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Knots, tangles and braid actions Watson, Liam Thomas 2004

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K N O T S , T A N G L E S A N D B R A I D A C T I O N S by LIAM THOMAS WATSON B.Sc. The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics - We accept this thesis as conforming -to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 2004 © Liam Thomas Watson, 2004 THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES Library Authorization B In p resen t ing th is thes is in part ia l fu l f i l lment o f the requ i remen ts for a n a d v a n c e d d e g r e e at the Univers i ty o f Bri t ish C o l u m b i a , I a g r e e that the L ibrary shal l m a k e it f ree ly ava i lab le for re fe rence a n d study. I fur ther ag ree that pe rmiss ion for ex tens ive copy ing o f th is thes is for scho lar ly p u r p o s e s m a y be g r a n t e d by t h e h e a d of m y d e p a r t m e n t or by his or her representa t ives . It is u n d e r s t o o d tha t copy ing or publ icat ion o f th is thes is for f inanc ia l ga in shal l not be a l l owed w i thou t m y wr i t ten pe rm iss ion . L I A H VJkr**o*^ N a m e of A u t h o r (please print) (p . JO • Z o o * / Date (dd /mm/yyyy ) Ti t le o f Thes i s : \<^C5TS T A ^ U R . ~&£Aj\> A A T l O M ^ Degree : Year : ZOO*/ D e p a r t m e n t o f M A T K F M A Y I ^ T h e Univers i ty o f Br i t ish C o l u m b i a V a n c o u v e r , B C C a n a d a g r a d . u b c . c a / f o r m s / ? f o r m l D = T H S page 1 of 1 last updated: 20-M-04 A b s t r a c t Recent work of Eliahou, Kauffmann and Thistlethwaite suggests the use of braid actions to alter a link diagram without changing the Jones polynomial. This technique produces non-trivial links (of two or more components) having the same Jones polynomial as the unlink. In this paper, examples of distinct knots that can not be distinguished by the Jones polynomial are constructed by way of braid actions. Moreover, it is shown in general that pairs of knots obtained in this way are not Conway mutants, hence this technique provides new perspective on the Jones polynomial, with a view to an important (and unanswered) question: Does the Jones polynomial detect the unknot? ii Table of Contents Abstract » Table of Contents iii List of Figures v Acknowledgement vii Chapter 1. Introduction 1 Chapter 2. Knots, Links and Braids 3 2.1 Knots and Links 3 2.2 Braids 5 Chapter 3. Polynomials 8 3.1 The Jones Polynomial 8 3.2 The Alexander Polynomial 9 3.3 The HOMFLY Polynomial 13 Chapter 4. Tangles and Linear Maps 21 4.1 Conway Tangles 21 4.2 Conway Mutation • 25 4.3 The Skein Module 27 4.4 Linear Maps 31 4.5 Braid Actions .... 32 Chapter 5. Kanenobu Knots 37 5.1 Construction 37 5.2 Basic Examples 41 5.3 Main Theorem 42 5.4 Examples 46 5.5 Generalisation 50 5.6 More Examples 53 5.7 Observations • • 59 iii Table of Contents iv Chapter 6. Thistlethwaite Links 62 6.1 Construction 62 6.2 Some 2-component examples 69 6.3 A 3-component example 70 6.4 Closing Remarks 71 Bibliography 72 iv List of Figures 2.1 Diagrams o f the Trefoil Knot 3 2.2 The Hopf Link 4 2.3 The braid generator CTJ and its inverse 7 2.4 The link /? formed from the closure of /3 7 4.1 Some diagrams of Conway tangles 21 4.2 The generator ej 30 4.3 The tangle e S2 33 5.1 The Kanenobu knot K(T, U) 37 5.2 Distinct knots that are not Conway mutants 43 5.3 Example 1 47 5.4 Example 2 49 5.5 Another diagram of the Kanenobu knot K(T, U) 50 5.6 The link \R\ 50 5.7 The closure of a Kanenobu braid 51 5.8 The braid c r f c T a 1 ^ 53 5.9 The Kanenobu braid (<T^CT^"1cri)(o-^"3(740-^1) 53 5.10 The knot |(CT?CT2"1CTI)(CT5"3CT4CT5 x ) | .54 5.11 The knot K(0,0) ~ 5 2 # 52* 55 5.12 The knot K{a2,a^1) 55 5.13 The braid aja^ai 56 5.14 The Kanenobu braid ( c r^a^c r j^^o - lo -g 1 ) 56 5.15 The knot \{(y\a^ai){a^2ala^l)\ 57 5.16 The knot i f (0,0) ~ 6i # 6 f 58 5.17 The knot K{a2,o^l) 58 5.18 The 2-bridge link obtained from (3 € 5 3 59 5.19 The square knot 3i # 3f 60 5.20 The knot \{aia^lai){a^la^l)\ 60 6.1 The Thislethwaite link H{T,U) 63 6.2 The link H{0,0) 63 6.3 The braid u € B3 , . . . . . . . . . . . . : .« 66 6.4 The link Uw~l) .:". 1 . 6 7 6.5 The result of w 2 acting on H(T,U) 68 6.6 The result of a;4 acting on H{T, U) 68 v List of Figures vi 6.7 A non-trivial, 2-component link 69 6.8 A non-trivial, 2-component link 70 6.9 A non-trivial, 3-component link 71 Acknowledgement It's a strange process, writing down the details to something you've been work-ing on for some time. And while it is a particularly solitary experience, the final product is not accomplished on you're own. To this end, there are many people that should be recognised as part of this thesis. First, I would like to thank my supervisor Dale Rolfsen for ongoing patience, guidance and instruction. It has been a privilege and a pleasure to learn from Dale throughout my undergraduate and masters degrees, and I am grateful for his generous support, both intellectual and financial. I have learned so much. Of course, I am indebted to all of my teachers as this work comprises much of what I have learned so far. However, I would like to single out Bill Casselman, as the figures required for this thesis could not have been produced without his guidance. To my parents, Peter and Katherine Watson, family and friends, I am so fortunate to have a support network that allows me to pursue mathematics. In particular, I would like to thank Erin Despard for continued encouragement, support and perspective, everyday. vii C h a p t e r 1 I n t r o d u c t i o n The study of knots begins with a straight-forward question: Can we distinguish between two closed loops, embedded in three dimensions? This leads naturally to a more general question of links, that is, the ability to distinguish between two systems of embedded closed loops. Early work by Alexander [1, 2], Artin [3, .4], Markov [24] and Reidemeister [29] made inroads into the subject, developing the first knot and link invariants, as well as the combinatorial and algebraic languages with which to approach the subject. The subtle relationship between the combinatorial and algebraic descriptions continue to set the stage for the study of knots and links. With the discovery of the Jones polynomial [15] in 1985, along with a two variable generalization [12] shortly thereafter, the study of knots was given new focus. These new polynomial invariants could be viewed as combinatorial objects, derived directly from a diagram of the knot, or as algebraic objects, resulting from representations of the braid group. However, although the new polynomials were able to distinguish between knots that had previously caused difficulties, they led to new questions in the study of knots that have yet to be answered. In particular, we are led to the phenomenon of distinct knots having the same Jones polynomial. There.are.many..examples of families of knots that share common Jones polynomials.' Such examples have given way to a range of tools to describe this occurrence [30, 31]. In particular, it is unknown if there is a non-trivial knot that has trivial Jones polynomial. This question motivates the understanding of knots that cannot be distinguished by the Jones polynomial, as well as the development of examples of such along with tools to explain the phenomenon. The prototypical method for producing two knots having 1 Chapter 1. Introduction 2 the same Jones polynomial is known as Conway mutation. However, it is well known that this method will not alter an unknot to produce a non-trivial knot. Recent work of Eliahou, Kauffman and Thistlethwaite [9] suggests the use of braid group actions in the study links having the same Jones polynomial. Revisiting earlier work of Kanenobu [18], new families of knots are described in this work. Once again, there is a subtle relationship between the combinatorics and the algebra associated with such examples. As a result, the study of knots obtained through braid actions can be restated in terms of fixed points of an associated group action. The study of this braid action certainly merits attention, as the work of Eli-ahou, Kauffman and Thistlethwaite [9] explores Thistletwaite's discovery [33] of links having the trivial Jones polynomial, settling the question for links having more than one component. As a result, only the case of knots is left unanswered as of this writing. This thesis is a study of families of knots sharing a common Jones polynomial. In chapter 1 the classical definitions and results of knot theory are briefly reviewed, developing the necessary background for the definitions of the Jones, Alexander and HOMFLY polynomials in chapter 2. Then, in chapter 3, the linear theory of tangles (due to Conway [8]) is carefully reviewed. Making use of this linear structure, we define a new form of mutation by way of an action of the braid group on the set of tangles. The main results of this work are contained in chapter 5. We produce exam-ples of distinct knots that share a common Jones polynomial, and develop a generalization of knots due to Kanenobu [18]. Moreover, it is shown (theorem 5.3) that knots constructed in this way are not related by Conway mutation. We conclude by restating the results of Eliahou, Kauffman and Thistlethwaite [9] in light of this action of the braid group, giving examples of non-trivial links having trivial Jones polynomial in chapter 6. C h a p t e r 2 Knots, Links and Braids 2.1 Knots and Links A knot K is a smooth or piecewise linear embedding of a closed curve in a 3-dimensional manifold. Usually, the manifold of choice is either M3 or S3, so that the knot K may be denoted S1 «-»• R3 C S3. While it is important to remember that we are dealing with curves in 3-dimensions, it is difficult to work with such objects. As a result, we deal primarily with a projection of a knot to a 2-dimensional plane called a knot diagram. In this way a knot may be represented on the page as in figure 2.1. a s Figure 2.1: Diagrams of the Trefoil Knot In such a diagram the indicates that the one section of the knot (the broken line) has passed behind another (the solid line) to form a crossing. In general there will not be any distinction made between the knot K and a diagram representing it. That is, we allow a given diagram to represent a knot 3 Chapter 2. Knots, Links and Braids 4 and denote the diagram by K also. It should be pointed out, however, that there are many diagrams for any given knot. Indeed, K and K' are equivalent knots (denoted K ~ K') if they are related by isotopy in S 3 . Therefore, the diagrams for K and K' may be very different. An n-component link is a collection of knots. That is, a link is a disjoint union of embedded circles U s . 1 i = l where each C § J is a knot. Of course, a 1-component link is simply a knot, and a non-trivial link can have individual components that axe unknotted. Figure 2.2: The Hopf Link To study links by way of diagrams, it is crucial to be able to alter a link diagram in a way that reflects changes in the link resulting from isotopy in S 3 . To this end, we introduce the Reidemeister Moves defined in [28, 29]. 5^ ~ 5^-5^ (Ri) (R 2 ) (Ra) In each of the three moves, it is understood that the diagram is unchanged outside a small disk inside which the move occurs. Theorem (Reidemeister). Two link diagrams represent the same link iff the diagrams are related by planar isotopy, and the Reidemeister moves. Chapter 2. Knots, Links and Braids 5 Assigning an orientation to each component of a link L gives rise to the oriented link L. Def in i t i on 2.1. Let C be the set of crossings of a diagram L. The writhe of an orientation L is obtained taking a sum over all crossings C cec where w(c) = ± 1 is determined by a right hand rule as in w (><) = 1 and w (><) = While writhe is not a link invariant, it does give rise to the following definition. Def in i t i on 2.2. For components Ly and L2 of L let C C C be the set of crossings of L formed by the interaction of L\ and L2. The linking number of Lx and L2 is given by c e c The linking number is a link invariant. Note that, for the Hopf link of figure 2.2, there are two distinct orientations. One orientation has linking number 1, the other linking number —1 and hence there are two distinct oriented Hopf links. 2.2 Braids There are many equivalent definitions of braids (see [6], [10], [27]). In this setting it is natural to start from a geometric point of view. Let B e l 3 denote the yz-plane and let E' denote its image shifted by 1 in the a; direction. Consider the the collection of points -P = { l , . . . , n } = { ( 0 , 0 , l ) , . . . , ( 0 , 0 , n ) } c £ ; " and denote by 7>' = { ( l , 0 , l ) , . . . , ( l , 0 , n ) } the image of V in E'. Chapter 2. Knots, Links and Braids 6 Definition 2.3. A (n-strand) braid is a collection of embedded arcs (or strands) ca : [0,1] ^ [0,1] x E C R 3 such that (a) ai(0) = ieV (b) Oi(l)eV (c) oti fi oij = 0 as embedded arcs for i / j. (d) oti is monotone increasing in the x direction. As with knots, it will be convenient to consider the diagram of a braid by projecting to the xy-plane. Also, we may consider equivalence of braids via isotopy (through braids), although we will confuse the notion of a braid and its equivalence class. In [3, 4] Artin showed that there is a well defined group structure for braids. The identity braid is represented by setting each arc to a constant map oti (x) = (x,0, i) so that each strand is a straight line. Multiplication of braids is defined by concatenation, so that inverses are constructed by reflecting in the xz-plane. The n-strand braid group has presentation OiOj = OjOi \i - j\ > 1 \ OiUjOi = o-jOiOj \i — j\ = 1 j where the generators correspond to a crossing formed between the i and i +1 strand as in figure 2.3. Just as group elements are formed by words in the generators, a braid diagram for a given element can be constructed by concatenation of braids of the form shown in figure 2.3. If E and E' of an n-strand braid are identified so that V — P'. pointwise, the result is a collection of embeddings of S 1 in R 3 and we obtain a link. Given any braid f3 we can form a link ft by taking the closure in this way. It is a theorem of Alexander [1] that every link, arises as the closure of some braid. Given a link diagram L, it is always possible to construct a braid /? such that /3 = L . Two such constructions (there are many) are due to Morton [26] and . Vogel [34]. • • -crn_i Chapter 2. Knots, Links and Braids 7 Figure 2.3: The braid generator <TJ and its inverse @ Figure 2.4: The link (3 formed from the closure of B. Now it should be noted that the group operation o~iO~~x = 1 = O j r l ( 7 i corresponds exactly to the Reidemeister move R2, while the group relation corresponds to the Reidemeister move R3. This suggests the possibility of studying equivalence of links through braid representatives. To this end we define the Markov moves. Suppose B 6 Bn and write B = (8,n). Then (Pifan) ~ M {foPun) ( M i ) (B,n) ~u{Bet\n+1) (M 2) where ~ M denotes Markov equivalence. The following theorem, due to Markov [24], is proved in detail in [6]. Theorem (Markov). Two links B\ and B2 are equivalent iff By ~M @2-C h a p t e r 3 P o l y n o m i a l s 3.1 The Jones Polynomial Define the Kauffman bracket (L) of a link diagram L recursively by the axioms ( 0 ) = 1 (3-1) ( x ) = a ( = ) + a " 1 ( D c ) (3.2) ( i u O ) = * ( i ) (3-3) where a is a formal variable and x -2 2 o = -a — a so that (L) is an element of the (Laurent) polynomial ring Z[o, a~1]. In some cases a is specified as a non-zero complex number, in which case (L) G C. The Kauffman bracket is invariant under the Reidemeister moves R2 and R3. To get invariance under R i , we recall definition 2.1 for the writhe of an orien-tation L of the diagram L. The writhe of a crossing is ±1 and is determined by a right hand rule. That is w (><) =1 and w (x) = -1 so that w(L) € Z. Now is invariant under R i and gives rise to an invariant of oriented links. 8 Chapter 3. Polynomials 9 Definition 3.1. The Jones Polynomial [15, 19] is given by ' L where t is a commuting variable a=t~~i Note that it will often be convenient to work with t = a - 4 , and the polynomial obtained through this substitution will be referred to as the Jones Polynomial also. As we shall see, there are many examples of distinct links having the same Jones Polynomial. However, the following is still unknown: Question 3.2. For a knot K, does Vg(t) = 1 imply that K ~ Q ? 3.2 The Alexander Polynomial For any knot K, let F be an orientable surface such that dF = K. Such a surface always exists [32], and is called a Seifert surface for the link K. The homology of such a surface is given by #!(F ,Z) = 0 Z where g is the genus of the surface F. Let {aj} be a set of generators for H^F, Z) where i € {1,.. . , 2g} . Let D2 = {z G C : \z\ < 1} and consider a tubular neighborhood N{K) = K x D2 of the link K. That is, an embedding such that ii" is the restriction to. S 1 x {0}.'f Now consider the surface F in the complement X = § 3 \ N(K). Here F is being confused with its image in the compliment X, by abuse. For a regular neighborhood F x [1,1] c S 3 Chapter 3. Polynomials 10 there are natural inclusions i ± : F 4 f x { ± l } where F = F x {0} is the Seifert surface in X . Therefore a cycle x G H\ (F, Z) gives rise to a cycle x± = if{x) G Hi(X,Z). Def in i t i on 3.3. The Seifert Form is the bilinear form v : fli(F,Z) x Hi{F,Z) -> Z and is represented by the Seifert Matrix .V = (lk{oi,afj) where y+ = i+ (y). The aim is to construct X , the infinite cyclic cover [25, 32] of the knot com-plement X = § 3 \ jV(iO- To do this, start with a countable collection of for some small e G (0,1). The boundary of this space contains two identical copies of F denoted by F± = Fx {±e}, and the infinite cyclic cover of X is defined by identifying Ff C dXi with F^+1 C dXi+i for each i G Z . The space obtained corresponds to the short exact sequence 1 *~ it\X ^rtiX *Hi{X,-Z) -0 a\——Ik so that the infinite cyclic group Hi(X,Z) = (t) gives the covering translations of X\X. Now H~x(X',Z)r although typically not finitely generated as an abelian group, is finitely generated as a Z[i, i_1]-module by the {ai}. The variable t corresponds to the (i)-action taking Xi to Xi+i. Chapter 3. Polynomials 11 Definition 3.4. H~i(X,Z) is called the Alexander module and has module presentation V — tVT, where V is the Seifert matrix. This is a knot invariant. This gives rise to another polynomial invariant due to Alexander [2]. Definition 3.5. The Laurent polynomial AK{t) = det(V - tVT) € Z[t, r 1 ] is an invariant of the knot K called the Alexander polynomial. It is defined up to multiplication by a unit ± t ± n (indicated by =). This knot invariant is of particular interest due to this topological construction. Question 3.6. Is there a similar topological interpretation for the Jones poly-nomial? On the other hand, it is easy to generate knots K such that A/<-(£) = 1 (see for example, [32]). Theorem 3.7. For any knot K, AK(t) = AK(t~l). P R O O F . Given the n x n Seifert matrix V, AK = det{V -tVT) = det{VT - tV) = ( -* ) n de t (V r - r 1 V T ) = AK{rl). • Theorem 3.8. For any knot K, AK(l) = ±1. P R O O F . Setting t — 1 and using the standard (symplectic) basis for Hi(F,Z) Chapter 3. Polynomials 12 gives V — VT = (lk{ai,a+) - lk(aj,af)) - (lk(ai,dj) - lk(aj,ai)j = (lk(ai,a^~) -/A;(oi,a^)j 0 1 -1 0 0 1 -1 0 0 1 -1 0 therefore AK{1) = ±det{V - VT) = ±1 . Corollary 3.9. For any knot K A/c(*) = c 0 + cx{rl + tl) + c2{rl + t1)2 + ••• where c% G Z . PROOF. The symmetry given by Ax(t) = A/ f ( i _ 1 ) gives rise to the form • k=0 where Cm-r = ± c r , and the same choice of sign is made for each r. Now m must be even, since m odd gives" rise to A# (1) even, contradicting theorem 3.8. Further, if Cm-r = — cr then c m / 2 = 0 and m A*(i) = J > = o, fe=0 again contradicting theorem 3.8. Therefore cm—r = cv, and A/f(t) = c 0 + c ^ r 1 + i 1 ) + c2(<"1 +11)2 + ••• as required. • Chapter 3. Polynomials 13 W e w i l l see t h e f o r m g i v e n i n c o r o l l a r y 3.9 i n t h e n e x t s e c t i o n . T o g e t h e r w i t h t h e n o r m a l i z a t i o n A # - ( l ) = 1, i t i s s o m e t i m e s re fe r red t o as t h e Alexander-Conway polynomial as i t has a r e c u r s i v e d e f i n i t i o n , o r i g i n a l l y n o t i c e d b y A l e x a n -d e r [2] a n d l a t e r e x p l o i t e d b y C o n w a y [8]. It s h o u l d b e n o t e d t h a t t he re a re g e n e r a l i z a t i o n s o f t h i s c o n s t r u c t i o n t o i n v a r i -an t s o f o r i e n t e d l i n k s t h a t have b e e n o m i t t e d . N e v e r t h e l e s s , we s h a l l see t h a t t h e r e c u r s i v e d e f i n i t i o n o f Ax{t) is de f ined for a l l o r i e n t e d l i n k s . 3.3 The H O M F L Y Polynomial A t w o v a r i a b l e p o l y n o m i a l [12, 16] t h a t r e s t r i c t s t o each o f t h e p o l y n o m i a l s i n t r o d u c e d m a y b e de f ined , a l b e i t b y v e r y different m e a n s . T h e n - s t r a n d b r a i d g r o u p Bn genera tes a g r o u p a l g e b r a Hn over Z[q, q~l] w h i c h has r e l a t i o n s (i) OiOj — OjOi for \i — j\ > 1 (i i ) OiO-jOi = o-jUiO-j for \i — j\ — 1 ( i i i ) a2 = {q-l)ui + q V i G { 1 , . . . , n - 1 } c a l l e d t h e Hecke a l g e b r a . B y a l l o w i n g q t o t a k e va lues i n C , Hn c a n b e seen as a s u b a l g e b r a o f t h e g r o u p a l g e b r a CBn. J u s t as {1} < B2 < B3 < B4 < • • • . we have t h a t N o t e t h a t for q = 1, t h e r e l a t i o n ( i i i ) r educes t o o f = 1 a n d we o b t a i n t h e r e l a t i o n s for t h e s y m m e t r i c g r o u p Sn. Definition 3.10. Sets of p o s i t i v e p e r m u t a t i o n b r a i d s may be'defined recur-sively via S 0 = . { 1 } _ . S j = {1} U & - j E j _ i for i>0. A monomial m € Hn is called n o r m a l if it has the form m = mim2 . . . mn-i where mi G S j . Chapter 3. Polynomials 14 T h e n o r m a l m o n o m i a l s f o r m a bas i s for Hn, a n d i t fo l lows t h a t d i m z [ g i 9 - ! ] ( # „ ) = n\. M o r e o v e r , t h i s bas i s a l l o w s us t o p resen t a n y e l emen t o f Hn+i i n t h e f o r m Xl + X 2 C T N X 3 for Xi G Hn. T h e r e l a t i o n (i) i m p l i e s t h a t XCTJI — 0~jiX w h e n e v e r x G Hn-i, g i v i n g r i se t o t h e d e c o m p o s i t i o n H n + 1 ^Hn® (Hn ®Hb_1 Hn) . N o w w e def ine a l i n e a r t r a c e f u n c t i o n t r : f f n — > Z [ g ± 1 , z ] CTj I > Z t h a t i s n o r m a l i z e d so t h a t t r ( l ) = 1. Theorem 3.11. tr(xiX2) = ti(x2Xi) for x^ G Hn. P R O O F . B y l i n e a r i t y , i t suffices t o s h o w t h a t t r (mi?7 i2 ) = t r ( m 2 W i i ) for no r -m a l m o n o m i a l s m,- G Hn. S i n c e t h e t h e o r e m is c l e a r l y t r u e for t h e n o r m a l m o n o m i a l s o f i f 2 , we p r o c e e d b y i n d u c t i o n . S u p p o s e first t h a t m i = m[o~nm" w h e r e m ' ^ m ' / G Hn a n d rri2 G Hn ( t ha t i s , rri2 c o n t a i n s n o on). T h e n t r ( m i m 2 ) = tr(m[anm'(m2) = z tx(m'lm'{m2) = 2 ; t r ( m 2 m ' i m i ) = t r (m2mi<7 n m' i ' ) = t r ( m 2 T O i ) . N o w m o r e g e n e r a l l y w r i t e ' • -mi = Tn[anrri'l a n d 7712 w h e r e m ^ , m" G Hn. I n t h i s case we w i l l m a k e use o f t h e f o l l o w i n g : b y i n d u c t i o n = m!2anrri2 Chapter 3. Polynomials 15 (1) tr(/XiCT„/i20-n) = t r ( c r n / i i c r n / i 2 ) (2) tr(/Lti fT„M20nM3) = t r ( / i 3^ i (7 n )U2<7n) w h e r e / i j € Hn are i n n o r m a l f o r m so t h a t p,{ = \x\an-\n" w i t h p,'i,p,1- € . f f n - i -(1) tr( / i i<T n /U 2 <T„) = t r ( / i iC r n / i 2<7„_ i / i 2 CT n ) = t r (Mi J t i 2 o- n <r n _icr n /L i2) u s i n g ( i) = t r ( / i l / / 2 < 7 n _ l C r „ C r n _ i / / 2 ) u s i n g (") = ^ t r ^ i / i ^ C T 2 . ! ^ ' ) = 2 tr(/ii/i'2[(g - l ) a n - i + g]/4') u s i n g ( " 0 = z{q - 1) tr(niiJ,'2an-iP2) + zqtr{p,ip!2ti2) = z{q - 1) t r ( /U i / x 2 ) + zqti{p,[an-ip"p,2ti2) = z ( g - 1) t r ( / i i / / 2 ) + ^ g t r ^ i / X i j U ^ n - i M ^ ' ) i n d u c t i o n = z{q - 1) t r ( / u i / i 2 ) + z q t r ^ V i ' ^ ) = «tr(/ij[(g - l ) a n - i + gK7»2) = z t r ^ i c r 2 , ^ ' / ^ ) u s i n g ( i i i ) = ^ ( / / i c T n - i a n C T n - i j u " ^ ) = t r ( / * i<r„ (T n _ i<T n ^ i /U2) u s i n g ( i i ) = t r ( a n / x i C T „_ i / u / 1 / C T n / x 2 ) u s i n g (i) = t r ( c r n M i c r n / i 2 ) (2) t r (^ iCT„M20- n M3) = t r ( / / i / u 2 < 7 n / i 3 C T n ) a p p l y i n g (1) = t r ( ^ l / i 2 C T n | 4 e r „ _ i ^ 3 < 7 n ) = z t r ( / i i M 2 ^ 3 c r 2 J _ 1 / L i 3 ' ) as above = z{q - 1) t r ( / i i / z 2 / 4 C T n _ i / 4 ' ) + * g tr(/ii^2A*3A«3) u s i n S ( h i ) = z(g - 1) t r ( M 3 c r „ - i / i 3 / i i M 2 ) + z g t r ( / i 3 / i 3 V i M 2 ) i n d u c t i o n = ztr(M3CTn_i/4W2) u s i n g ( i i i ) = t r ( a n / i 3 < 7 n _ i / i 3 V n / i i / i 2 ) • ' a s above = t r ( < r n ^ 3 c r n / z i / x 2 ) = t r ( ^ 3 M i a „ M 2 C T n ) a p p l y i n g (1) Chapter 3. Polynomials 16 Now the proof is complete, since tr(mim2) = tr(m[anrnirn2o-nm2) = tr(m2'm'1crnm/1'm2an) by (2) = tx{anm2rn\anrn'lrn!2) by (1) = tr(m2CT„m'2m'1CT„m'1') by (2) — tr(m2mi). • Now since elements of i?n+i a r e of the form x\ + x2anx% where xi € Hn, the trace function may be extended from Hn to Hn+i by ti{xx + X20nxz) = tr(xi) + ztx{x2X3). The aim is to use the trace function to define a link invariant. In particular we would like to make use of this trace on braids in the composite Bn —+• Hn —> Z[q±l,z}. Xq(l-q) To do this, we introduce a change of variables A — — where 7 ° qz • Z = - l ^ X q W = 1-Xq so that x = l ~ q + z . zq Definition 3.12. The HOMFLY polynomial is given by where 0 E Bn is a monomial in Hn and e — e(/3) is the exponent sum of /? (equivalently, the abelianization Bn —>Z). Note that the closure of the identity braid in Bn gives the n component unlink o o - o J Chapter 3. Polynomials 17 and the HOMFLY polynomial for this link is given by 1-Xq x n ~ l VX(l-q) Theorem 3.13. Let f3 = L then XL(q, A) e Z ant. ±1 ' (3.4) is a link invari-PROOF. By Markov's theorem, we need only check that Xi,(q,X) is invariant under Mi and M 2 . The fact that tr(/?i/32) = tr(/?2#i) from Theorem 3.11 gives invariance under Mi, so it remains to check invaxiance under M 2 . Suppose then that /? G Bn. With the above substitution we have tr(<rn) = 1-g 1-Xq so that = v / A VX(l-q) 1 - Xq VX(l-q) i - g 1 - Xq xp{q, A) = Xp(q,X). Further, from (iii) we can derive o f = {q- + 9 Vi = {q - 1) + go,"1 = ffi + 1 - g erf1 = q-lai + g"1 - 1 Chapter 3. Polynomials 18 hence t r ( a - 1 ) = t T ( q - 1 a i + q - l - l ) = q'1 tr{<7i) + q'1 - 1 1 f - l + q + l-Xq 1-Xq 1 - X - l + Xq = - A 1 - Xq 1 - A g ' T h u s ^ ^ ) = ( - v & ) " ( v x ) ' " t r ( ^ l ) V I V V X ( 1 - , ) J I 1 - W ' a n d Xp(q, A) is a l i n k i n v a r i a n t . • B o t h t h e s i n g l e v a r i a b l e p o l y n o m i a l s m a y b e r e t r i e v e d f r o m t h e H O M F L Y p o l y n o m i a l v i a t h e s u b s t i t u t i o n s VL(t) = XL(t,t) AL{t) = xL(t,t-1). A n o t h e r d e f i n i t i o n o f t h e H O M F L Y p o l y n o m i a l is p o s s i b l e . F o r / 3 6 Bn s u p p o s e t h a t L = /3, o r i e n t e d so t h a t t h e gene ra to r ai i s a p o s i t i v e c r o s s i n g ( tha t i s , w(oi) = 1) . S u p p o s e t h a t 8 c o n t a i n s s o m e <7j 1 a n d w r i t e '• -8 = 7i<Ti72 a s imi lar cons t ruc t ion is possible for c r " 1 Chapter 3. Polynomials 19 for 7i € Bn. By applying M i we can define L ~ L 0 = Wi where 7 = 7172- Let L+ = 7cr? and L _ = 7. The relation (iii) gives so that - ?tr(7) = (g - 1) tr(7<7j). Let e = e(7) be the exponent sum of 7. Then y/q V VA ( l -g ) ,e+l 1 - Ag 1 A ? 1 ( t r ( 7 ^ 2 ) - g t r ( 7 ) ) and this shows that Xi,(q, A) satisfies the skein relation -±^=XL+(q,X) ~ v V ^ L - M ) = ( y q - ± ^ XLo(q,X). By introducing the substitution ; t — y/q\/X and x = Jq 3— Chapter 3. Polynomials 20 we c a n def ine PL(t,x)=XL{q,\) w h e r e P j , ( i , x ) € Z ^ 1 , ^ 1 ] is c o m p u t e d r e c u r s i v e l y f r o m t h e a x i o m s P (*,*) = 1 (3.5) rlPL+{t,x) - tPL_{t,x) = xPLo{t,x). (3.6) I n t h i s s e t t i n g , L+, L- a n d LQ axe d i a g r a m s t h a t axe i d e n t i c a l e x c e p t for i n a s m a l l r e g i o n w h e r e t h e y differ as i n L+ L _ LQ B y a s i m p l e a p p l i c a t i o n o f (3.6) , t h e p o l y n o m i a l o f t h e n c o m p o n e n t u n l i n k i s x (3.7) i n t h e s k e i n d e f i n i t i o n o f t h e H O M F L Y p o l y n o m i a l . T h i s agrees w i t h (3.4) u n d e r t h e s u b s t i t u t i o n s t = y/q\/\ a n d x = Jq A s i n d i c a t e d ea r l i e r , b o t h t h e J o n e s p o l y n o m i a l a n d the A l e x a n d e r p o l y n o m i a l m a y b e c o m p u t e d r e c u r s i v e l y as t h e y each sa t i s fy a s k e i n r e l a t i o n b y s p e c i f y i n g VL{t) = PL (it,-iVt + ^j AL(t) = P L [ l , V t - ^ j w h e r e i = C h a p t e r 4 Tangles and Linear Maps 4.1 Conway Tangles I n t h e r e c u r s i v e c o m p u t a t i o n o f t h e K a u f f m a n b r a c k e t o f a l i n k , t h e o r d e r i n w h i c h t h e c ros s ings a re r e d u c e d is i m m a t e r i a l . I n m a n y cases i t w i l l b e conve-n i e n t t o g r o u p c ross ings t oge the r i n t h e course o f c o m p u t a t i o n . F r o m C o n w a y ' s p o i n t o f v i e w [8], s u c h g r o u p i n g s o r tangles f o r m the b u i l d i n g b l o c k s o f k n o t s a n d l i n k s . I n a d d i t i o n , t h i s p o i n t o f v i e w w i l l a l l o w us t o t ake a d v a n t a g e o f t h e w e l l - d e v e l o p e d t o o l s o f l i n e a r a l g e b r a . Definition 4.1. Given a link L in §3 consider a S-ball B3 C S 3 such that dB3 intersects L in exactly 4 points. The intersection B3 D L is called a C o n w a y t a n g l e (or simply, a tangle,) denoted by T. The exterior of the tangle §3 x £ 3 p| L JS caned a n e x t e r n a l w i r i n g , denoted by L \ T. N o t e t h a t , as S 3 \ B3 is a b a l l , t he e x t e r n a l w i r i n g L \ T i s a t a n g l e a l so . F i g u r e 4 .1 : S o m e d i a g r a m s o f C o n w a y t ang le s A t a n g l e , as a subse t o f a l i n k , m a y b e c o n s i d e r e d u p t o e q u i v a l e n c e u n d e r i so topy . W h e n a d i a g r a m o f the l i n k L is c o n s i d e r e d , a t a n g l e m a y b e repre -sen ted b y a d i s k i n t h e p r o j e c t i o n p l a n e , w i t h b o u n d a r y i n t e r s e c t i n g t h e l i n k 21 Chapter 4. Tangles and Linear Maps 22 i n 4 p o i n t s . E q u i v a l e n c e o f t a n g l e d i a g r a m s t h e n , is g i v e n b y t h e R e i d e m e i s t e r m o v e s , w h e r e t h e fou r b o u n d a r y p o i n t s are fixed. F u r t h e r , t h e K a u f f m a n b r a c k e t o f a t a n g l e T m a y b e c o m p u t e d b y w a y o f t h e a x i o m s (3.2) a n d (3 .3) . T h u s t h e t h e K a u f f m a n b r a c k e t o f a n y t a n g l e m a y b e w r i t t e n i n t e r m s o f t a n g l e s h a v i n g n o c ross ings o r c lo sed l o o p s . T h e r e a re o n l y t w o s u c h t ang l e s , a n d t h e y are d e n o t e d b y 0 = Q a n d oo = T h e s e t a n g l e s are f u n d a m e n t a l i n t h e sense t h a t t h e y f o r m a bas i s for p r e s e n t i n g t h e b r a c k e t o f a g i v e n t a n g l e T . T h a t is T)=X0(Q w h e r e x c ^ o o G Z [ a , a Definition 4.2. Let T be a Conway tangle and T) = [x0 Zoo] where rroj^oo € Z [ a , a -n The b r a c k e t v e c t o r of T is denoted br{T) = [ s 0 *oo] . I n t h i s way , t h e K a u f f m a n b r a c k e t d i v i d e s C o n w a y t ang les i n t o e q u i v a l e n c e classes c o m p l e t e l y d e t e r m i n e d b y br(T). F o r e x a m p l e , - « ( © ) + . - ' ( ® a n d br(Q) = [a a-*] W e c a n def ine a p r o d u c t for t ang les t h a t is s i m i l a r t o m u l t i p l i c a t i o n i n t h e b r a i d g r o u p . G i v e n C o n w a y t ang les T a n d U t h e p r o d u c t TU is a C o n w a y t a n g l e o b t a i n e d b y c o n c a t e n a t i o n : N o t i c e t h a t w h e n T G B2 t h i s is e x a c t l y b r a i d m u l t i p l i c a t i o n . Chapter 4. Tangles and Linear Maps 23 Definition 4.3. The K a u f f m a n b r a c k e t s k e i n m o d u l e Sbr is the Z [ a , a - 1 ] -module generated by isotopy classes of Conway tangles, modulo equivalence given by axioms (3.2) and (3.3) defining the Kauffman bracket. T h e t a n g l e s { 0 , oo} p r o v i d e a m o d u l e bas i s for S1" so t h a t t h e e l emen t s T E S1" m a y b e r e p r e s e n t e d b y br(T). S u p p o s e t h e t a n g l e T is c o n t a i n e d i n some l i n k L. T h e n w r i t i n g L = L(T) a n d c o n s i d e r i n g T € Sbr g ives r i se t o a Z [ a , a _ 1 ] - l i n e a r m a p f:Sbr r a , a (4.1) br(T) (L(0))' L<L(oo)>. w h e r e L ( 0 ) = i ( Q ) a n d L ( o o ) = L ( ( g ) ) . T h i s m a p is s i m p l y a n e v a l u a t i o n m a p c o m p u t i n g t h e b r a c k e t o f L(T) s i nce (L(T)) = br(T) W)) <L(oo)> f(T). G i v e n a t a n g l e T , one m a y f o r m a l i n k i n a n u m b e r o f w a y s b y c h o o s i n g a n e x t e r n a l w i r i n g . A s w i t h t h e p r e v i o u s c o n s t r u c t i o n , t he r e a re o n l y t w o s u c h e x t e r n a l w i r i n g s w h i c h d o n o t p r o d u c e a n y n e w c ros s ings . Definition 4.4. For any Conway tangle T we may form the n u m e r a t o r c l o s u r e TN = and the d e n o m i n a t o r c l o s u r e N o w r e t u r n i n g t o t h e l i n k £ ( T ) , r e c a l l t h a t t h e e x t e r n a l w i r i n g L \ T is i t s e l f a t a n g l e . A g a i n , a l l c ro s s ings a n d c l o s e d l o o p s m a y b e e l i m i n a t e d u s i n g t h e b r a c k e t a x i o m s so t h a t (L \ T) = 6 r ( L \ T ) = [(TN) (TD)] br{L \ T ) T Chapter 4. Tangles and Linear Maps 24 This gives rise to another Z[a,a ^-linear evaluation map «S 6 r —>Z[a,a- 1} L ^ T ^ [(T N) (T D)] br(L \ T ) T In fact, combining the linear maps (4.1) and (4.2) forms a bilinear map f : 5 t r x 5 i r - 4 Z [ o , a - 1 ] ( T , L x T ) , - M L ( T ) > (4.2) (4.3) where <L(T)> = [(T N) (T D)] br(L x T) T © N XQ D = br{T) N © " ) ( © br(L \ T ) T + ^ o o ( @ D D br(L \ T) D br{L \ T) T 1 1 5 = br(T) so that given T,U € <S6r we have F(T, (7) = br{T) '5 1 1 5 br{U) T Definition 4.5. For Conway tangles T and U define the link J (T, U) = (TU) N and call this the join ofT and U. We have that by definition, and further {J(T,U)) = F(T,U) L(T) ~ J ( T , L N T) . Definition 4.6. For Conway tangles T and U define the connected sum T D#U D = {TU) D. Chapter 4. Tangles and Linear Maps 25 Since any link L may be written as TD for some tangle T this definition gives rise to a connected sum for links. It follows that provided orientations agree. A similar argument gives such an equality for the HOMFLY polynomial, and hence the Alexander polynomial as well. 4.2 Conway Mutation Consider a link diagram containing some Conway tangle T. We can choose the coordinate system so that T is contained in the unit disk, for convenience of notation. Further, we can arrange that the 4 points of intersection between the link and the boundary of the disk are Let p be a 180 degree rotation of the unit disk about any of the three coor-dinate axis. Note that p leaves the external wiring unchanged and, for such a projection, p fixes the boundary points as a set. Definition 4.7. Given a link L(T) where T 6 Sbr define the Conway mutant denoted by L(pT). Notice that (Li)(L2), and > Q - Q 0 0 = 0 so that (L(T)} = br(T) (HO)) (L(oo)) = br{PT) (L(0)) (L(oo)) = (L(PT)). Chapter 4. Tangles and Linear Maps 26 M o r e o v e r , w i t h o r i e n t a t i o n d i c t a t e d b y t h e e x t e r n a l w i r i n g w{T) = w{pT) so we have t h e f o l l o w i n g t h e o r e m . Theorem 4.8. Ify T ) = VLl<pT). W h i l e i t m a y b e t h a t L(T) *> L{pT), i t is c e r t a i n t h a t t h i s m e t h o d does n o t p r o v i d e a n a n s w e r t o q u e s t i o n 3.2: I t c a n b e s h o w n t h a t a C o n w a y m u t a n t o f t h e u n k n o t i s a l w a y s u n k n o t t e d [30]. T h e o r e m 4.8 i s i n fact a c o r o l l a r y o f a s t r o n g e r s t a t e m e n t . Theorem 4.9. P L ( T ) = PL(PT)-PROOF. U s i n g t h e s k e i n r e l a t i o n (3.6) d e f i n i n g t h e H O M F L Y p o l y n o m i a l , i t is p o s s i b l e t o d e c o m p o s e a n y t a n g l e T i n t o a l i n e a r c o m b i n a t i o n o f t h e f o r m T = a i @ + a 2 @ + a 3 ( g ) w h e r e a ; G Z\t±l,x±l]. T h e r e f o r e , these t ang les p r o v i d e a bas i s for p r e s e n t i n g t h e H O M F L Y p o l y n o m i a l o f a t a n g l e T. T h u s , we c a n def ine a Z[t±l,x±l}-m o d u l e <S F g e n e r a t e d b y i s o t o p y classes o f t ang les u p t o e q u i v a l e n c e u n d e r t h e s k e i n r e l a t i o n . M o r e o v e r , i f L = L(T) t h e n we have a Z [ i ± 1 , x ± 1 ] - l i n e a x e v a l u a t i o n m a p S p — > Z [ i ± 1 , a : ± 1 ] T' ^ PL(T) or , m o r e gene ra l ly , t h e b i - l i n e a r e v a l u a t i o n m a p SpxSp-^Z[t±\,x±l] ( T , U ) ^ P M U ) . S i n c e t h e bas i s { © • © • © } is p - i n v a r i a n t , i t fo l lows t h a t PT a n d PPT are e q u a l hence SpxSp pxid Z[t±l,x±l) Chapter 4. Tangles and Linear Maps 27 commutes and PL(T) = P L ( p T ) -• 4.3 The Skein Module Everything that has been said regarding tangles to this point can be stated in a more general setting [30, 31]. Definition 4.10. Given a link L in S 3 consider a 3-ball B3 C S 3 such that dB3 intersects L in exactly 2n points. The intersection B3 D L is called an re-tangle denoted by T. As before, the exterior of the n-tangle §3 \ B3 n L is another n-tangle L \ T called an external wiring. In this setting, Conway tangles arise for re = 2 as 2-tangles. Let M.n be the (infinitely generated) free Z[a,a_1]-module generated by the set of equivalence classes of re-tangles. The axioms (3.2) and (3.3) defining the bracket give rise to an ideal Tn c Mn generated by (^><)-a(pZ^-a-l(pc) (4-4) (TUO}-6(T) (4.5) where 6 = -a"2 — a2 and the () indicate that the rest of the tangle is left unchanged. Definition 4.11. The Z[a, a~l]-module is called the (Kauffman bracket) skein module. Note that S2 = <S6r. Due to the form of ln it is possible to choose representatives for each equiv-alence class in <Sn that have neither crossings nor closed loops. These tangles form a basis for <Sn. We have seen, for example, that S2 is 2-dimensional as a module, with basis-given by.the fundamental Conway tangles Q and Chapter 4. Tangles and Linear Maps 28 Theorem 4.12. Sn has dimension (2n) ! C n n!(n + l ) ! as a module. PROOF. S i m p l y p u t , we n e e d t o d e t e r m i n e h o w m a n y n - t a n g l e s t he r e a re t h a t have n o c ros s ings o r c l o s e d l o o p s . T h a t i s , g i v e n a d i s k i n t h e p l a n e w i t h 2n m a r k e d p o i n t s o n t h e b o u n d a r y , h o w m a n y w a y s c a n t h e p o i n t s b e c o n n e c t e d b y n o n - i n t e r s e c t i n g a rcs ( up t o i s o t o p y ) ? C l e a r l y , C\ = 1, a n d as d i s c u s s e d e a r l i e r C2 = 2. N o w s u p p o s e n > 2 a n d c o n s i d e r a d i s k w i t h 2 n p o i n t s o n t h e b o u n d a r y . S t a r t i n g a t s o m e c h o s e n b o u n d a r y p o i n t a n d n u m b e r i n g c l o c k w i s e , t h e p o i n t l a b e l e d 1 m u s t c o n n e c t t o a n even l a b e l e d p o i n t , say 2k. T h i s a r c d i v i d e s t h e d i s k i n t w o : O n e d i s k h a v i n g 2(k — 1) m a r k e d p o i n t s , t h e o t h e r w i t h 2(n — k). T h e r e f o r e n Cn = Ck-\Cn-k fe=l = CoCn-\ + C\Cn-2 H 1- Cn-iCo w h e r e C o = 1 b y c o n v e n t i o n . N o w c o n s i d e r t h e g e n e r a t i n g f u n c t i o n f(x) = J2Cixi i=0 a n d n o t i c e so t h a t a n d ' i=0 \ fe= l / .. x(f(x))2 = f(x) - 1 2x T o d e d u c e t h e coeff ic ients o f f(x), first c o n s i d e r t h e e x p a n s i o n o f yfz a b o u t 1. 00 i=Q Chapter 4. Tangles and Linear Maps 29 where , _ 1 1 * di - < ( - i y - i ( 2 i - 3 ) ! ! • Q Therefore with z = 1 - 4x oo 2=0 OO = l + ^2i-l)i2i2idixi z = i and 1 0 0 i = l oo 1=1 so that d = ( - l ^ + ^ ' d , •i+i _ ( 1 ) t g i + 1 , - ( - l ) i (2 i - l ) ! ! V J 2*+i(* + l)! _ 2*(2i - 1)!! (» + l)! for i > 1. Finally, the fact that 2 n ( 2 n - ! ) ! ! _ (2n)! (n + 1)! n!(n+l)! follows by induction since 2 n + 1 (2(n + l) -1)!! _ 2(2n + l )2" (2n- l ) ! ! (n + 2)! ~ n + 2 (n + 1)! _- 2(n + l)(2n+ 1) • "• (n + 2)(n + l) n _ (2ra + 2)(2n + l) (2n)! • ' (2(n + l))! (n + l)!(n + 2)!' • Chapter 4. Tangles and Linear Maps 30 As an example of theorem 4.12, the 5-dimensional module S3 has basis given by { © , © ' Q , © , 0 } - (4-6) Let a given n-tangle diagram be contained in the unit disk so that, of the 2n boundary points, n have positive ^ -coordinate while the remaining n have negative x-coordinate. With this special position, multiplication of tangles by concatenation, as introduced for Conway tangles, extends to all n-tangles. When two n-tangles are in fact n-braids, we are reduced to multiplication in Bn. With this multiplication, <Sn has an algebra structure called the Temperly-Lieb algebra [21, 22]. The n-dimensional Temperly-Lieb algebra TCn over Z[a, a~l] has generators ei, e2, • • •, e n _i and relations Sei (4.7) ejei for \i - j\ > 1 (4.8) ei for \i - j\ = 1. (4.9) The multiplicative identity for this algebra is exactly the identity in Bn, and the generators are tangles of the form shown in figure 4.2. ( Figure 4.2: The generator For example, 7T3 is generated by {ei,e2}, while the basis for the module £ 3 is the set of elements {1, ei, e%, e\e<z, e2e{\ as in (4.6). Notice that there is a representation of the braid group via the Kauffman ei ej e^ — n *+l i 1 Chapter 4. Tangles and Linear Maps 31 bracket given by B, n •n Oi i—> a + a &i aei + a - l 4.4 Linear Maps Let L = L(T\,..., Tfe) be a link where {Tj} is a collection of subtangles Tj C L. If T[ is a tangle such that Tj = T[ as elements of Sn then, in the most general setting, V = L(T{,... ,T'k) is a mutant of L (relative to the Kauffman bracket). Therefore when w(L) = w(L'), we have that Of course, it may be that L ^  L' and this approach has been used in attempts to answer question 3.2 [30, 31]. Let's first revisit Conway mutation in this context. We have, given the bilinear evaluation map F and a 180 degree rotation p, the commutative diagram since F = F o (p x id). We saw that the linear transformation p was in fact the identity transformation on c>2, and as a result the link L = J(T,U) and the mutant V = J{pT, U) have the same Kauffman bracket. A possible generalization arises naturally at this stage. As was pointed out earlier, it is possible to construct a link from two tangles in many different and complicated ways. Starting with T C S | C S 3 and U C Bfj C S 3 , the link L(T, U) is constructed by choosing an external wiring of § 3 \ (J9f. U Bfj). In VL = VV. <S2 x S2 S2 x S2 Chapter 4. Tangles and Linear Maps 32 this setting we have (L(T,U)) = br(T) (L(0,U)) [(L(oo,U))\ <L(0,0)) (L(0,oo)) (L(oo,0)) <L(oo,oo)) = br(T) Cbr{U)T which gives rise to the bilinear map G:S2xS2—>Z[a,a_1] (T,U) H - > br(T)Cbr{U)T. If additionally there is a linear transformation r : <S2 x <S2 —> <S2 x <S2 (T,?7)h-> ( T lr,r 2r7) which acts as the identity as a linear transformation of modules, then we have the commutative diagram S2 x S2 T = T l XT2 Z[a, a - H S2 x S2 and finally, if w(L(T, U)) = W(L(TIT, T2{7)) we can conclude that . V L { T , U ) = V L ( T I T I T 2 U ) . . 4.5 Braid Actions The three strand braid group has presentation •^ 3 = {oi,<72|criO'20'l = cr2<7l.CT2) where CTI = L J and o2 = Chapter 4. Tangles and Linear Maps 33 F i g u r e 4.3: T h e t a n g l e e S2 G i v e n T G S2 a n d a 3 G B% we c a n def ine a n e w 2-tangle d e n o t e d as i n f igure 4.3. Proposition 4.13. The map S2 x B3 —> S2 (T,3)^T^ . is a well defined group action. PROOF. L e t id-B3 b e t h e i d e n t i t y b r a i d . T h e n for a n y t a n g l e T we have s ince ~ ® F o r 3,3' G £ 3 , t h e p r o d u c t 83' i s de f ined b y c o n c a t e n a t i o n so t h a t b y p l a n a r i s o t o p y o f t h e d i a g r a m Chapter 4. Tangles and Linear Maps 34 • Proposition 4.14. For T G <S2 and 6 € B3 br(Tai) = br(T) br(Ta2) = br(T) - a " 3 0 a - 1 a a a - 1 0 - a " 3 (4.10) (4.11) PROOF. A p p l y i n g t h e a c t i o n o f o~\ t o a n a r b i t r a r y t a n g l e T , we m a y d e c o m p o s e TCl i n t o t h e t ang le s a n d i n <S2- R e l a x i n g these d i a g r a m s gives r i se t o t h e f o l l o w i n g c o m p u t a t i o n i n £2: br{T^) = br{T) br(T) = br(T) a \ - a ~ 3 0' a - 1 a Chapter 4. Tangles and Linear Maps 35 S i m i l a r l y , for t h e a c t i o n o f o~2: br{TC2) = br{T) = br{T) = br{T) a a 1 0 - a ~ 3 T h i s g ives r i se t o a g r o u p h o m o m o r p h i s m I - a GL2{Z[a,a-1}) - 3 01 02 - 1 a a 0 - a " s m c e $(o-iO- 2o-i) = - a " 2 -a -4" - a " 3 0" 1 0 a - 1 a 0 a -3" - a " 3 0 a a - 1 a" " 2 - a - 4 " 0 - a " 3 1 0 • = $ ( < 7 2 0 - l C r 2 ) . Question 4.15. Is this representation of B3 faithful? W i t h t h e f r a c t i o n o n <S2, c o n s i d e r t h e l i n e a r t r a n s f o r m a t i o n g i v e n b y /3 : <S2 x <S2 —> S2 x S2 [ ( T , U) i—> (T^, U^1).-Chapter 4. Tangles and Linear Maps 36 F o r a l i n k o f t h e f o r m L ( T , [/), t h i s leads t o t h e d e f i n i t i o n o f a n e w l i n k L{TP,Uf>-). D e n o t e t h e evaluation matrix o f L(T, U) b y C = <L(0 ,0) ) <L(0 ,oo) ) ( X ( o o , 0 ) ) <L(oo ,oo) ) a n d s u p p o s e t h a t C G GL2(Z[a,a 1 ] ) , t h a t i s , d e t ( £ ) / 0. N o t e t h a t (L(T,U)) = br{T)Cbr(U)T {L(TP, U?'1)) = br(T)${8) C ($(8-l))Tbr(U)r.. S o , d e f i n i n g a s e c o n d ^ - a c t i o n £ 3 x GL2(Z[a,a~ 1]) G L 2 ( Z [ a , a - 1 ] ) we a re l e d t o a n a l g e b r a i c q u e s t i o n . W h e n a n o n - t r i v i a l 8 G Bz g ives r i se t o a fixed p o i n t u n d e r t h i s a c t i o n , t h e l i n e a r t r a n s f o r m a t i o n g i v e n b y 8 is t h e i d e n t i t y . T h u s G = G o 8 a n d we have t h e c o m m u t a t i v e d i a g r a m <S2 x S2 <S2 x « S 2 w h e r e C G F i x ( / 3 ) , so t h a t (L(T,U)) = (L(T^Up-1)). I n p a r t i c u l a r , we w o u l d l i k e t o s t u d y t h e case w h e r e <' L(T, U) oo L(TP, U^'1). Question 4.16. For a given link I*(T, U) with evaluation matrix C G GL2(Z[a, a' what are the elements 8 G B% such that C G F i x ( / 3 ) ? T h i s q u e s t i o n is t h e m a i n focus o f t h e f o l l o w i n g chap t e r s . Chapter 5 Kanenobu Knots 5.1 Construction Shortly after the discovery of the H O M F L Y polynomial, Kanenobu introduced families of distinct knots having the same H O M F L Y polynomial and hence the same Jones and Alexander polynomials as well [18]. It turns out that these knots are members of a much larger class of knots which we wil l denote by K{T,U) for tangles T,U € S2. Proposition 5.1. Suppose x is a non-trivial polynomial in Z[a,a l] so that where 6 = —a 2 — a2. Then $(<T2) X <&(CT2 1 ) T = X and X 6 Fix(cr 2) under the Bz-action on GL(71[a,a^1]). PROOF. Since Figure 5.1: The Kanenobu knot K(T, U) X = i eGLiZfaa-1]) *(o-2) = a 0 -a: a - 3 and <&(<T2 X ) = —a' a 3 37 Chapter 5. Kanenobu Knots 38 we have , - 3 x 6 6 82 a —a ax + a lS a.6 + a l52~\ la 1 0 -a~38 -a~382 J [ a -a3 x + a-25 + a25 + 62 -aA6 - a262' -a~45-a-252 52 x + 8{a-2 + a2 + S) 5(-a 4 - a2S) <K-a"4 - a~2<5) 62 x 5{-aA + 1 + a4) <5(-cr4 + a" 4 + 1) 52 x S 8 82 with 5 = — a 2 — a 2. • To compute the bracket of the Kanenobu knot K(T, U), we need the evaluation matrix K = Since If (0,0) is the connected sum of two figure eight knots (the figure eight is denoted 4 i as in [32]), we can compute ( 4 i ^ = (^a2aTl a2aVl ^ using the Temperly-Lieb algebra 7 X 3 . We have o2o7 l = 1 + a2ei + aT2e2 + e2ei as an element of TC3 so that (cr2crfx)2 = (1 + a2ei + aT2e2 + e 2ei) 2 = 1 + (3a 2 +a 4 £)e! + (3a~2+a~45)e2 + exe2 + {A+a26+a'2S)e2ei. Chapter 5. Kanenobu Knots 39 Then the Kauffman bracket is given by ( ( W i " 1 ) 2 ) = 5 2 + ( 3 ° 2 + a i S ) S + ( 3 a _ 2 + a~iS)5 + 1 + (4 + a2^ + a-2rj)(l) = 5{-a-6 + 2a~4 + 2o4 - a6) + 5 = a - 8 - a - 4 + 1 - a 4 + a 8 and (4i#4i) = (a"8 - a"4 + 1 - a 4 + a 8) 2 = a ' 1 6 - 2a- 1 2 + 3a~8 - 4a~4 + 5 - 4a4 + 3a8 - 2a1 2 + a 1 6 . In addition, it can be seen from the braid closure o~io~\ 1C2Cri1 ~ 4i that w(4i) = 0 and hence w (if (0,0)) =w(4i#4i) = 0. Thus the Jones polynomial of K(0,0) is given by VK{0,0) = fl_16 - 2 a ~ U + 3a - 8 - 4a"4 + 5 - 4a4 + 3a8 - 2a1 2 + a 1 6 . Now the evaluation matrix for K(T, U) is given by Ua~8 - a" 4 + 1 - a 4 + a 8) 2 6' IL~ [ 8 8 2 since the three entries for K(0,00), K(oo, 0) and i f (00,00) are all equivalent to unlinks with no crossings via applications of the Reidemeister move R2 (recall that R2 leaves the Kauffman bracket unchanged). For any tangle diagram T, denote by T* the tangle diagram obtained by switch-ing each crossing of T. That is, for any choice of orientation w(T*) = -w{T). This can be extended to knot diagrams if, where K* is the diagram such that w{K*) = -w{K) so that K* is the mirror image of K. When U = T*, : • ' ' '• •w{K{TM))=0 . ' Chapter 5. Kanenobu Knots 40 t h e b i l i n e a r e v a l u a t i o n m a p for t h e b racke t G(T, U) = br(T) IC br(U)T c o m p u t e s t h e J o n e s p o l y n o m i a l VK{T>u) = br(T)rCbr(U)T. S i n c e K. i s o f t h e f o r m g i v e n i n p r o p o s i t i o n 5 .1 , t h e b i l i n e a r m a p d e f i n e d b y t h e b r a i d cr" € B% {T,U) >-> {Tu*,Ua*n) is t h e i d e n t i t y t r a n s f o r m a t i o n for every n € Z . M o r e o v e r , w h e n U = T* w (K{Ta*,Uazn)) = 0 so we have t h e f o l l o w i n g t h e o r e m . Theorem 5.2. When U = T*, the family of knots given by K ( T a Z , U a z n ) for n € Z are indistinguishable by the Jones polynomial. O f cou r se , t h a t these a re i n fact d i s t i n c t k n o t s r e m a i n s t o b e seen. K a n e n o b u ' s o r i g i n a l k n o t f a m i l i e s [18] c a n b e r ecove red f r o m Kn>m = K(T#;T*r) w h e r e n , m e Z a n d T is t h e 0- tangle . Theorem (Kanenobu). Kn~iTn and Kn^mi have the same HOMFLY polyno-mial when \n—m\ = \n' — m'\. Moreover, when ( n , m ) and (n',m') are pairwise distinct, these knots are distinct. T h e k n o t s o f K a n e n o b u ' s t h e o r e m are d i s t i n g u i s h e d b y t h e i r A l e x a n d e r m o d u l e s t r u c t u r e [18]. Chapter 5. Kanenobu Knots 41 5.2 Basic Examples C o n s i d e r t h e K a n e n o b u k n o t s a n d n o t i c e t h a t b y a p p l y i n g t h e a c t i o n o f o2 we have so t h a t b y c o n s t r u c t i o n , these k n o t s have t h e s a m e J o n e s p o l y n o m i a l . F i r s t r e c a l l t h a t t h e H O M F L Y p o l y n o m i a l o f t h e n - c o m p o n e n t u n l i n k O O - O is g i v e n b y x A s t h i s p o l y n o m i a l w i l l b e u s e d of ten , we def ine o x N o w , u s i n g t h e s k e i n r e l a t i o n (3.6) d e f i n i n g t h e H O M F L Y p o l y n o m i a l P ( i , z ) , we c a n c o m p u t e a n d Chapter 5. Kanenobu Knots 42 Combining these tangles pairwise, we have :@r.) = (3©:, :©:) + ( 3 © , giving rise to equality among HOMFLY polynomials PK2 = PK0 + - tx)P0 - x2P02 = PK0 + {t~lx - tx) = P K 0 -This common polynomial1 is PKo(t, x) = ( i " 4 - IT2 + 3 - 2 i 2 + i 4 ) + ( - 2 r 2 + 2 - 2i 2 )x 2 + a;4. Notice that this is in agreement with Kanenobu's theorem. On the other hand K \ has HOMFLY polynomial PKl (t, x) = (2i~ 2 - 3 + 2i 2) + (3t~2 - 8 + 3i 2 )x 2 + (t~2 - 5 + t2)xA - x% and we can conclude that KQ and K \ are distinct knots despite having the same Jones polynomial: o" 1 6 - 2a" 1 2 + 3a" 8 - 4a~4 + 5 - 4a4 + 3a8 - 2a 1 2 + a 1 6 Further, applying theorem 4.9, these knots cannot be Conway mutants as they have different HOMFLY polynomials. 5.3 M a i n Theorem Theorem 5.3. For each 2-tangle T there exists a pair of external wirings for T that produce distinct links that have the same Jones polynomial. Moreover, the links obtained are not Conway mutants. C o m p u t e d using K N O T S C A P E Chapter 5. Kanenobu Knots 43 Figure 5.2: Distinct knots that are not Conway mutants P R O O F . Take U — T* (that is, the tangle such that w{U) = -w(T)) and define Kanenobu knots for the pair (T, U) K = K(T, U) and Kn = K(Ta2,Ua2). Then, by construction, we have that VK = VK°I • It remains to show that these are in fact distinct knots. To see this, we compute the HOMFLY polynomials P K and PK°2 • Now with the requirement that the tangle U = T*, there are two choices of orientations for the tangles that are compatible with an orientation of the knot (or possibly link, in which case a choice of orientation is made) K(T, U). They are Type 2 so we proceed in two cases. Type 1. Using the skein relation we can decompose Chapter 5. Kanenobu Knots 44 w h e r e a r , &T, at/ , bjj € Z[t; , a; ]. C o m b i n i n g p a i r w i s e we o b t a i n ( : © : • : © : ) = ° ™ n g r ) ( a g e . a g e ) ( 3 g r , 3 g : ) + ^ ( a g i : . i g r ) + &rar j so t h a t PK = aTauPKl + (arbu + brau)Po + brbuPo aTauPKl + [arbu + krau) arauPxr + R t-l-t + brbu X w h e r e R = (arbu + brau) ^ + brbu ^ *) . N o w a p p l y i n g t h e a c t i o n o f <T2 we have T " 7 2 = a n d C7 C T2 1 = the re fore (T»,U^l)=aTau (3@T,3gr:) +aTbv ( 3 ^ , j @ t ) : © t ) + ( : @ r , i g n ) = arau so t h a t = aTauPK2 + {arbu + brau)Po + brbuP^ = PK0+R Chapter 5. Kanenobu Knots 45 since PK2 — PK0- However, as PKX PK0 w e n a v e that PK + PK^ giving rise to distinct knots. Type 2. As before, the tangles are decomposed via the HOMFLY skein relation for some other ay, &r, at/, fy/ € Zfi*,a^]. Combining pairwise + w ( u g c , : © : ) + ( a g e . a g e ) and PR: = aTavPKo + (aTbu + brau)Po + brbrrPo + brbu = aTauPKo + R with Pi € Z ^ , ^ ] as before. Again, applying the action of a2 we have {T>\V°?} = a T a a (-jgr;, rjgt) +aTbu (-jgr., -jgc) +frra«/ (J © : > + ^  ( J @ D 3@t) +brau ( n g D 3 g r ) + b>h> ( j g n , 3 g r ) and the HOMFLY polynomial Pftr-2 = qfauPK, + (aTbu + brau)Po + ' 6 T 6 [ / P 0 2 - . ,. = aTaTjPKl+R. Chapter 5. Kanenobu Knots 46 O n c e a g a i n , as PK^ # PK0 we have t h a t PK ± P K ° * g i v i n g r i se t o d i s t i n c t k n o t s . F i n a l l y , as K a n d K172 have d i s t i n c t H O M F L Y p o l y n o m i a l s i n b o t h cases, i t fo l lows f r o m t h e o r e m 4.9 t h a t these k n o t s c a n n o t b e C o n w a y m u t a n t s . • 5.4 Examples 1. C o n s i d e r t h e K a n e n o b u k n o t K ( T , U) w h e r e as i n figure 5.3. Definition 5.4. A tangle is called r a t i o n a l if it is of the form T@ where 0 £ B3 and the tangle T is either the 0-tangle or the oo-tangle. This is equivalent to Conway's definition for rational tangles [8]. A s a r e s u l t , t h e c o m p u t a t i o n o f t h e b r a c k e t for r a t i o n a l t a n g l e s i s s t r a i g h t f o r -w a r d . I n t h i s e x a m p l e we have so t h a t T ^ © * 1 a n d C/ = @ C T l br{T) = [0 i l r-a-3 a " 1 - [0 1] : - a 3 a 1 = [0 1] a - 1 -- = ra--1 _ a 3 +a7 ' - a 3 0 " " - a 3 0 ' a a " 1 a a - 1 0 " Chapter 5. Kanenobu Knots 47 a n d br(U) = [0 1] = [0 1] N o w - a " u a - 1 a 3 o: —a a 1 a —a - 3 - a ~ 3 0 a 7 - a 1 + a a 3 ] (K{T,U)) = br{T)Kbr{U)T [ ( a - 8 - a " 4 + 1 - a 4 + a 8 ) 2 6 r ( T ) -24 - a " 2 - a 2 - a " 2 - a 2 a " 4 + 2 + a 4 a - - 4 a " 2 0 + 1 0 a ~ 1 6 - 1 9 a ~ 1 2 + 2 7 a ~ 8 - 3 3 a - 4 + 37 - 3 3 a 4 + 2 7 a 8 - 1 9 a 1 2 + 1 0 a 1 6 - 4 a 2 0 + a 2 4 br(U) hence V, K(T,U) = a - 2 4 - 4 a " 2 0 + 1 0 a - 1 6 - 1 9 a " 1 2 + 2 7 a " 8 - 3 3 a " 4 + 37 - 3 3 a 4 + 2 7 a 8 - 1 9 a 1 2 + 1 0 a 1 6 - 4 a 2 0 + a 2 4 . F i g u r e 5.3: E x a m p l e 1 T h i s g ives a c o l l e c t i o n o f k n o t s h a v i n g t h e s a m e J o n e s p o l y n o m i a l V K ( T , U ) = V K { r n ^ - n ) Chapter 5. Kanenobu Knots 48 since w{K(Ta2, U** )) = 0 for all n e Z . The Knots K{T, U) and K{T"2, U^) are distinct as they have different HOM-FLY polynomials2: XK{T,u) = - 1 + (6<~2 - 12 + 6t2)x2 + (9t~2 - 24 + 9 i 2 ) z 4 + (5i~2 - 19 + 5*V + (t~2 - 7 + t2)x8 - x 1 0 X _! =(t~A -At'2 + 7 -4t2 +t4) + (2r 4 - i r 2 + io - 7i2 + 2ti)x2 + {t~4 -6t~2 + 8-6t2 +t4)x4 + (-2t - 2 + 4 - 2 i 2 )x 6 + x 8 In particular, these knots cannot be Conway mutants in view of theorem 4.9. 2. Now consider the case when T, U are not rational tangles. For this example take T = fUA-in and U = in K{T, U). We can compute br(T) = [a-5 - 2a'1 + a 3 - a 7 - a - 1 1 + 2a~7 - 2a"3 + a] hr{U) = [-a~7 + a~3 - 2a + a 5 a" 1 - 2a3 + 2a7 - a11] so that {K(T, U)) = -a'28 + 5a" 2 4 - 15a-20 + 31a"16 - 52a~12 + 73a~8 - 88a"4 + 95 - 88a4 + 73a8 - 52a12 + 31a16 - 15a20 + 5a2 4 - a 2 8 . Again we have a family of knots (distinct from those of example 1) such that since w(K(Ta2, Ua>n)) = 0 for all n 6 Z. Computed using KNOTSCAPE Chapter 5. Kanenobu Knots 49 F i g u r e 5.4: E x a m p l e 2 T h e k n o t s K(T, U) a n d K(T<T2,U(T2 *) o f t h i s e x a m p l e a re d i s t i n c t as c a n b e seen b y t h e H O M F L Y p o l y n o m i a l s 3 : X K ( T , U ) =(-2i - 6 + 9 r 4 - 22i"2 + 31 - 22i2 + 9i4 - 2t6) + ( -3r 6 + 22r 4 - 58i~2 + 82 - 58i2 + 22i4 - 3t6)x2 + (-t~6 + 21i"4 - 67r 2 + 94 - 67t2 + 21£4 - t6)x4 + (8i - 4 - 44t"2 + 62 - 44i2 + 8t4)x6 + ( r 4 - 15r 2 + 28 - 15i2 + t4)x8 + (-2t'2 + 8 - 2t 2)a; 1 0 + x12 X - i =(-4i"4 + 12t"2 - 15 + 12i2 - 4i4) + (-12r 4 + 56t~2 - 84 + 56i2 - 12i 4)z 2 + (-13r 4 + 99i"2 - 176 + 99i2 - 13t4)z4 + (-6i~4 + 87t~2 - 197 + 87i2 - 6i 4)a; 6 + ( - r 4 + 41i"2 - 130 + 41i2 - « 4 ) x 8 + (iof-2 - 51 + l o t V ° + (t~2 - i i + 1 V 2 - V 4 O n c e a g a i n , f r o m t h e o r e m 4.9 i t fo l lows t h a t these k n o t s a re n o t r e l a t e d b y C o n w a y m u t a t i o n . : i computed using K N O T S C A P E Chapter 5. Kanenobu Knots 50 5.5 Generalisation W e s a w i n p r o p o s i t i o n 5.1 u n d e r t h e a c t i o n B3 x GL2(Z[a,a-1]) —> GL2{Z[a,a-1}) t h a t x 6 5 S2 G GL< ( Z K ^ 1 ] ) } C F i x ( c r 2 ) -A s a final t a s k for t h i s c h a p t e r , w e ' l l def ine a f a m i l y o f l i n k s t h a t gene ra te s u c h e v a l u a t i o n m a t r i c i e s . C o n s i d e r a s l i g h t l y different d i a g r a m o f t h e k n o t K ( T , f7), g i v e n i n figure 5.5. F i g u r e 5.5: A n o t h e r d i a g r a m o f t h e K a n e n o b u k n o t K ( T , U) F r o m t h i s d i a g r a m , w e are l e d t o def ine a r a t h e r e x o t i c b r a i d c l o s u r e t h a t w i l l b e o f use. T h a t i s , for t h e p a i r o f t ang les ( T , U) a n d a n a p p r o p r i a t e l y c h o s e n b r a i d 3 G B Q w e def ine a l i n k \ B \ as i n figure 5.6. F i g u r e 5.6: T h e l i n k \ B \ I t r e m a i n s t o d e s c r i b e w h i c h b r a i d s i n Be g ive r i se t o a n e v a l u a t i o n m a t r i x o f t h e a p p r o p r i a t e f o r m . F o r t h i s we w i l l n e e d t w o b r a i d h o m o m o r p h i s m s . L e t N 3> n a n d def ine , for e ach n o n - n e g a t i v e m G Z , t h e i n c l u s i o n h o m o m o r -p h i s m im '• Bn B N Chapter 5. Kanenobu Knots 51 for e a c h k E { l , . . . , n — 1 } . N o t e t h a t w h e n m = 0, t h i s i s r educes t o t h e n a t u r a l i n c l u s i o n Bn < B N - N o w t h e g r o u p B3 © B3 ar ises as a s u b g r o u p o f i?6 b y c h o o s i n g B3 © B$ — > BQ (a,0)—>»o(a)» 3G8). N o t i c e t h a t t h e i m a g e io{ct)i3{3) c o n t a i n s n o o c c u r r e n c e o f t h e gene ra to r cr 3 a n d hence i n Be- N o w def ine t h e switch h o m o m o r p h i s m s : B3 —-> B3 0-2 -1 a n d n o t e t h a t , g i v e n a 180 degree r o t a t i o n p i n t h e p r o j e c t i o n p l a n e , p(sd) = r 1 -Definition 5.5. F o r eacft a € B3 define the K a n e n o b u b r a i d io(a)i3(sa) E B§. F i g u r e 5.7: T h e c l o s u r e o f a K a n e n o b u b r a i d Theorem 5.6. Let 3 be a Kanenobu braid. The evaluation matrix X associ-ated with the link \8\ is an element of ¥0.(02). P R O O F . W e n e e d t o c o m p u t e X = ( i f (0 ,0 ) ) (K (0,oo)>-(K(oo,0.)). (K(00,00)) Chapter 5. Kanenobu Knots 52 w h e r e i f ( T , U) = \8\ for s o m e K a n e n o b u b r a i d 8. S i n c e p(sa) = a 1 , t h e l i n k i f (oo, oo) is o f t h e f o r m IhHID w h i c h is s i m p l y t h e (usua l ) b r a i d c l o s u r e a a - 1 . H e n c e ( i f ( o o , o o ) ) = ( a a - 1 ) = 52 s ince t h e g r o u p o p e r a t i o n OiO~l c o r e s p o n d s t o t h e R e i d e m e i s t e r m o v e R 2 , so t h a t t h e K a u f f m a n b r a c k e t is u n c h a n g e d . S i m i l a r l y , t h e l i n k K(oo,0) ( equ iva -l e n t l y , i f (0, oo)) is o f t h e f o r m w h i c h r educes , c a n c e l i n g a 1a, t o the l i n k m so t h a t ( i f ( o o , 0 ) ) = ( i f (0, oo)) = 6. F i n a l l y , le t t h e p o l y n o m i a l ( i f (0 ,0 ) ) = x so t h a t . X x S d 62 is a n e l emen t o f Fix (<72). • Chapter 5. Kanenobu Knots 53 5.6 More Examples F o r t h e f o l l o w i n g e x a m p l e s , we i n t r o d u c e t h e s h o r t h a n d ^ ( a ^ a ^ 1 ) r e f e r r i n g t o t h e k n o t o b t a i n e d f r o m t h e a c t i o n W e c o n t i n u e n u m b e r i n g o f e x a m p l e s f r o m s e c t i o n 5.4. 3. T a k i n g t h e b r a i d o\o2 1ai E B$ F i g u r e 5.8: T h e b r a i d a\a2 la\ we c a n f o r m t h e K a n e n o b u b r a i d (cr3<72 lo~i)(o~5 3 0 4 c r 5 1 ) € F i g u r e 5.9: T h e K a n e n o b u b r a i d (of<7 2 1Oi)(a5 3 0 4 O 5 1 ) a n d t h e g e n e r a l i z e d K a n e n o b u k n o t j r ( T , ^ ) = = | ( a ? a 2 - V ) ( < T 5 - 3 o - 4 ( 7 5 - 1 ) | K(T, U) has e v a l u a t i o n m a t r i x K = x S 5 S2 w h e r e x = _ a - 2 0 + 2 0 -16 _ 4 o - 1 2 + 6 a - 8 _ ? a - 4 + 9 ' - 7 a 4 + 6 a 8 - 4 a 1 2 + 2 a 1 6 - a 2 0 . Chapter 5. Kanenobu Knots 54 Figure 5.10: The knot | (afa^  lai)(a^aAa^ 1) | Since w(K(T, U)) = 0 when U = T*, the Jones Polynomial is given by VK(?,u) = br{T)lCbr(U). In particular, ^ ( 0 , 0 ) = x and we have that Vul„ „ - \ \ = x. K(<T2,<r2 ) It fact i f (0,0) ~ 52 # 5 £ (following the notation in [32]). These knots are distinct and not Conway mutants, as can be seen from the HOMFLY polynomial4 of i f (0,0) ( -4r 2 + 9 - 4t2) + (-8t - 2 + 20 - 8t2)x2 + (-5t~2 + 18 - 5t 2)x 4 + ( - r 2 + 7 - t2)x6 + x8 while the HOMFLY polynomial of i f ( c^ , ^ 1 ) is (-t"4 + 3 - i4) + (-r4 + r 2 + 4 - ( 2 + t 4 - i V + ( r 2 + 2 + t2)xA "computed us ing K N O T S C A P E Chapter 5. Kanenobu Knots 55 Chapter 5. Kanenobu Knots 56 4. T a k i n g t h e b r a i d o\o2 3 c i E B$ F i g u r e 5.13: T h e b r a i d o\o2 3oi we c a n f o r m t h e K a n e n o b u b r a i d [p\a2 3a\)(a5 2cr3a5 1) E B§ F i g u r e 5.14: T h e K a n e n o b u b r a i d {a\a2 3a\)(a5 2cr3a5 1) a n d t h e g e n e r a l i z e d K a n e n o b u k n o t K(T,U) = | (aJV^)K^4 3 0| • K(T, U) has e v a l u a t i o n m a t r i x K = x 8 6 82 w h e r e x =a-2i - 2a~20 + 4 a - 1 6 - 7 a ~ 1 2 + 9 a ~ 8 - 1 1 a " 4 + 13 - 1 1 a 4 + 9 a 8 - 7 a 1 2 + 4 a 1 6 - 2 a 2 0 + a 2 4 . N o t e t h a t i f (0 ,0 ) ~ 6i # 6f ( f o l l o w i n g t h e n o t a t i o n i n [32]). S i n c e w(K(T: U)) — 0 w h e n U — T*, t h e J o n e s P o l y n o m i a l i s g i v e n b y VK(T,u) = br(T)lCbr(U). I n p a r t i c u l a r , a n d we have t h a t 'VK-(O,O) = X K ( < T 2 , c r 2 ) Chapter 5. Kanenobu Knots 57 F i g u r e 5.15: T h e k n o t ^ala^ (TX){a^2 alo^)\ O n c e a g a i n we o b t a i n d i s t i n c t k n o t s . T h e H O M F L Y p o l y n o m i a l 5 o f i f ( 0 ,0 ) i s ( 4 r 2 - 7 + At2) + ( 1 6 i - 2 - 36 + m2)x2 + (I7t~2 - 50 + 17t2)x4 + (7t~2 - 31 + 7t2)xe + ( i - 2 _ 1 + t 2 ) a . 8 _ a . 1 0 W h i l e t h e H O M F L Y p o l y n o m i a l o f K(a2, a^1) is ( t - 6 _ t - 4 _ t-2 + 3 _ t2 _ t 4 + i 6 ) + (_2r4 - r 2 + 2 - 1 2 - 2 * V + (r2 + 2 + * V O n c e a g a i n we have d i s t i n c t k n o t s t h a t have t h e s a m e J o n e s p o l y n o m i a l b u t are n o t r e l a t e d b y C o n w a y m u t a t i o n . 5 computed. .us ing K N O T S C A P E -Chapter 5. Kanenobu Knots 58 Chapter 5. Kanenobu Knots 59 5.7 Observations Definition 5.7. If a link is equivalent to a 3-braid, closed as in figure 5.18, it is called a 2 - b r i d g e link. F i g u r e 5.18: T h e 2 -b r idge l i n k o b t a i n e d f r o m 6 £ B$. F o r a g e n e r a l i s e d K a n e n o b u k n o t K(T,U), t h e k n o t i f (0 ,0 ) is a l w a y s o f t h e f o r m w h e r e L is a 2 -b r idge l i n k . I n t h e case w h e r e i f (0 ,0 ) ~ i f # i f * is a c o n n e c t e d s u m o f 2 - b r i d g e k n o t s w i t h m o r e t h a t 3 c ross ings , s u c h a i f is g e n e r a t e d b y t a k i n g t h e 2 - b r i d g e c l o s u r e o f a n e l emen t a € B%. S u c h a b r a i d genera tes t h e K a n e n o b u b r a i d io(a)i2(sa), a n d t a k i n g t h e c l o s u r e \i0(a)i2{sa)\=K(T,U) w i t h U = T* g ives r i se t o t h e e v a l u a t i o n m a t r i x .IR_\(K#K*) 6' *- ~ [ 8 8\ s ince K#K* = i f ( 0 , 0 ) . N o w K. £ F i x ( c r 2 ) , a n d w i t h t h e s p e c i f i c a t i o n t h a t U = T*, t h e f a m i l l y o f k n o t s i f ( T * 2 , ^ 1 ) sha re t h e c o m m o n J o n e s p o l y n o m i a l V AN -N = (K#K*). B y r e c y c l i n g t h e a r g u m e n t o f t h e o r e m 5.3, we c a n r e d u c e t h e c o m p a r i s o n o f t h e k n o t s K{T,U) a n d i f (T°\UA^ t o t h e c o m p a r i s o n o f t h e H O M F L Y p o l y n o m i a l s P K m a n d PK(nia-iy Chapter 5. Kanenobu Knots 60 A s s h o w n b y t h e p r e v i o u s e x a m p l e s , t h i s genera tes fu r t he r p a i r s o f d i s t i n c t k n o t s t h a t a re n o t C o n w a y m u t a n t s d e s p i t e s h a r i n g t h e s a m e J o n e s p o l y n o m i a l . T h e n o t a b l e e x c e p t i o n is t h e squa re k n o t , o b t a i n e d f r o m t h e c o n n e c t e d s u m o f t r e f o i l k n o t s 3 i # 3*. T h i s is t h e c o n n e c t e d s u m o f 2 - b r i d g e k n o t s . I t c a n b e seen as i f (0 ,0 ) i n t h e c l o s u r e K(T,U) = \(a1a^1a1)(a^a^1)\ b u t a n o t h e r v i e w is g i v e n i n figure 5.19. @> F i g u r e 5.19: T h e squa re k n o t 3 i # 3 * . F r o m t h e d i a g r a m i n figure 5.20 i t c a n b e seen t h a t t h e a c t i o n o f o2 c a n c a n c e l a l o n g a b a n d c o n n e c t i n g t h e t ang les . F i g u r e 5.20: T h e k n o t \{oio2 1ai){o^1aia^l)\. T h i s c a n c e l a t i o n i s o f t h e f o r m :xn£Dc~:©: so i n t h e case t h a t T is a r a t i o n a l t ang l e , t h e i r k n o t t y p e i s u n a l t e r e d , w h i l e a m o r e g e n e r a l t a n g l e r e su l t s i n a C o n w a y m u t a n t o f t h e o r i g i n a l d i a g r a m . I n p a r t i c u l a r , t he re i s n o change t o t h e Jones p o l y n o m i a l . I n g e n e r a l however , t h e set o f t ang les S2 t oge the r w i t h t h e set o f 2 - b r i d g e k n o t s (gene ra t ed b y Bz) p r o v i d e a r ange o f k n o t s ( a n d e v e n l i n k s ) h a v i n g e v a l u a t i o n m a t r i c i e s c o n t a i n e d i n F i x ( c r 2 ) . I n t h e cases d i s c u s s e d a n d t h e e x a m p l e s p r o d u c e d , we have seen t h a t t h e H O M F L Y p o l y n o m i a l m a y b e u s e d t o d i s t i n g u s h these k n o t s . T h u s , we c o n c l u d e t h a t t h i s m e t h o d o f p r o d u c i n g Chapter 5. Kanenobu Knots 61 f a m i l l i e s k n o t s s h a r i n g a c o m m o n J o n e s p o l y n o m i a l i s d i s t i n c t from C o n w a y m u t a t i o n . I n t h e n e x t c h a p t e r , t h e r e su l t s o f E l i a h o u , K a u f f m a n a n d T h i s t l e t h w a i t e [9] w i l l b e r e s t a t e d u s i n g t h e b r a i d a c t i o n s i n t r o d u c e d -in t h i s p a p e r . Chapter 6 T h i s t l e t h w a i t e L i n k s 6.1 Construction T h e g r o u p a c t i o n o f b r a i d s o n t ang les p r e sen t ed i n t h i s w o r k was o r i g i n a l l y d i s c u s s e d b y E l i a h o u , K a u f f m a n a n d T h i s t l e t h w a i t e [9] i n t h e cou r se o f s t u d y o f t h e r e c e n t l y d i s c o v e r e d l i n k s d u e t o T h i s t l e t h w a i t e [33]. W h i l e i t is s t i l l u n k n o w n w h e t h e r t he re is a n o n t r i v i a l k n o t h a v i n g J o n e s p o l y n o m i a l V = 1, T h i s t l e t h w a i t e ' s e x a m p l e s a l l o w us t o a n s w e r t h e q u e s t i o n for l i n k s h a v i n g m o r e t h a n 1 c o m p o n e n t . Theorem (Thislethwaite). For n > 1 there are non trivial n-component links having trivial Jones polynomial V = 5n~l. I n t h e e x p l o r a t i o n o f these l i n k s [9], i t i s s h o w n t h a t t h i s is i n fact a c o r o l l a r y o f a m u c h s t r o n g e r s t a t e m e n t . Theorem (Eliahou, Kauffman, Thislethwaite). For every n-component link L there is an infinite family of (n +1)-component links V such that Vy — SVL. W h i l e these a s se r t ions a re d i s c u s s e d a t l e n g t h i n [9], t h e g o a l o f t h i s c h a p t e r is t o p resen t s o m e o f t h e e x a m p l e s i n l i g h t o f the g r o u p a c t i o n s d i s c u s s e d i n t h i s w o r k . Definition 6 . 1 . A T h i s l e t h w a i t e l i n k H(T, U) is an external wiring of tangles T,U £ S2 modeled on the Hopf link. O u r f i rs t t a s k is t o c o m p u t e t h e e v a l u a t i o n m a t r i x \(H(0,0)) (H(0,oo))-[ ( i f ( o o , 0 ) ) " (#(00,00)) -62 Chapter 6. Thistlethwaite Links 63 Figure 6.1: The Thislethwaite link #(T, U) It is easy to see, by applications of the Reidemeister move R 2 , that (#(0,00)) = (.ff(oo,0)) = 52 and (#(oo,oo)) = 5. For the non trivial link #(0,0), the computation of (#(0,0)) requires a little more work. Figure 6.2: The link #(0,0) For this, the following switching formula (stated in [20]) will be useful. Switching Formula. The equality - > = ( ° 4 - « - 4 K < = + (a 2 - a" 2) + holds for the Kauffman bracket, giving rise to equivalence in TC±. Chapter 6. Thistlethwaite Links 64 PROOF OF T H E SWITCHING FORMULA. F i r s t no t e t h a t t h e d o u b l e c r o s s i n g m a y b e v i e w e d as t h e b r a i d 020103(72 € B4. T h u s , as G{ 1—> a + a _ 1 e i g ives a r e p r e s e n t a t i o n o f t h e b r a i d g r o u p i n TCn, we c a n represen t t h i s e l e m e n t o f Bi as ( o 2 + ei + e 2 + a _ 2 e 2 e i ) ( a 2 + e 2 4- e 3 4- a " 2 e 3 e 2 ) = a 4 + a 2 e 2 + a 2 e 3 + e 3 e 2 + a 2 e i + eie2 + e i e 3 + a~ 2 e ie 3 e2 + a2e2 + e 2 + e2e3 + a - 2e2e 3e2 + e 2ei + a _ 2e2eie2 + a T 2 e 2 e i e 3 4- a _ 4e2eie 3e2 =o 4 + o 2ei + ( 2 a ~ 2 + 5 + 2a 2 ) e 2 + a 2 e 3 + a~2t\eze<L + a _ 2e2eie3 + a - 4 e 2 e i e 3 e 2 4- eie2 4- eie3 + e^e^ + e 2 e 3 + e 3e 2. S i m i l a r l y , t h e d o u b l e c r o s s i n g m a y b e v i e w e d as ( 0 2 0 1 o ^ o ^ ) - 1 € B4 so t h a t o7l ' — • a - 1 + aei gives t h e r e p r e s e n t a t i o n ( a - 2 + ei 4- e 2 4- a 2 e 2 e 1 ) ( a " 2 4- e2 4- e3 4- a 2 e 3 e 2 ) = a " 4 4- a'2ei 4- ( 2 a ~ 2 + S + 2 a 2 ) e 2 + a ~ 2 e 3 + a 2 e i e 3 e 2 + a 2 e 2 e i e 3 + a 4 e 2 eie 3 e2 4- eie2 4- eie 3 + e2ei 4- e2e3 4- e3e2-Chapter 6. Thistlethwaite Links 65 Therefore ' x > - ( X = ( a 2 + e i -+ e 2 + a _ 2 e 2 e i ) ( a 2 + e 2 + e 3 + a ~ 2 e 3 e 2 ) - ( a - 2 + e i + e 2 + a 2 e 2 e i ) ( a ~ 2 + e 2 + e 3 + a 2 e 3 e 2 ) = a 4 - a - 4 + ( a 2 - a ~ 2 ) e i + ( a 2 - a ~ 2 ) e 3 + ( a - 2 - a 2 ) e i e 3 e 2 + ( a - 2 - a 2 ) e 2 e i e 3 + ( a - 4 - a 4 ) e 2 e i e 3 e 2 = ( a 4 - a _ 4 ) ( l - e 2 e i e 3 e 2 ) + ( a 2 - c T 2 ) ( e i + e 3 - e i e 3 e 2 - e 2 e i e 3 ) = ( « • - - ) ( ( = ) - < x ) ) + ( ^ - 0 - ) ( ( = > + < = ; as required. Applying the switching formula twice, = * 3 + ( f l 4 _ f l - 4 ) ^ , = <53 + ( a 4 - a " 4 ) • [ - ( a 4 - o - 4 ) ( 5 - <53) - ( a 2 - a - 2 ) ( 2 - 2 5 2 ) ] = 63 + ( a 4 - a-A)(S2 - 1) [<5(a4 - a ~ 4 ) + 2 ( a 2 - a ~ 2 ) ] = 83 + ( a 4 - eT 4 )(<5 2 - 1) [ a " 6 - a " 2 + a 2 - a 6 ] . a - 1 4 _ a - 6 _ 2 a - 2 _ 2 a 2 _ a 6 _ a 1 4 so the evaluation matrix is U = - a - - a " 6 - 2 a " 2 - 2 a 2 - a 6 - a 1 4 <52 Now.consider the-braid-a; = cr 2av l.a 2-. Chapter 6. Thistlethwaite Links 66 F i g u r e 6.3: T h e b r a i d w € B3 g i v i n g r i se t o t h e m a t r i c e s $ ( w ) = ^ ( u r 1 ) = a - 1 + a 3 - a 7 a 7 + a 3 - a - a " " 1 1 + 2 a " 7 - 2 a " 3 + 2 a - a 5 a - i 3 _ a - 9 + a - 5 - a " 5 + 2 a - 1 - 2 a 3 + 2 a 7 - a 1 1 a 5 - a 9 + a 1 3 I t c a n b e c h e c k e d ( u s i n g M A P L E , for e x a m p l e ) t h a t $(u)H$(u-l)r =H so t h a t 7i € F i x ( w ) , g i v i n g r i se t o a c o m u t a t i v e d i a g r a m <S2 x «S2 Z [ a , a _ 1 ] S2 x S2 a n d (H(T,U)) = ( f l ' ( T w , C r )). A s a first e x a m p l e , c o n s i d e r t h e t ang le s T = ( 1 ( 7 7.1 a n d U = f o r m i n g a n u n l i n k H(T, U). U n d e r t h e a c t i o n o f w , w e have @ ) -Chapter 6. Thistlethwaite Links 67 T h e r e f o r e , t h e l i n k H(TW,UW 1) is t w o l i n k e d t re fo i l s a n d , d e p e n d i n g o n o r i -e n t a t i o n , w(Jf(rw,[/w~ 1)) = ± 8 F i g u r e 6.4: T h e l i n k H(Tw, U" l) so t h a t (H{T",UU~1)) = 5 a n d • j , ± 2 4 r VH(T",UU~1) ~ ~ U H o w e v e r , t h e a c t i o n o f w 2 leaves t h e w r i t h e u n c h a n g e d . T h i s g ives r i se t o a f a m i l y o f 2 - c o m p o n e n t T h i s l t l e t h w a i t e l i n k s , a l l h a v i n g J o n e s p o l y n o m i a l 5. T a k i n g t a n g l e s T , U as i n t h e p r e v i o u s e x a m p l e , t h e l i n k s H(T"2n,U"~2n) have J o n e s p o l y n o m i a l 6 for a l l n 6 Z . M o r e o v e r , for n ^ 0 t h e l i n k s o b t a i n e d a re n o n - t r i v i a l , s i nce e a c h c o m p o n e n t is t h e n u m e r a t o r c l o s u r e o f a t a n g l e , g i v i n g r i se t o a p a i r o f 2 -b r idge l i n k s t h a t a re g e o m e t r i c a l l y e s sen t i a l t o a p a i r o f l i n k e d s o l i d t o r i [32]. T h e f i rs t t w o l i n k s i n t h i s sequence (for n — 1,2) a re s h o w n i n figures 6.5 a n d 6.6. Chapter 6. Thistlethwaite Links 68 r<xxxx^pooocpocpo F i g u r e 6.6: T h e r e su l t o f w 4 a c t i n g o n H(T, U) Chapter 6. Thistlethwaite Links: 69 6.2 Some 2-component examples Thistlethwaite's original discovery [33] consisted of links that had fewer cross-ings than those of the infinite sequence constructed above. Starting with the pair of tangles we obtain a trivial link H(T,U) such that w(H(T,U)) = —3. Applying the action of u to this link gives rise to a non-trivial link Figure 6.7: A non-trivial, 2-component link such that (H(T,U)) = (H{T',UU~)) and w(H(TJ,U"J~1)) = - 3 , The result is a non-trivial link with trivial Jones polynomial 6. Similarly, starting with the pair of tangles (T,f/) = gives rise to another trivial link H(T, U), in this case having w(H(T, U)) = —1. Applying the action of CJ to this link gives rise to a non-trivial link such that (H(T,U)) = (H(T",U"~1)) and w ^ r , ^ " 1 ) ) = - l . Chapter 6. Thistlethwaite Links 70 F i g u r e 6.8: A n o n - t r i v i a l , 2 - c o m p o n e n t l i n k A g a i n , t h e r e s u l t is a n o n - t r i v i a l l i n k w i t h t r i v i a l J o n e s p o l y n o m i a l 8. I t has b e e n s h o w n t h a t these e x a m p l e s are a l so m e m b e r s o f a n i n f i n i t e f a m i l y o f d i s t i n c t 2 - c o m p o n e n t l i n k s h a v i n g t r i v i a l J o n e s p o l y n o m i a l s [9]. 6.3 A 3-component example It is p o s s i b l e t o c o n s t r u c t a 16-cross ings n o n - t r i v i a l l i n k w i t h t r i v i a l J o n e s p o l y n o m i a l i f w e c o n s i d e r l i n k s o f 3 c o m p o n e n t s . S t a r t i n g w i t h t h e p a i r o f t ang le s g ives a 3 c o m p o n e n t t r i v i a l l i n k H(T, U). I n t h i s case, w(H(T, U)) — —2 a n d a p p l y i n g t h e a c t i o n o f u, t h e o r i e n t a t i o n o f t h e r e s u l t i n g l i n k m a y b e c h o s e n so t h a t a l so . T h u s , w i t h t h i s o r i e n t a t i o n , V — /S2 VH(TU,UU'1) ~ 0 • I n fact , w i t h o r i e n t a t i o n s chosen a p p r o p r i a t e l y , t h i s cho ice o f t ang le s p r o d u c e s a n o t h e r i n f i n i t e f a m i l y o f l i n k s for n € Z , e ach h a v i n g t r i v i a l Jones p o l y n o m i a l [9]. T h e 16 -c ros s ing e x a m p l e is i n t e r e s t i n g , as i t is m a y b e c o n s t r u c t e d b y l i n k i n g t w o s i m p l e l i n k s : t h e W h i t e h e a d l i n k ( 5 2 ) , a n d t h e t r e f o i l k n o t ( 3 i ) . H{Twn,U"J~n) Chapter 6. Thistlethwaite Links 71 F i g u r e 6.9: A n o n - t r i v i a l , 3 - c o m p o n e n t l i n k 6.4 Closing Remarks W h i l e t h e s e a r c h for a n a n s w e r t o q u e s t i o n 3.2 c o n t i n u e s , t h e m e t h o d o f m u t a -t i o n d e v e l o p e d i n t h i s w o r k p r o v i d e s a n e w t o o l i n t h e p u r s u i t o f a n e x a m p l e o f a n o n - t r i v i a l k n o t h a v i n g t r i v i a l Jones p o l y n o m i a l . N o t o n l y has t h i s t y p e o f m u t a t i o n p r o d u c e d T h i s t l e t h w a i t e ' s e x a m p l e s , i t i s a l so a b l e t o p r o d u c e p a i r s o f d i s t i n c t k n o t s s h a r i n g a c o m m o n Jones p o l y n o m i a l t h a t are n o t r e l a t e d b y C o n w a y m u t a t i o n ( t h e o r e m 5.3) . I n l i g h t o f t h e fact t h a t C o n w a y m u t a t i o n c a n n o t a l t e r a n u n k n o t so t h a t i t i s k n o t t e d , i t i s d e s i r a b l e t o have m o r e g e n e r a l f o r m s o f m u t a t i o n s u c h as t h i s b r a i d a c t i o n at o u r d i s p o s a l . W e have p r o d u c e d p a i r s o f k n o t s s h a r i n g a c o m m o n J o n e s p o l y n o m i a l . A s these e x a m p l e s c a n b e d i s t i n g u i s h e d b y t h e i r H O M F L Y p o l y n o m i a l s , t h e y c a n n o t b e C o n w a y m u t a n t s . I n o u r d e v e l o p m e n t , i t is s h o w n t h a t f u r t h e r s u c h e x a m p l e s m a y b e o b t a i n e d e i t h e r b y a l t e r i n g the cho ice o f t ang les m a d e , o r b y f o r m i n g a s p e c i a l c l o s u r e |/?| o f a K a n e n o b u b r a i d /? € B§. I n a d d i t i o n , i t i s s h o w n t h a t s u c h a 0 m a y b e p r o d u c e d f r o m a n y g i v e n 3 - b r a i d . I t i s h o p e d t h a t f u r t h e r s t u d y o f t h i s n e w f o r m o f m u t a t i o n w i l l l e a d t o a b e t t e r u n d e r s t a n d i n g o f t h e p h e n o m e n o n o f d i s t i n c t k n o t s s h a r i n g a c o m m o n J o n e s p o l y n o m i a l . A s w e l l , i t i s p o s s i b l e t h a t a b e t t e r g e o m e t r i c u n d e r s t a n d i n g o f t h i s b r a i d a c t i o n c o u l d g ive r i se t o a b e t t e r u n d e r s t a n d i n g o f t h e J o n e s p o l y n o m i a l i t se l f . B i b l i o g r a p h y J . W . A l e x a n d e r . A l e m m a o n s y s t e m s o f k n o t t e d c u r v e s . Proc. Nat. Acad. Sci. USA, 9 :93 -95 , 1923. J . W . A l e x a n d e r . T o p o l o g i c a l i n v a r i a n t s o f k n o t s a n d l i n k s . TYans. AMS, 3 0 ( 2 ) : 2 7 5 - 3 0 6 , 1928. ' E m i l A r t i n . T h e o r i e de r Zopfe . Abh. Math. Sem. Univ. Hamburg, 4 : 4 7 - 7 2 , 1925. E m i l A r t i n . T h e o r y o f b r a i d s . Ann. of Math., 4 8 : 1 0 1 - 1 2 6 , 1947. D r o r B a r - N a t a n . T h e k n o t a t l a s , w w w . m a t h . t o r o n t o . e d u / ~ d r o r b n . J o a n S. B i r m a n . Braids, Links and Mapping Class Groups. N u m b e r 82 i n A n n a l s o f M a t h e m a t i c s S t u d i e s . P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1974. W . B u r a u . U b e r Z o p f g r u p p e n u n d g l e i c h s i n n i g v e r d r i l l t e V e r k e t t u n g e r . Abh. Math. Sem. Hanischen Univ., 11 :171-178 , 1936. J . H . C o n w a y . A n e n u m e r a t i o n o f k n o t s a n d l i n k s . I n J . L e e c h , e d i t o r , Computational problems in abstract algebra, pages 3 2 9 - 3 5 8 . P e r g a m o n P r e s s , 1970. S h a l o m E l i a h o u , L o u i s K a u f f m a n , a n d M o r w e n T h i s t l e t h w a i t e . I n f i n i t e f a m i l i e s o f l i n k s w i t h t r i v i a l Jones p o l y n o m i a l , p r e p r i n t . R o g e r F e n n . A n e l e m e n t a r y i n t r o d u c t i o n t o t h e t h e o r y o f b r a i d s . N o t e s b y B e r n d G e m e i n . R o g e r F e n n . H e c k e a lgebras , 2004. N o t e s f r o m ' K n o t s i n V a n c o u v e r ' . P . F r e y d , D . Y e t t e r , J . H o s t e , W . L i c k o r i s h , K . M i l l e t , a n d A . O c n e a n u . A n e w p o l y n o m i a l i n v a r i a n t o f k n o t s a n d l i n k s . Bull. AMS, 12 :183 -312 , 1985. [13] B . H a r t l e y a n d T . 0 . H a w k e s . Rings, Modules and Linear Algebra. C h a p -m a n a n d H a l l M a t h e m a t i c s Ser ies . C h a p m a n a n d H a l l , 1971 . 72 Bibliography 73 J i m H o s t e a n d J o s e f H . P r z y t y c k i . T a n g l e surger ies w h i c h p rese rve Jones -t y p e p o l y n o m i a l s . Int. J. of Math., 8 (8 ) :1015-1027 , 1997. V a u g h a n F . R . J o n e s . A p o l y n o m i a l i n v a r i a n t o f k n o t s v i a v o n N e u m a n n a lgeb ra s . Bui. Amer. Math. 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