

Definition 4.1.4 (Bennequin number) The Bennequin number of a front F , alsocalled Bennequin-Tabachnikov number or Thurston-Bennequin number, is defined by:

β(F ) = w(F )− 1

2cusps(F )

where w(F ) and cusps(F ) are the writhe and the number of cusps of F . The max-imum of β(F ) among all the fronts of a topological link L is called the Bennequinnumber of L, and denoted by β(L).

Remark. We have a priori defined the writhe only on standard diagrams (Defini-tion 2.2.1) but it can be naturally extended on front projections.

Remark. β(L) is indeed unchanged by the three Legendrian Reidemeister movesof Figure 4.5. Given a front F , the Bennequin number can always be decreased asmuch as we want by adding zig-zags as in Figure 4.6. Showing that β(L) is welldefined actually requires that β(F ) is indeed bounded above. This result is due toEmmanuel Ferrand [F⁀02]:

Lemma 4.1.2 (upper bound for β(F )) Let F be a front of a topological link L.Then:

β(F ) < − degaFK

where F denotes the two variable Kauffman polynomial.

4.2 Grid form and front projection

We will see now the connection between the front projection of a Legendrian knot andthe grid form of its standard image. These two notions are in effect very analogous.This observation will lead us to the main result of this section, namely a relationbetween the Bennequin numbers of a knot and its mirror image, and their arc index.

The point is that on a grid form, all the over-crossings are vertical. On the otherhand, on a front projection, all the over-crossings have a negative slope of −45 up tosmall rotations. Thus, rotating a grid form of a link L of +45 (i.e. anti-clockwise)and then smoothing up and down edges and turning left and right edges into cuspsgives us a Legendrian realisation of L.

Rotating a grid form of −45 (i.e. clockwise) gives us almost a Legendrian reali-sation. The trouble is the crossings are all ‘bad’, c.f. Section 4.1.2. Thus, switchingall the crossing types will produce a Legendrian realisation of the mirror image L ofL.

We illustrate this point in Figure 4.9. In the middle is a grid form of the left-handed trefoil. On the left is a realisation of the left-handed trefoil, simply obtained


by rotating the grid of +45, and arranging the edges. On the right is a realisationof the right-handed trefoil, obtained by rotating the grid of −45 and switching allthe crossings.

+45

←−

−45

−→

Figure 4.9: A Legendrian realisation of a knot and its mirror image from a grid form

This leads us to state the following Proposition:

Proposition 4.2.1 (Bennequin number and arc index) Let L be a topologicallink. The following holds:

β(L) + β(L) ≥ −α(L)

PROOF. Let L be link. Let F (resp. F ) be a front of L (resp. L) implied by itsminimal grid form as described above. Recall that β(F ) = w(F ) − 1

2cusps(F ). A

knot and its mirror image have exactly opposite writhe, because we obtain one fromthe other one by switching the crossings, which multiplies by −1 all the signs. Thus:

β(F ) + β(F ) = w(F )− 1

2cusps(F ) + w(F )− 1

2cusps(F )

= −1

2(cusps(F ) + cusps(F ))

Notice that cusps(F ) + cusps(F ) coincides with the number of edges of the gridform. Besides, there are exactly two edges per vertical part in a grid form, and thenumber of vertical parts equals the arc index of L. That is to say: β(F ) + β(F ) =−1

2(2α(L)) = −α(L), this implies β(L) + β(L) ≥ −α(L).

As an aside, let mention that the following is straightforward:

Corollary 4.2.2 Amphicheiral links have even arc index.

4.2 Grid form and front projection 45

PROOF. Let L be an amphicheiral link, and F the front obtained from its grid form.L is equal to its mirror image hence:

α(L) = −2β(F )

Chapter 5

Braid index of satellites

We shall define in this section the notions of braids, especially closures of braids, andsatellites.

5.1 Closed braids

We shall define here braids, closures of braids, and give the Alexander’s theoremwhich states that each link can be obtained as the closure of a braid.

Consider two lines y = 0, z = 1 and y = 0, z = 0 and choose m points on eachof these lines having abscissas 1, . . . , m.

Definition 5.1.1 (braid) An m−strand braid is a set of m non-intersecting (strictly)monotonic smooth paths connecting the chosen points on the first line with the oneson the second line. The paths are called strands of the braid.

Two 3−strand braids are given in Figure 5.1.

Similarly to knots and links, braids are considered up to ambient isotopy. Adiagram of a braid is a regular projection of the braid on the plane x = 0 endowedwith an over/under-structure at double-points.

Definition 5.1.2 (closed braid) The closure of a braid B is a link Cl(B) obtainedfrom B by connecting the lower ends of B with its upper ends.

Keep in mind that a closure of a braid is not necessarily a knot. More precisely,it is a link whose number of components is computed as follow. An m−braid definesa permutation σ ∈ Sm such that the ith point of ordinate 1 is sent on the σ(i)th

point of ordinate 0. For example, the two braids of Figure 5.1 define respectively the

5.1 Closed braids 47

•

•

•

•

•

•1 2 3

1 2 3

•

•

•

•

•

•1 2 3

1 2 3

Figure 5.1: Two 3−strand braids

permutations

(1 2 33 1 2

)and

(1 2 33 2 1

). The multiplicity of the closure of a braid

of permutation σ is given by the number of orbits of σ.For example, the closure of the first braid in Figure 5.1 is a knot, whereas the

second one is a 2−link.

We can now state the main theorem:

Theorem 5.1.1 (Alexander) Each link can be represented as the closure of abraid.

PROOF. The idea of the proof is to transform any link into a kind of a string bracelet.As we saw in Section 1.4, it suffices to prove the statement for the case of polygonallinks. Consider a diagram D of a polygonal link L, and a point A in the plane ofD. If L is not oriented, let us give it any orientation. We say that an edge of D ispositive if it is seen from A as counterclockwise-oriented, and negative otherwise. Adiagram with only positive edges is called braided around A. It is clear that once wehave got a diagram braided around any point A (let us call it a bracelet), then wecan deduce a braid whose closure is L: it suffices to cut the bracelet radially along ahalf-line from A which does not run through any crossing, and then to open it, seeFigure 5.2.

Let us choose a point A. If D if braided around A, it is finished. Otherwise,we will apply the so-called Alexander’s trick to negative edges. Consider a negativeedge c1c2 of D. We apply a ∆-move (see Definition 1.4.3) on c1c2 with a ∆-trianglewhose third vertex is such that A lies in the interior of ∆ (in the plane of D). Giventhat c1c2 is a negative edge, the two other edges, with the corresponding orientation,will be positive.

48 Braid index of satellites

Let us precise the construction. If c1c2 has no crossing then we can perform thetrick by forming the two new edges above all the others. If c1c2 has more that onecrossing, then we can divide it so as to come down to the case of a negative edgewith one crossing. If the edge is the over-crossing (resp. under-crossing), then weapply the ∆-move above (resp. bellow) to all the other edges.

We shall use this operation until we get a diagram braided around A.

⊙• ••

A

ooooooooooooooooo

•

•

•

•

•

•

Figure 5.2: Cutting a diagram braided around a point, getting a braid

Any link is the closure of a braid. This arises the question of the minimal numberof strands required:

Definition 5.1.3 (braid index) Let L be a link. The minimal number of strandsof the braids whose closures are L is called its braid index, and denoted s(L).

As an aside, let us mention a result which is due to Joan Birman and WilliamMenasco:

Theorem 5.1.2 For any two knots K1 and K2:

s(K1 # K2) = s(K1) + s(K2)− 1

Remark. Actually, the ‘≤’ part is quite simple to see with a peculiar way to connectK1 and K2.

5.2 Satellites

Let W be a full torus. In simple words, we call a meridional disc of W a section ofW which is a disc.

A general definition of the satellite construction can be given as follow:

5.2 Satellites 49

Definition 5.2.1 (satellite construction) Let W be a full torus. Let P be a linkin W which meets every meridional disc in W at least once. Let C be a knot and let Vdenote a solid tubular neighbourhood of C. Let h : W −→ V be a homeomorphism.

Then the link h(P ) is called a satellite with pattern P and companion C, and isdenote Sh.

Let us specify this definition to our study. We need the notion of annulus:

Definition 5.2.2 (annulus) An annulus A is a disc minus a smaller disc in itscenter (see Figure 5.3).

PWQVRUSTpwqvrust

Figure 5.3: Annulus

In the sequel we will only look at a peculiar pattern called reverse parallel . It isa 2−link which lies in the annulus without any crossing, as shown in Figure 5.4.

OO OO

(

6G

QQY_fmmw

(

6G

QQ Y _ f mmw

6QQ_mm

6QQ _ mm

Figure 5.4: Reverse parallel pattern in an annulus

A satellite of a companion C with this pattern is called a reverse parallel satellite,or simply a reverse parallel of C. Figure 5.5 shows a trefoil companion in dashedline, and a reverse parallel of the trefoil.


1

` W QQ LEE

<1

17

>NN_pp

mm g ^

2

(

##

1KK

W [ _ _

GG>

1

wwqkkf_ _ c g

ss

(-

2

K1

K2

oo

FF

//

XX

Figure 5.5: A trefoil pattern and a reverse parallel satellite

For a given companion, the reverse parallel construction is only governed by aninteger. Indeed the construction consists in cutting the annulus radially, i.e. gettinga ribbon, knotting it around the companion C, and then gluing both ends of theoriginal annulus. However, before gluing the ends together, one can twist the ribbonas much as one want, as shown in Figure 5.6.

OO

MM

KKII

Figure 5.6: Adding 1, 2 or 3 half-twists before gluing the ribbon

In order to determine how to define the ‘number of half-twists’ before gluing, letus define:

Definition 5.2.3 (linking number) Let K1 and K2 be two components of a link.

5.3 Braid index and arc index 51

The linking number of K1 with K2, denoted by lk(K1, K2), is the sum of crossingsigns of all the crossings where K1 over-crosses K2 in a given diagram D.

Remark. Considering Reidemeister moves, we show that this number is independentof the diagram D. Moreover, lk is a symmetric function.

The sign of a crossing was given in Figure 1.5.

Now we are in a position to measure the number of times one component ofthe reverse parallel winds around the other one once the annulus is knotted in thecompanion shape:

Definition 5.2.4 (framing) Let K1 and K2 be the two components of a reverseparallel satellite S. Then we call the linking number of K1 and K2 lk(K1, K2) theframing of S, denoted by f . We specify the notation Sf .

In Figure 5.7, the three crossings where, say the exterior K1 component over-crosses the interior one K2 are positive. Thus the framing of this satellite is 3. Wedenote it S3.

oo

FF

+1

//

+1

XX

+1

Figure 5.7: The reverse parallel trefoil of framing +3

5.3 Braid index and arc index

The two previous notions of braids and satellites are related, and once more the gridform is at the root of the link between them.

Given an arc presentation A of a link L, recall from Section 3.3.2 the constructionwhere the link avoids the binding axis by running around a cylinder. We rememberthat the choice of the arc which replaces an horizontal wedge was given by the onewhich does not meet a given half-plane. Here the choice will be slightly different: we


choose the arc which runs anti-clockwise around the cylinder. In that way, we arenot far to get a closed braid of the link L. Let us see what happens at the level ofthe grid form of A.

5.3.1 Closed braid of the companion

Recall the grid form obtained from an arc presentation: we get a diagram of a link.Now with the construction described above, cutting between any two half-planes ofthe arc presentation will produce a braid. Actually, we do not get formally a braid,because the strands are not strictly monotonic in the left-right direction, becauseof the vertical parts. However it is a simple technicality to make them strictlymonotonic, by slightly tilting each of the vertical parts: rotating rising (resp. goingdown) parts clockwise (resp. anti-clockwise).

A braid for the trefoil is given in Figure 5.8.

2

1

3

2

4

3

5

4

15

2

1

3

2

4

3

5

4

15

Figure 5.8: Constructing a braid from a grid form of the companion

5.3.2 Closed braid of the satellite

Let us apply the same construction. We notice first that since the two strands of thereverse parallel pattern have reverse orientations, at each level one string will runaround one side of the cylinder, and the other string around the other side. A secondremark is that the value of the framing is determined by the choice of the crossingsat each vertical part. Let us see what happens in the case of the reverse parallel ofthe trefoil in Figure 5.9. The choice here is that all the crossings in the vertical partsare positive.


2

1

3

2

4

3

5

4

15

Figure 5.9: Constructing a braid from a grid form of a reverse parallel satellite

Remark. This diagram is not formally a braid diagram because it is not regular :there are indeed singular points of multiplicity 3. Once again, a simple technicalityallows us to overcome this trouble: horizontal edges are known to be always under-crossings in grid forms. In our case, the horizontal edge under-crosses both othercrossings at triple points.

So we get a braid of the reverse parallel satellite. Recall that the length of aside of the minimal grid form of a link L coincides with the arc index α(L). Thisconstruction leads us to foresee the following upper bound for the arc index. Let Sbe a reverse parallel of companion C, then we get a grid form of S of length α(C).So s(S) ≤ α(C).

We should keep in mind that the framing could interfere in this result. It turnsout indeed that the previous inequality is only true in a certain range of framing.

Let us compute the framing of the previous satellite. Firstly, label the twodifferent strings by ∗ and ⋆ (see Figure 5.10). The permutation of this braid is(

1 2 3 4 52 1 4 5 3

)thus we label the first two levels with ∗ and the last three with ⋆,

and call the corresponding components K∗ and K⋆. We have f = lk(K∗, K⋆). Wehave labeled positive and negative over-crossings of the strand labelled with 1 with⊞ and ⊟, and the ones of the strand labelled with 2 with ⊕ and ⊖.

We have a framing +6 for the left of Figure 5.10, and a framing +1 for theright, where we have chosen on the contrary only negative crossings at vertical parts.Depending on the choices of the five crossings at vertical parts, we get a certain rangeof framing for which the braid index is bounded above by the arc index.

In general, a grid form of a reverse parallel satellite of companion C has α(C)crossings at vertical parts. So we have 2α(C) choices of crossings, as mentioned on


⊞

⊞

⊞

⊕⊕⊕

⋆

⋆

∗

∗

∗ 2

1

3

2

4

3

5

4

15

⊞

⊞

⊟

⊟

⊖ ⊕

⊕⋆

⋆

∗

∗

∗2 3 4 5 1

Figure 5.10: Two different framings

Figure 5.11. That means that only a certain range of framing, say [f1; f2] can bereached for a number of braids equal to α(C).

//

//

//

//

Figure 5.11: Two different crossings at each vertical part of the braid

We can now state a proposition:

Proposition 5.3.1 (bound on braid index) Let C be a companion. There existsa range of framing [f1; f2] such that:

s(Sf) ≤

α(C) + f1 − f if f < f1,

α(C) if f ∈ [f1; f2],

α(C) + f − f2 if f > f2,

i.e. the number of extra strings in the initial presentation of C must be at most f−f1

if f < f1 or f − f2 if f > f2.

PROOF. We have discussed the case where f ∈ [f1; f2] above. Now adding a newstring can introduce a crossing of a needed sign. So if we want a framing f out of[f1; f2], adding f − f1 or f − f2 strands whether f < f1 or f > f2 allows to produce


a braid with the required framing.

For further work and conjectures on more general satellites, see [N⁀99].

Chapter 6

Ropelength of knots

6.1 Introduction

In this section we shall show some of the advantages provided by arc presentation,namely:

• basically it gives an explicit realisation of any knot which lies on a cylinder,

• moreover, this structure is ‘compact’ in the terms of ropelength.

J. Cantarella, X. W. Faber and C. A. Mullikin focused on these propertiesin [CFM03] to present great improvements for upper bounds for ropelength in termsof the crossing number.

As in earlier papers, the bounds given in [CFM03] grow with the square of thecrossing number; however, the constant involved is a substantial improvement onprevious results. The proof depends essentially on writing links in terms of their arcpresentations, and has as a key ingredient Bae and Park’s theorem that an n-crossinglink has an arc presentation with less than or equal to n + 2 arcs.

The ropelength of a space curve is defined to be the quotient of its length by itsthickness, where thickness is the radius of the largest embedded tubular neighborhoodaround the curve. For a knot or link type L, we define the ropelength Rop (L) tobe the minimum ropelength of all curves with the given link type. This minimumropelength is a link invariant which measures the topological complexity of the link.

We will tackle in this section the following problem: given a link type L ofcrossing number c (L), can we guarantee the existence of a representative curve withropelength less than some function of c (L)? That is, can we find upper bounds onropelength in terms of crossing number?

The main result in [CFM03] provides an upper bound O(c (L)2), with a leading

coefficient equal to 1.7, whereas the better bound in earlier papers had coefficient 24.

6.2 Definition of ropelength 57

6.2 Definition of ropelength

Let us begin with two definitions given in [CFM03]. Firstly, the thickness of acurve should be thought as the radius of the largest embedded tubular neighborhoodaround the curve. For C2 curves, this radius is locally controlled by curvature andglobally controlled by distances of self-approach between various regions of the curve:

Definition 6.2.1 (thickness) The thickness of a C2 curve c is given by

τ (c) = min

min

s

1

κ (s),dcsd (c)

2

,

where κ (s) is the curvature of c at s, and dcsd (c) is the shortest doubly-criticalself-distance of c; that is, the length of the shortest chord of c which is perpendicularto the tangent vector c′ at both endpoints.

Figure 6.1 shows two curves of unit thickness in the plane with their largest em-bedded tubular neighborhoods. In the left curve, thickness is controlled by curvaturewhile in the right curve, thickness is controlled by the length of the doubly-criticalchord shown.

Figure 6.1: Thickness of a curve

We can extend this defintion to C1,1 curves by adjusting our idea of the radius ofcurvature as follows:

Definition 6.2.2 (infimal radius of curvature) Let s be a point on a C1,1 curve.Consider a decreasing sequence of open neighborhoods Un of s. The infimal radiusof curvature at s is given by

infUn

inft∈Un

1

κ (t)

,

where the inner infimum is restricted to t in Un such that κ (t) exists.

Lastly:

58 Ropelength of knots

Definition 6.2.3 (ropelength) The ropelength of a curve c is given by the quo-tient:

Rop (c) =λ (c)

τ (c)

where λ (c) is the length of the curve c.

6.3 Upper bound for ropelength

We would like to take an arc presentation A for L as a template for constructing anembedding of L with unit thickness. We will then bound the length of this embeddingin terms of the arc-index and the total skip of A.

Recall the polygonal arc presentation described in Section 3.3.1. Let us change itslightly. Let us have a look on the radial extension of the polygonal arc presentation.We construct a right regular polygonal prism P×[0, 2α], where P is a regular polygonwith α sides of length 2.

1. Firstly, we impose that the height of each level equals 2 instead of 1. In thisway, horizontal parts, called bins in [CFM03], can have doubly-critical self-distance at most 2. We construct a bin as an arc of a circle whose tangentsat the end points are perpendicular to the corresponding faces of the prism, asshown in Figure 6.2.

Figure 6.2: Possible bins (arc number equal to 9)

2. We impose that the vertical parts of the presentation, called fins in [CFM03],lay in the axis of 2α×2×2 rectangular boxes adjacent to the faces of the prism.This is shown in Figure 6.3: we look at the construction from above. This isthe case of arc number equal to 6. The basis of the prism is shown in plainlines, and the square bases of the boxes in dot lines. Fins are represented with⊙. A tubular extension of radius 1 of each fin is represented by a circle. In this

6.3 Upper bound for ropelength 59

way, vertical parts can have doubly-critical self-distance at most 2. Moreover,if a vertical part links levels xi and yi, we make it begin one unit higher thanthe bottom level, and one unit lower than the top level, so that next statementis possible.

1111111111

1111111111

hoinjmklhoinjmkl

hoinjmkl

hoinjmklhoinjmkl

hoinjmkl

⊙

⊙

⊙

⊙

⊙

⊙

Figure 6.3: Fins in boxes (an arc number equal to 6)

3. Lastly, we have to link horizontal and vertical segments. We use quarter-circlesof radius 1 lying in corresponding half planes.

Definition 6.3.1 (Tubular handle presentation) The specific construction de-scribed above is called a tubular handle presentation and denoted by T .

A tubular handle presentation of the right-handed trefoil is given in Figure 6.4.

This construction can be accomplished with a unit thickness curve: by definitionits doubly-critical self-distance at most 2. Moreover the sides of the polygon havelength 2, so each bin is an arc of a circle of radius at least 1; lastly, quarter-circleshave precisely radius 1.

Let us now compute the length of a tubular handle presentation T correspondingto an arc presentation A. Let us denote by λhoriz (T ), λvert (T ) and λquarter−circle (T )the lengths of corresponding parts in T .

Lemma 6.3.1λvert (T ) = 2 Skip (A)− 2α.

60 Ropelength of knots

Figure 6.4: Tubular handle presentation of the right trefoil

PROOF. Suppose that Fi travels from floor xi to floor yi of the prism. The totalvertical distance covered by the fin is 2|xi − yi| (recall that each floor has height2). However, the quarter-circles on each end of the fin cover a vertical distance of 2units. Thus, the straight segment has length 2|xi − yi| − 2, and the total length ofthe fin is π − 2 + 2|xi − yi|. Summing over i = 1, . . . , α and using Definition 3.2.3proves the lemma.

Lemma 6.3.2λquarter−circle (T ) = πα.

PROOF. There are 2 quarter-circles on each end of each fin. The quarter-circles haveradius 1. So their length is 2α 1

42π = απ.

Lemma 6.3.3

λhoriz (T ) ≤ 2α

tan (π/α).

PROOF. Each of these circular arcs is contained in a sector of the circle inscribedwithin the polygonal cross-section of the prism as shown in Figure 6.2. Since eacharc is convex, its length is bounded above by the diameter of the inscribed circle.This diameter is exactly 2 cot (π/α). Summing over i = 1, . . . , α proves the lemma.

6.3 Upper bound for ropelength 61

For a given arc presentation A we can construct a realisation of the knot in spacewith ropelength bounded in terms of Skip (A) and α (A):

Lemma 6.3.4 An arc presentation A can be realised with ropelength smaller than

2α (A)

tan (π/α (A))+ (π − 2) α (A) + 2 Skip (A) .

PROOF. It remains simply to combine the first three lemmas and to notice that thetubular handle presentation has radius 1.

Moreover, we have seen in Section 3.6 that any non-split link L admits an arcpresentation with α (L) ≤ c (L) + 2. This result, when coupled with the previouslemma, gives the following main theorem:

Theorem 6.3.5 If L is a non-split link, then

Rop (L) ≤ 1.7 c (L)2 + 7.7 c (L) + 6.8

PROOF. From Lemma 3.4.1 we deduce this larger bound for Skip (A):

Skip (A) ≤ α (A)2

2

Taylor’s theorem gives the approximation 1tan(x)

≤ 1/x − x/3 for x > 0. ViaPropositions 1 and 2 we gather that

Rop (L) ≤ 2α

tan (π/α)+ (π − 2)α + α2

≤ (2/π + 1)α2 + (π − 2)α− 2π/3.

By Theorem 3.6.3, for any non-split link L we have α (L) ≤ c (L) + 2. Inserting thisinto the above bound for ropelength yields

Rop (L) ≤ (2/π + 1) c (L)2 + (8/π + 2 + π) c (L) + (8/π + 4π/3)

Each of these constants are smaller than the approximations given in the statementof the theorem.

Appendix A

Knot Table

We have listed the prime knots up to 6 crossings, and the torus knot with 7 crossings.Amphicheiral knots are denoted by (A). As for chiral ones, only the left-handed knot(L) is given. All are reversible (c.f. Table 1.1).

01 (A) 31 (L) 41 (A)

51 (L) 52 (L) 61 (L)

62 (L) 63 (A) 71 (L)

Appendix B

Conjectures

Below is given a list of unsolved problems related to notions tackled in this report.Most of them come from Robion Kirby homepage, see [K⁀95].

Crossing number

This first conjecture illustrates the difficulty to prove results involving crossing num-ber:

Conjecture 1 The crossing number c(n) is additive with respect to the connectedsum, that is for any two knots K1 and K2:

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

Remark. Given two diagrams of K1 and K2, it is straightforward to find a diagramof K1 # K2 with c(K1)+c(K2) crossings. That shows a first inequality c(K1 # K2) ≤c(K1) + c(K2). Besides we have seen (Kauffman-Murasugi Theorem) that reduceddiagrams of alternating knots, or connected-sum of alternating knots, are crossing-minimal. Thus the conjecture is true for alternating knots.

Counting knots

Let C(n) (resp. A(n)) be the number of prime (resp. amphicheiral) knots K ofcrossing number c(K) = n. The first values of C(n) and A(n) are given in Table B.1.

Question 2 What is the asymptotic behaviour of C(n)?

64 Conjectures

n 3 4 5 6 7 8 9 10 11 12 13

C(n) 1 1 2 3 7 21 49 165 552 2176 9988

A(n) 0 1 0 1 0 5 0 13 0 58 0

Table B.1: First values of C(n) and A(n)

Remark. Using the Jones polynomial, D. W. Sumners has proved that C(n) growsat least exponentially.

Conjecture 3 C(n) is a supermultiplicative function, that is:

∀n, m ≥ 3 : C(n) C(m) ≤ C(n + m)

Amphicheirality

Question 4 Are there amphicheiral knots K of any crossing number c(K) ≥ 15?

Question 5 Are there (prime, alternating) amphicheiral knot of every even crossingnumber?

Arc index

Remember that Proposition 3.6.5 states that if a link L is non-split and non-alternating,then α(L) ≤ c(L) + 1. A better inequality is conjectured:

Conjecture 6 If a link L is non-split and non-alternating, then α(L) ≤ c(L).

Satellites

The following question seems obvious, but has still not been proved:

Question 7 Is the crossing number of a satellite bigger than that of its companion?

Bibliography

[BP00] Y. J. Bae and C. Y. Park. An upper bound of arc index of links. Math.Proc. Cambridge Philos. Soc., 129(3):491–500, 2000.

[B⁀02] E. Beltrami. Arc index of non-alternating links. Journal of knot theoryand its ramifications, 11(3):431–444, 2002.

[CF63] R. H. Crowell and R. H. Fox. Introduction to knot theory. New York:Ginn & Co, 1963.

[CFM03] J. Cantarella, X. W. Faber, and C. A. Mullikin. Upper bounds forropelength as a function of crossing number. Topology and its applications,(135):253–264, 2003.

[C⁀95] P. R. Cromwell. Embedding knots and links in an open book i: basicproperties. Topology and Its Applications, (64):37–58, 1995.

[C⁀98] P. R. Cromwell. Arc presentations of knots and links. Knot theory, pages57–64, 1998.

[C⁀04] P. R. Cromwell. Knots and links. Cambridge University Press, 2004.

[F⁀02] E. Ferrand. On legendrian knots and polynomial invariants. Proc. Amer.Math. Soc., 4(130):1169–1176, 2002. arXiv:math.GT/0002250.

[K⁀88] L. H. Kauffman. New invariants in the theory of knots. SocieteMathematique de France, (Asterisque 163-164):137–219, 1988.

[K⁀95] R. Kirby. Problems in low-dimensional topology.http://www.math.berkeley.edu/∼kirby, 1995.

[L⁀80] H. C. Lyon. Torus knots in the complements of links and surfaces. Michi-gan Math. J., (27):39–46, 1980.

[MB98] H. R. Morton and E. Beltrami. Arc index and the Kauffman polyno-mial. Math. Proc. Camb. Phil. Soc., (123):41–48, 1998.

66 BIBLIOGRAPHY

[M⁀04] V. O. Manturov. Knot theory. CRC Press, 2004.

[N⁀84] L. P. Neuwirth. *-projections of knots. Algebraic and Differential Topol-ogy: Global Differential Geometry, (70):198–205, 1984.

[N⁀99] I. Nutt. Embedding knots and links in an open book iii: on the braidindex of satellite links. Math. Proc. Camb. Phil. Soc., (126):77–98, 1999.

[R⁀92] L. Rudolph. Quasipositive annuli (constructions of quasipositive knotsand links iv). Journal of Knot Theory and its Ramifications, (4):451–466,1992.

[R⁀05] D. Rutherford. The bennequin number, kauffman polynomial, and rul-ing invariants of a legendrian link: the fuchs conjecture and beyond, 2005.arXiv:math.GT/0511097.

List of Figures

1.1 Trivial knot, trefoil and figure-eight knot . . . . . . . . . . . . . . . . 41.2 Hopf link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Local structure of a crossing . . . . . . . . . . . . . . . . . . . . . . . 61.4 A splitting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Positive and negative crossings . . . . . . . . . . . . . . . . . . . . . . 71.6 A ∆-move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 The three Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . 101.8 Types of ∆-triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 Connected sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Computing the bracket polynomial with a rooted binary tree . . . . . 14

3.1 Arc presentation for the trefoil . . . . . . . . . . . . . . . . . . . . . . 213.2 Piece of a square pattern . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 A grid form deduced from a diagram . . . . . . . . . . . . . . . . . . 233.4 Three grid forms deduced from an arc presentation . . . . . . . . . . 243.5 Spoked form for the trefoil . . . . . . . . . . . . . . . . . . . . . . . . 283.6 Collapsing the edge e . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7 Configurations of connexions . . . . . . . . . . . . . . . . . . . . . . . 313.8 Creating a cut vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.9 Two steps in contracting edges . . . . . . . . . . . . . . . . . . . . . . 32

4.1 The plane of front projection . . . . . . . . . . . . . . . . . . . . . . . 364.2 Left and right cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Recovering a standard diagram from a front projection . . . . . . . . 374.4 Forbidden configuration on a front diagram . . . . . . . . . . . . . . . 374.5 Legendrian Reidemeister moves . . . . . . . . . . . . . . . . . . . . . 384.6 Zig-zags in a front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.7 Constructing a ‘good’ crossing from a ‘bad’ crossing . . . . . . . . . . 394.8 Standard and Legendrian right-trefoil . . . . . . . . . . . . . . . . . . 39

68 LIST OF FIGURES

4.9 A Legendrian realisation of a knot and its mirror image from a gridform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Two 3−strand braids . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Cutting a diagram braided around a point, getting a braid . . . . . . 445.3 Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Reverse parallel pattern in an annulus . . . . . . . . . . . . . . . . . 455.5 A trefoil pattern and a reverse parallel satellite . . . . . . . . . . . . . 465.6 Adding 1, 2 or 3 half-twists before gluing the ribbon . . . . . . . . . . 465.7 The reverse parallel trefoil of framing +3 . . . . . . . . . . . . . . . . 475.8 Constructing a braid from a grid form of the companion . . . . . . . 485.9 Constructing a braid from a grid form of a reverse parallel satellite . 485.10 Two different framings . . . . . . . . . . . . . . . . . . . . . . . . . . 495.11 Two different crossings at each vertical part of the braid . . . . . . . 50

6.1 Thickness of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 Possible bins (arc number equal to 9) . . . . . . . . . . . . . . . . . . 536.3 Fins in boxes (an arc number equal to 6) . . . . . . . . . . . . . . . . 546.4 Tubular handle presentation of the right trefoil . . . . . . . . . . . . . 54

List of Tables

1.1 Chirality and reversibility . . . . . . . . . . . . . . . . . . . . . . . . 7

B.1 First values of C(n) and A(n) . . . . . . . . . . . . . . . . . . . . . . 58

Index

VL(.), 17#, 11∆-move, 9∆-triangles, 10∆-move, 8Rop (.), 52α(.), 20Skip(A), 21β(.), 40cusps(.), 40dcsd (.), 52〈.〉, 13lk(., .), 46spr(.), 18, 19spr.(.), 27⊔, 12τ (.), 52c (.), 6f , 47p (.), 8s(.), 44w(.), 16

achiral, see amphicheiralalternating− diagram, 19− link, 19

amphicheiral, 7annuli, see annulusannulus, 44arc index, 20, 35

arc number, 20

Bennequin number, 35, 40bin, 53binding, 20, 22bracket polynomial, 13, 14braid, 42braid index, 44branch− of a link, 5

breadth, 27bridge, 22

chiral, 7closed braid, 42closure, see closed braidcombinatorial isotopy, 9companion, 44complicated knot, 12component− of a link, 4

composition, see connected sumconcatenation, see connected sumconnected shadow, 6connectedness, 6contact structure, 35crossing− of a digram, 6−, positive, 7

crossing index, 6crossing number, 6

INDEX 71

diagram, 6−, alternating, 7, 19−, connected, 6−, minimal, 6−, polygonal, 8−, positive, 7−, reduced, 6

doubly-critical self-distance, 52

factor, 11fin, 53first Tait conjecture, 19flip, 17framing, 44, 47, 49front projection, 36

graph−, n-connected, 28

grid form, 23

Hopf link, 4

index−, arc, 20−, braid, 44−, crossing, 6

innermost loop, 28isotopy−, ambient, 5

Jones polynomial, 17

knot, 3, see link−, polygonal, 8−, tame, 8−, trivial, 3−, wild, 8

Lagrangian projection, 36Laurent degree, 18Legendrian link, 35link, 4−, alternating, 7, 19, 27

−, oriented, 7−, polygonal, 8−, positive, 7−, split, 12, 29−, tame, 8−, trivial, 4−, wild, 8

link type, 5linking number, 46

meridional disc, 44minimal diagram, 6, 19mirror image, 6, 35, 41multiplicity, 4

number−, Bennequin, 40−, arc, 20−, framing, 47−, linking, 46

orbit, 42over-crossing, 5

pattern, 44−, reverse parallel, 45

point−, double, 5−, singular, 5−, transverse, 5

polygon, 8polygon index, 8polynomial−, Jones, 17−, bracket, 13

presentation−, arc, 20−, bridge, 22−, square-bridge, 23−, tubular handle, 53

prime knot, 12projection

72 INDEX

−, Lagrangian, 36−, front, 36

regular point, 5regular projection, 5Reidemeister moves, 10, 15, 16reverse, 7reverse parallel, 45reversible, 7ropelength, 52

satellite, 44sattelite−, reverse parallel, 45

shadow, 6singular point, 5skip−, total, 21

splitting point, 6spoked form, 28spread, 27stacked tangles, 27strand, 42sublink, 4sum−, connected, 11−, disconnected , 12

thickness, 52trefoil, 3

under-crossing, 5unknot, 3

vertex− of a graph, 5−, cut, 6

writhe, 16, 40

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