Braid Group Representations

BRAID GROUP REPRESENTATIONS

A Thesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Science in the Graduate School of the Ohio State University

By Craig H. Jackson, B.S. *****

The Ohio State University 2001

Approved by

Master’s Examination Committee: Dr. Thomas Kerler, Advisor Advisor Dr. Henry Glover Department Of Mathematics

ABSTRACT

It is the purpose of this paper to discuss representations of the braid groups and some of the contexts in which they arise. In particular, we concentrate most of our attention on the Burau representation and the Krammer representation, the latter of which was recently shown to be faithful ([3], [14]). We deﬁne these representations as actions of Bn on the homology of certain covering spaces. Explicit matrices for these representations are then calculated. We also develop the theory of braided bialgebras and show how representations of the braid groups arise in this context in a systematic way. It is proved that both the Burau representation and the Krammer representation are summands of representations obtained from the quantum algebra Uq(sl2). Calling attention to the important connection between braids and links, we use geometrical methods to show that the Burau matrix for a braid can be used to ﬁnd a presentation matrix for the Alexander module of its closure. Hence, the Alexander polynomial of a closed braid is shown to arise from the Burau matrix of the braid.

In the ﬁnal chapter we consider representations of Bn arising from the Hecke algebra. This is put into a tangle theoretic context which allows for the extension of these representations to string links. It is also shown how the Conway polynomial arises from the constructions of this chapter. Preceding all of this, however, is a survey of many of the important results of braid theory. The emphasis throughout is on making the concepts and results clear and approachable. In general, geometrical methods are emphasised.

ii To Mark Burden

iii ACKNOWLEDGMENTS

I wish to thank my advisor, Thomas Kerler, for the many ideas he gave me and for his suggestion that I write this thesis. Thanks also to Tom Stacklin for giving me a copy of the osuthesis document class which I used to typeset this paper.

iv VITA

1975 ...... Born in Columbia, Missouri

1999 ...... B.S. in Mathematics, The University of Alaska, Anchorage

1999-Present ...... Graduate Teaching Associate, The Ohio State University

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Topology

v TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... ii

Acknowledgments ...... iv

Vita...... v

List of Figures ...... viii

CHAPTER PAGE

1 Introduction ...... 1

2 The Braid Groups Bn ...... 5 2.1 First Deﬁnitions ...... 5 2.2 A Presentation of Bn ...... 7 2.3 Bn and Aut(Fn)...... 8 2.4 Links ...... 11

3 Representations of Bn from Homology ...... 14 3.1 The Burau Representation ...... 14 3.2 Matrices for the Burau Representation ...... 19 3.3 The Krammer Representation ...... 23 3.4 Forks ...... 25 3.5 Matrices for the Krammer Representation ...... 27

4 The Alexander Polynomial ...... 32

4.1 Deﬁnition of ∆L(t)...... 32 ∞ 4.2 Finding a Presentation Matrix for H1(C ) ...... 34 4.3 Proof of Theorem 11 ...... 36

vi 5 Representations of Bn from Uq(sl2) ...... 38 5.1 Braided Bialgebras ...... 38 5.2 Uq(sl2) ...... 43 5.3 The Burau Representation from Uq(sl2) ...... 45 5.4 The Krammer Representation from Uq(sl2) ...... 46 5.5 The Verma Module ...... 50

6 Tangle Representations ...... 53

6.1 The Tangle Category ...... 53 6.2 An Algebra of Tangle Diagrams ...... 55 6.3 The Conway Polynomial ...... 60

Bibliography ...... 62

vii LIST OF FIGURES

FIGURE PAGE

2.1 A geometric braid on 4 strands...... 6

2.2 Generators of Bn...... 8

2.3 The space Dn and the generators of its fundamental group...... 8 2.4 The Dehn half-twist...... 9

2.5 The complement in S of a closed braid...... 12

2.6 Loops in D2n which generate its fundamental group ...... 13 3.1 Two leaves of the inﬁnite cyclic covering of X...... 16 ˜ 3.2 Calculating the action of ψrσi on H1(X) using level diagrams. . . . . 18

3.3 Two loops γ1 and γ2 in C. The point labeled s = 1/2 shows how γ1 is parametrized...... 24

3.4 A fork F and its parallel copy F 0 ...... 26

3.5 The nonstandard forks F1, F2, and F3 ...... 28

3.6 Deforming Fi−1,i+1 so that a small part of its tine edge, and also the tine edge of its parallel copy, lies in U. This allows us to write

[SFi−1,i+1 ] = [S1] + [S2] + [S3] + [S4]...... 29 4.1 The complement in S3 of a closed braid with its Seifert Surface . . . 35

6.1 A generic tangle ...... 54

6.2 (a) Showing how braids are viewed in the tangle category. (b) Showing how string links are viewed in the tangle category...... 55

viii 6.3 (a) A generic tangle diagram. Diagrams such as this one generate the space V². (b) A tangle diagram in V3...... 56

ix CHAPTER 1 INTRODUCTION

Emil Artin [1] first introduced the braid groups Bn in 1925. He also proved many of the most fundamental results concerning them. He gave a finite presentation of Bn and solved the word problem for these groups. Perhaps the longest standing open question concerning the braid groups is whether or not they are linear. That is, does there exist a faithful representation of Bn into a group of matrices over a commutative ring. Interest in this problem originated with the discovery by Burau [5] in 1935 of a nontrivial n-dimensional linear representation of Bn. This representation was long considered a candidate for faithfulness. In fact, there are simple arguments which show this representation to be faithful for n ≤ 3 ([4], Thm. 3.15). However, J. Moody [18] discovered in 1991 that the Burau representation is unfaithful for n ≥ 9. Stephen Bigelow [2] later improved this result to n ≥ 5. The faithfulness of the Burau representation in the case n = 4 is still unknown. The recent work of Daan Krammer and Stephen Bigelow, however, has given an answer to the question of the linearity of the braid groups. In [13] Krammer considered a two-variable representation of Bn first constructed by Lawrence [15] and showed it to be faithful for n = 4. Soon after, Bigelow [3] used topological techniques to prove the faithfulness of this representation for all n. Extending the results of

1 his earlier paper, Krammer [14] then gave another proof of this result using different methods. These results have sparked renewed interest in braid groups and their representations. Recent papers by Lin/Tian/Wang [17] and Silverman/Williams [19] give generalizations of the Burau representation to the semigroup Stn of string links. One might similarly hope to find extensions of Krammer’s representation to more general objects such as string links or tangles. Additionally, a particularly exciting result would be to discover a use for the Krammer representation in constructing link invariants. The Burau representation itself is closely related to the Alexander polynomial link invariant as they both may be constructed via certain infinite cyclic coverings which are themselves closely related, one being a subcover of the other. In fact, for an open book decomposition of a knot the Alexander polynomial is explicitly given by the determinant of the (reduced) Burau matrix minus the identity ([4], Thm. 3.11):

n−1 (1 + t + ··· + t )∆βˆ(t) = det(ψrβ − I) (1.1)

It is known that the Alexander polynomial separates knots with a 3-braid open book decomposition but not for more complicated knots. One of the original motivations for this thesis was to find a link invariant that is related to the Krammer representation in somewhat the same way as the Alexander polynomial is related to the Burau representation. Due to the faithfulness of the representation one would expect this to be a very powerful invariant. To this end we carefully investigate in this thesis the relation between the Alexan- der polynomial and Burau representation using both the relation between the covering spaces used to define them as well as the categorical picture of tangles. This latter approach has proven to be very effective in the definition of many other knot polynomials in recent times. Particularly, we give a proof of the relation (1.1) without using the usual Fox

2 differential calculus of fundamental groups. Instead we consider cyclic coverings of tangle complements that we glue together and compute the homology of. In the last section we consider a fibre functor on a special tangle category. This functor that we define is a representation of the tangle category to the category of complex vector spaces. We will see that the spaces associated to the objects in the tangle category are the Hecke algebras of type An. In particular we show that any representation of the braid group which arises by passing through the Hecke algebra will extend to the semigroup of string links. This result contains the extension of the Burau representation to string links given by Lin [17] as a corollary. In addition, we point out how our constructions in this chapter encompass the Conway-normalized Alexander polynomial in their definition.

The Burau and Krammer representations of Bn are closely related to the first two in a series of representations of the braid group constructed by Lawrence. These representations are given via local coefficient systems on the configuration space of m ( = 1, 2,... ) points in an n-punctured plane. These representations factor though the Hecke algebra which is known to be related to the HOMFLYPT knot polynomial. Similar local systems and their homology have later been related by Felder [7] to a quantum group action, in particular the quantum group Uq(sl2), where the E and F generators can be thought of as acting by adding or deleting points in the configuration space – i.e., paths in the local system – and in particular commute with the representation of the braid group. A result of this paper is to make this relation between the Krammer representation and Uq(sl2) more precise by a computation. Namely, after developing the general theory of braided bialgebras and the braid group representations that they induce we show with explicit computations that both the Burau and Krammer representations are equivalent to summands of one of these induced representations. Since Uq(sl2) is used to define the Jones polynomial and, more generally, tangle functors we anticipate this to be of use in constructing link invariants from the Krammer representation.

3 Note that Zinno [22] also gives a relation of the Krammer representation with a quantum representation. Only here he uses a representation associated to the Birman-

−2 3 −1 Wenzl-Murakami algebra Cn(α, l). By identifying parameters q = −α and t = α l he shows the Krammer representation to be equivalent to the simple representation of the BMW algebra associated to the Young diagram with one row of n − 2 boxes

(see [22] or [20]). Generically, Cn(α, l) is the centralizer of Uq(g) where g is a Lie algebra of type C. There are some contraints on the parameter depending on the rank of g which essentially means that all ranks – or even continuous ones – have to be considered. In our description, using sl2, we only need to consider rank 1 instead of continuous rank at the price of admitting continuous highest weights. The majority of Chapter 2 is concerned with presenting results which are basic to braid theory. In Chapter 3 we give in detail the homological deﬁnitions of the Burau and Krammer representations and calculate their matrices. Chapter 4 is concerned with the relation between the Burau representation and the Alexander polynomial. In Chapter 5 we give the calculations that relate the Burau and Krammer representations to the braid group representations induced by the quantum algebra Uq(sl2). Finally, in Chapter 6 we investigate the tangle category and the representations arising from it.

4 CHAPTER 2

THE BRAID GROUPS Bn

In this chapter we give a brief overview of the basic deﬁnitions and results of braid theory. Our exposition will mainly follow [4] leaving out some of the proofs unless they are important for later results.

2.1 First Deﬁnitions

Let D be an oriented disk in the complex plane. Let

n C = {(z1, . . . , zn) ∈ D | zi 6= zj if i 6= j}

n where D is the Cartesian product of n copies of D. The symmetric group Sn on n ˆ letters acts on C by φ(z1, . . . , zn) = (zφ(1), . . . , zφ(n)) Let C = C/Sn be the identiﬁca- tion space of C under this action. Cˆ is then the collection of all unordered n-tuples ˆ {z1, . . . , zn} of elements of D such that zi 6= zj if i 6= j. The projection C → C is a regular n-fold covering map.

ˆ ˆ Deﬁnition 1. The fundamental group π1(C) of the space C is called the braid group on n strands and is denoted by Bn.

A geometrically more intuitive picture of Bn can be obtained as follows. Choose 0 ˆ 0 0 0 0 a base pointz ˆ in C. Let z = (z1, . . . , zn) be a lift ofz ˆ to C. Any element in ˆ 0 ˆ 0 π1(C, zˆ ) is represented by a loopα ˆ :(I, {0, 1}) → (C, zˆ ) which lifts to a unique path α :(I, {0}) → (C, z0). Moreover, sinceα ˆ(1) =z ˆ0 we have α(1) = φz0 for some

φ in Sn. We may write α = (α1, . . . , αn) where each αi is a path in C connecting

5 A3

A2 A1 A4

0 0 0 0 z1 z2 z3 z4

Figure 2.1: A geometric braid on 4 strands.

0 0 zi to zφ(i). These coordinate functions αi deﬁne arcs Ai = (αi(t), t) in D × I. Since

α(t) ∈ C for all t, the arcs Ai are disjoint. The union β = A1 ∪ · · · ∪ An is called a geometric braid on n strands. See Figure 2.1. A homotopy of loopsα, ˆ αˆ0 : I → Cˆ relative toz ˆ0 lifts to a homotopy F of paths α, α0 : I → C relative to {z0, φz0}. That is, we have a continuous map F : I × I → C such that

F (t, 0) = α(t)

F (t, 1) = α0(t)

0 0 0 F (0, s) = z = (z1, . . . , zn)

0 0 0 F (1, s) = φz = (zφ(1), . . . , zφ(n))

Let β and β0 be the geometric braids deﬁned byα ˆ andα ˆ0, respectively. Then the homotopy F deﬁnes a continuous family of geometric braids βs such that β0 = β and 0 β1 = β . Any two geometric braids are said to be equivalent if such a continuous 0 family βs exists and we call the family βs an equivalence of β with β .

Any geometric braid β = A1 ∪ · · · ∪ An with Ai = (αi(t), t) gives an elementα ˆ ˆ 0 of π1(C) in an obvious way. Similarly, an equivalence of β with β gives a homotopy

6 ofα ˆ withα ˆ0. Hence, the collection of geometric braids modulo equivalence gives ˆ an alternate description of Bn where composition of loops in π1(C) corresponds to concatenation of geometric braids. Because of this, from here on we write Bn to ˆ denote both π1(C) and the group of geometric braids on n strands. Since we are presently in the business of giving definitions, we give here the definition of the semigroup Stn of n-string links. This is a definition that will be needed in Chapter 6. String links are just like braids except we allow the strands to move up and down. To be precise, an n-string link is defined, up to isotopy, to be any

0 0 collection of n arcs Ai : I → D × I such that Ai(0) = zi and Ai(1) = zφ(i) for some

φ ∈ Sn. The point is that for string links the arcs Ai may be knotted whereas they can not be in the case of braids. The multiplication in Stn is given by concatenation and. The identity element of Stn is the same as the identity of Bn.

2.2 A Presentation of Bn

Let σ1, . . . , σn−1 ∈ Bn be the braids deﬁned in Figure 2.2. It should be clear that

σ1, . . . , σn−1 generate Bn and satisfy the following relations:

σiσj = σjσi |i − j| ≥ 2 (2.1)

σiσi+1σi = σi+1σiσi+1 1 ≤ i ≤ n − 2 (2.2)

In fact, the following theorem of E. Artin shows that these generators and relations are suﬃcient to deﬁne Bn.

ˆ Theorem 2. The braid group Bn = π1(C) admits a presentation with generators

σ1, . . . , σn−1 and relations given by (2.1)–(2.2).

For more details see [4], page 18.

7 σ : ...... σ -1 : ...... i i

0 0 0 0 0 0 0 0 0 0 z1 z2 zi zi+1 zn z1 z2 zi zi+1 zn

Figure 2.2: Generators of Bn.

p p p p 1 2 i n

Figure 2.3: The space Dn and the generators of its fundamental group.

2.3 Bn and Aut(Fn)

Fix a set P of n distinct points p1, p2, . . . , pn in the interior of the unit disk D and let d0 be a point on the boundary of D. Let Dn denote the n-punctured disk D − P . Let d0 ∈ ∂Dn be a base point. The fundamental group of Dn is isomorphic to Fn, where

Fn = hx1, x2, . . . , xni is the free group on n letters. The generators x1, x2, . . . , xn correspond to loops encircling the punctures p1, p2, . . . , pn, respectively (see Figure 2.3).

Let h : Dn → Dn be a homeomorphism of Dn ﬁxing ∂Dn pointwise. This induces an auotmorphism h∗ of the fundamental group π1(Dn, d0) = Fn. Let Mn denote the

8 α α

τ i

Figure 2.4: The Dehn half-twist.

subgroup of Aut(Fn) consisting of all such automorphisms h∗ where h is a homeo- ∼ morphism of Dn fixing ∂Dn pointwise. We wish to show Bn = Mn. ¯ Let h∗ ∈ Mn. The homeomorphism h extends uniquely to a homeomorphism h ¯ of D to itself which permutes P . This map h is isotopic to 1D through an isotopy that fixes the boundary of Dn (see [4], Lemma 4.4.1). Let F : D × I → D be such an isotopy. Then for each t ∈ I, F (x, t) is a homeomorphism of D to itself such that ¯ ¯ ¯ F (x, 0) = 1D and F (x, 1) = h. Define F : D × I → D × I by F (x, t) = (F (x, t), t). The image of P × I under F¯ is a geometric braid. This gives a well defined map

Aut(Fn) → Bn. To give a map in the other direction we need to deﬁne the so called “Dehn half-twists.”

Consider a simple closed curve α enclosing the punctures pi and pi+1. Identify the region enclosed by α with the twice-punctured disk D2 = D − {1/4, −1/4}. Identify 1 the annulus A = {z ∈ D2 | 1/2 ≤ |z| ≤ 1} with S × I. The Dehn half-twist τi −πit is deﬁned as follows: τi is the identity outside of D2; τi sends (s, t) to (e s, t) for 1 (s, t) ∈ S × I; and τi is rotation by an angle of π/2 on {z ∈ D2 | |z| ≤ 1/2}. See Figure 2.4.

9 The action on Fn of (τi)∗ (which we also denote by τi) is given by

−1 τixi = xixi+1xi (2.3)

τixi+1 = xi (2.4)

τixj = xj j 6= i, i + 1. (2.5) which can easily seen by drawing a few diagrams. From these relations we see immediately that τiτi+1τi = τi+1τiτi+1 for all i = 1, 2, . . . , n − 1, and that τiτj = τjτi for

|i − j| ≥ 2. That is, the elements τ1, τ2, . . . , τn−1 satisfy the braid relations (2.1) and

(2.2). Hence, we have a homomorphism Bn → Aut(Fn) given by φi 7→ τi.

These homomorphisms Aut(Fn) → Bn and Bn → Aut(Fn) are inverse to one another. This proves the following theorem.

Theorem 3. The homomorphism φi 7→ τi gives a faithful representation of the braid group Bn as a group of automorphisms of the free group Fn = hx1, x2, . . . , xni. This homomorphism induces an isomorphism of Bn with the subgroup Mn ≤ Aut(Fn) of all automorphisms of Fn induced by homeomorphisms h : Dn → Dn which ﬁx ∂Dn pointwise.

Because of this theorem we may identify Bn with Mn by setting φi equal to τi and so consider Bn to be contained in Aut(Fn). We end this section with a theorem due to Artin.

Theorem 4. Let β be an automorphism of Fn, then β ∈ Bn ⊆ Aut(Fn) if and only if β satisﬁes the conditions

−1 βxi = wixµi wi 1 ≤ i ≤ n (2.6)

β(x1x2 . . . xn) = x1x2 . . . xn (2.7) where (µ1, . . . , µn) is a permutation of (1, 2, . . . , n) and wi is any element of Fn.

The necessity of conditions (2.6) and (2.7) follows immediately from (2.3)–(2.5). For suﬃciency see [4], page 30.

10 Let β ∈ Bn be a braid with β acting on Fn as in (2.6) – (2.7). Suppose β is induced by a homeomorphism h : Dn → Dn. As above we consider the braid β to be embedded in D×I as the image of P ×I under the homeomorphism F¯ : D×I → D×I. ¯ ¯ ¯ Recall that F is deﬁned by F (x, t) = (F (x, t), t) where F is an isotopy of h with 1D.

Let α be the path in D × I − β deﬁned by α(t) = (d0, t). We specify (d0, 0) to 0 1 be the basepoint of D × I. Let j , j : Dn → D × I − β be the maps taking Dn 0 0 to D × {0} − P × {0} and D × {1} − P × {1}, respectively. We let xi = j xi and 1 1 −1 xi = α(j xi)α where x1, x2, . . . , xn are the free generators of Fn = π1(Dn, d0) as in 1 1 1 −1 1 Figure 2.3. Consider the curve wi xµi (wi ) in D × I − β where wi is obtained from 1 0 wi by replacing xi with xi . We show that the isotopy F induces a homotopy of xi 1 1 1 −1 with wi xµi (wi ) . ¯ Viewing xi as a map xi :(I, {0, 1}) → (D, d0) take the composition F (xi × 1I ): 0 ¯ 1 I ×I → Dn ×I. At s = 0 this map is xi . At s = 1 the map is (hxi, 1) = j hxi. Hence, 0 1 this gives us a homotopy γs(t): I × I → Dn × I with γ0 = xi and γ1 = j hxi = 1 −1 j (wixµi (wi) ). However, this homotopy moves the base point along the path α. −1 Hence, taking the composition α(st)γs(t)α (st) we have the desired homotopy from ¡ ¢ 0 1 −1 −1 1 1 1 −1 xi to α j (wixµi (wi) ) α = wi xµi (wi ) .

2.4 Links

A link L is the union of k ≥ 1 disjoint circles embedded in §3. A link with k = 1 is called a knot. There is a close connection between braids and links. In fact, one of the major reasons why braid theory is so extensively studied is because of its applications to the theory of knots and links. We give one such application in this section. Another will be given in Chapter 4. ˆ Any braid β ∈ Bn can be closed oﬀ to give a link β (see Figure 2.5). Moreover, as the next theorem shows, the action of the braid β on the group Fn gives a presentation 3 ˆ of the knot group π1(S − β).

11 A T B

y0 1 i yi

α d0

0 xi

1 xi

Figure 2.5: The complement in S of a closed braid.

Theorem 5. Let β ∈ Bn be a braid with β acting on Fn as in (2.6) – (2.7). Then 3 ˆ π1(S −β) admits a presentation with generators x1, x2, . . . , xn and relations given by

−1 xi = wixµi wi i = 1, 2, . . . , n (2.8)

Proof. Let β ∈ Aut(Fn) be induced by a homeomorphism h : Dn → Dn. As in section 2.3, we consider the braid β to be embedded in D × I as the image of P × I under the homeomorphism F¯ : D × I → D × I. Let S ∼= D × I be a closed cylinder containing βˆ in its interior. Looking at Figure 2.5 we see that the space S −βˆ can be viewed as having three components. There are the caps A and B at either end, which are both homotopy equivalent to Dn and there is the middle part T , which is the part in which all the braiding occurs. Furthermore, if we cut T down the middle, separating the braided strands from the unbraided

12 p p p p 1 n n+1 2n ...... xn yn

x1 y1

Figure 2.6: Loops in D2n which generate its fundamental group

strands, we obtain a copy of D × I − β. We consider D × I − β as embedded in S − βˆ in this way.

0 1 There is a copy of D2n at either end of T . Let j , j : D2n → S be maps identifying

D2n with these two copies. Let x1, . . . , xn, y1, . . . , yn be the loops pictured in Figure 0 0 2.6 (take note of how the yi’s are labeled and oriented). As above we set xi = j xi, 1 1 −1 0 0 1 1 −1 xi = α(j xi)α , yi = j yi, and yi = α(j yi)α . The discussion after Theorem 4 0 1 1 1 −1 shows that xi is homotopic to wi xµi (wi ) inside T . 0 0 Let U = T ∪B so that S = A∪U. Then π1(A) is freely generated by x1, . . . , xn and 1 1 π1(U) is freely generated by x1, . . . , xn. The fundamental group of the intersection 0 0 0 0 A∩U is freely generated by x1, . . . , xn and y1, . . . , yn. The inclusion A∩U ⊆ A induces 0 0 0 a map on the fundamental groups given by xi , yi 7→ xi for each i = 1, 2, . . . , n. 0 Similarly, A ∩ U ⊆ U induces a map on the fundamental groups given by xi 7→ 1 1 1 −1 0 1 ˆ wi xµi (wi ) and yi 7→ xi . From the Seifert–Van Kampen theorem π1(S − β) has the ˆ ∼ 3 ˆ presentation we seek. Hence the theorem follows since π1(S − β) = π1(S − β).

13 CHAPTER 3

REPRESENTATIONS OF Bn FROM HOMOLOGY

In this chapter we allow Bn to act on certain topological spaces. Passing to homology will then give the Burau and Krammer representations of the braid group. Explicit matrices for these representations will be given.

3.1 The Burau Representation

Let D = {z ∈ C | |z| ≤ 1} be the unit disk in the complex plane. Let P =

{p1, p2, . . . , pn} be n discrete points in D and let Dn = D − P . For any loop α in

Dn based at d0 there is a unique integer φα called the total winding number for α which counts the number of times α winds around the punctures p1, p2, . . . , pn. To be more speciﬁc, if a loop α is given by Πm x²k , then φα = Σm ² . This gives a k=1 ik k=1 k homomorphism φ : π1(Dn) → Z. ˜ Let Dn be the regular covering space of Dn corresponding to the kernel of φ. ˜ Z = hti acts on Dn as a group of deck transformations. Proposition 6 will show that ˜ −1 H1(Dn) is a free Z[t, t ] module of rank n − 1.

Let β ∈ Bn be induced by a self-homeomorphism h of Dn. That is, β = h∗. Then for any loop γ in Dn we have φ(hγ) = φγ so that h will lift to a homeomorphism ˜ ˜ ˜ h of Dn to itself. Passing to homology gives us an automorphism of H1(Dn). If 0 0 −1 0 h is any other self-homeomorphism with h∗ = β then h∗ h∗ = 1 so that since ˜ ˜−1˜0 ˜ π1(Dn) → π1(Dn) is injective we have h∗ h∗ = 1 as a map on π1(Dn). Passing to ˜0 ˜ ˜ homology gives h∗ −h∗ = 0. Hence, the homomorphism ψr : Bn → GL(H1(Dn)) given

14 ˜ by ψr : β 7→ h∗ is well deﬁned. It is called the reduced Burau representation of Bn. It is said to be reduced because there is a nice n-dimensional representation of which the reduced Burau representation is an irreducilble summand. This n-dimensional representation, which we will also give a construction for, is what is usually called the (unreduced) Burau representation. 1 Before we give the matrices for the reduced representation we need to ﬁrst verify the following proposition.

˜ −1 Proposition 6. H1(Dn) is a free module of rank n − 1 over Z[t, t ].

To aid us in the proof of this proposition we ﬁrst give a geometric description of ˜ Dn which will also be of importance later when we examine the connection between the Burau representation and the Alexander polynomial.

Enlarge the punctures of Dn slightly so that we are no longer removing points from

D but rather small open disks. Call this new space X. Draw arcs A1,A2,...,An from the centers of the removed disks to the boundary of X. Cut X along these arcs so

+ − ∗ that we have two disjoint copies Ai and Ai of each arc Ai. Call this space X . Let + − ∗ ∗ hi : Ai → Ai be a homeomorphism. Take countably many copies Xj of X . For ∗ ∗ ˜ each j let gj : Xj → X be a homeomorphism. The space X is deﬁned by taking the ∗ + ∗ − ∗ −1 disjoint union of the Xj ’s and identifying Ai ⊆ Xj with Ai ⊆ Xj+1 via gj+1higj. See Figure 3.1. ˜ ∗ −1 There is a natural Z = hti action on X given by Xj 3 x 7→ gj+1gjx. This action can be thought of as shifting every point in X˜ one level upward. The orbit space X/˜ Z of this action is precisely X. Let K be the kernel of the map π1(X) → Z which takes

1Actually, the reduced representation which we have described here via homology is the transpose of what is usually refered to as the reduced Burau representation, for instance in [4] and [6]. We will not worry ourselves about this, however, since it follow by the braid relations that the transpose of any representation of Bn is also a representation. See [19] for the diﬀerences between the usual Burau representation and the one we have given here as well as a two variable representation which generalizes both.

15 * Xj+1

- - - A1 A2 An

+ + + A1 A2 An

* Xj

Figure 3.1: Two leaves of the inﬁnite cyclic covering of X.

a loop to its total winding number. Evidently the space X˜ is the regular covering space of X coresponding to K.

Proof of Proposition 6. From the construction above it is clear that the inclusion

X,→ Dn is a homotopy equivalence which induces an isomorphism of K onto ker φ. ˜ ˜ Thus, the inclusion X,→ Dn lifts to a map X → Dn which induces an isomorphism of fundamental groups and, thereby, of homology. Choosing a generator t ∈ X/K˜ ∼= ˜ −1 Dn/ker φ, the induced map on homology becomes a Z[t, t ]-module isomorphism. Hence, we need only to calculate the homology of X. S S ∗ ∗ ˜ Let A = j∈Z X2j and B = j∈Z X2j+1 as subspaces of X. The Mayer-Vietoris long exact sequence gives us the following:

˜ δ γ 0 −−−→ H1(X) −−−→ H0(A ∩ B) −−−→ H0(A) ⊕ H0(B) where the zero on the left comes from the fact that both A and B are homotopy

16 ˜ ∼ equivalent to a discrete space. Hence, H1(X) = ker γ. Now H0(A ∩ B) is isomorphic L n ∗ ∗ to i∈Z Z since for each j the intersection Xj ∩ Xj+1 is homotopy equivalent to a space of n discrete points. Let {aj,1, aj,2, . . . , aj,n}j∈Z be a Z-basis for H0(A ∩ B). It ˜ ˜ is easily seen that {aj,1 − aj,2, . . . , aj,n−1 − aj,n}j∈Z is a basis for ker γ. Let d0 ∈ X ˜ ∗ ˜ be a lift of the basepoint d0 ∈ X (say d0 ∈ X0 ). Denote by vi the element of H1(X) −1 represented by the lift of the loop xixi+1. We have δvi = a0,i − a0,i+1 and for all j ∈ Z j ˜ we have δ(t vi) = aj,i − aj,i+1. Hence, {v1, v2, . . . , vn−1} is a basis for H1(X) as a Z[t, t−1]-module.

Our statement in the above proof that δvi = a0,i−a0,i+1 deserves some elaboration. ˜ ˜ Letα ˜ be a loop in X representing a cycle in H1(X). Projectα ˜ to a loop α in X. Draw a diagram consisting of horizontal lines (one for each level of the covering space ˜ X) with the numbers 1, 2, . . . , n marked along the bottom (one for each curve Ai). ∗ ∗ As α crosses Ai going to the right, the loopα ˜ passes from Xj to Xj+1 for some j. For each such crossing draw an arrow in the ith column of our diagram going up from

th st the j to the (j + 1) line. Similarly, each time α crosses Ai going to the left draw a down arrow. See for instance Figure 3.2. As a quick examination of the deﬁnition of the connecting homomorphism δ will show, this diagram tells us the element in

th H0(A∩B) to whichα ˜ is mapped by δ. Each upward pointing arrow in the i column th st from the j line to the (j + 1) line represents the element aj,i. Similarly, downward pointing lines represent negative elements.

17 Ai-1 Ai Ai+1 ψ σ Ai-1 Ai Ai+1 r i :

i -1 i i+1

d0 d0

Ai-1 Ai Ai+1 ψ σ Ai-1 Ai Ai+1 r i :

i -1 i i+1

d0 d0

A A A ψ σ A A A i i+1 i+2 r i i i+1 i+2 : i i+1 i+2

d0 d0

˜ Figure 3.2: Calculating the action of ψrσi on H1(X) using level diagrams.

18 3.2 Matrices for the Burau Representation

With this last observation of the previous section we can easily obtain matrices for ˜ the Burau representation. As Figure 3.2 shows, the action of σi on H1(X) is given by   vj + tvj+1, j = i − 1   −tvj, j = i ψ σ (v ) = (3.1) r i j  v + v , j = i + 1  j−1 j   vj, otherwise

Hence, the matrices for the representation relative to the basis {v1, v2, . . . , vn−1} are given by   I         −t 1  1 0 0  I             ψrσ1 =  0 1  , ψrσi =  t −t 1  , ψrσn−1 =  1 0  (3.2)         I  0 0 1  t −t   I

It can be checked directly that the matrices ψrσi satisfy the braid relations (2.1)- (2.2). As mentioned above, the reduced Burau representation is a summand of an n- dimensional representation called the unreduced Burau representation. This unreduced representation arises in much the same way as the reduced representation. Con- sider Dn+1 ⊆ Dn obtained by removing a point of Dn. We have Bn ⊆ Aut(Fn+1) in an obvious way, namely, β acts on xi by (2.6) and ﬁxes xn+1. Hence, we get an action of ˜ Bn on the regular covering space Dn+1 of Dn+1 corresponding to the kernel of the total winding number map. Arguments similar to those used in the proof of Proposition 6 ˜ −1 show that the ﬁrst homology of Dn+1 is a Z[t, t ]-module with basis {u1, u2, . . . , un} −1 ˜ where ui is a lift of xixn+1. By passing the action of Bn on Dn+1 to homology we ob- ˜ ∼ −1 tain the unreduced Burau representation ψ : Bn → GL(H1(Dn+1)) = GL(n, Z[t, t ]).

19 A calculation similar to the one carried out in Figure 3.2 will give the action of σi on ˜ H1(Dn+1):   (1 − t)uj + tuj+1, j = i  ψσ u = (3.3) i j uj+1, j = i + 1   uj, otherwise

Hence, the matrices for the representation relative to the basis {u1, u2, . . . , un} are given by   I      1 − t 1    ψσi =   (3.4)  t 0    I where the term 1 − t occurs in the ith row and column.

0 0 0 0 0 Consider the collection {u1, u2, . . . , un} where un = un and ui = ui − ui+1 for ˜ i < n. This collection is another basis for H1(Dn+1). The braid action relative to this basis can be calculated using (3.3) as follows:

0 ψσi(ui−1) = ψσi(ui−1 − ui) = ui−1 − ((1 − t)ui + tui+1) (3.5) 0 0 = ui−1 − ui + t(ui − ui+1) = ui−1 + tui

0 ψσi(ui) = ψσi(ui − ui+1) = (1 − t)ui + tui+1 − ui (3.6) 0 = tui+1 − tui = −tui

0 ψσi(ui+1) = ψσi(ui+1 − ui+2) = ui − ui+2 (3.7) 0 0 = ui − ui+1 + ui+1 − ui+2 = ui + ui+1

0 0 ψσi(uj) = uj forj 6= i − 1, i, i + 1 (3.8)

0 0 0 Notice that we have ψσn−1un = un−1 + un even though the calculation in (3.7) 0 does not apply in this case. Indeed, ψσn−1un = ψσn−1un = un−1 = un−1 − un + un = 0 0 un−1 + un.

20 Comparing these equations with (3.1) we see that for any braid β ∈ Bn we have   *    .   ψ β .   r  ψβ =   (3.9)  *    0 ··· 0 1 so that the reduced Burau representation is a summand of the unreduced Burau representation.

th Consider the (n−1)×n matrix Mβ obtained by deleting the n row of the Burau matrix ψβ in (3.9). To aid us in future calculations we deﬁne a map v : Bn → Ln−1 −1 th i=1 Z[t, t ] which takes a braid β ∈ Bn to the n column of the matrix Mβ. This allows us to write   ψrβ v(β) ψβ =   (3.10) 0 1 Note that v is not a homomorphism. Instead we have

v(β1β2) = vβ1 + (ψrβ1)(vβ2) (3.11)

which follows by virtue of the fact that ψ and ψr are homomorphisms. Also, (3.5) -

(3.8) give the following:   1, i = n − 1 v(σ ) = (3.12) i  0, i < n − 1

We should remark that the unreduced Burau representation arises by letting Bn ˜ act on spaces other than Dn+1. For instance, consider the 2n-punctured disk D2n. Let

π1(D2n) = F2n be generated by the loops x1, . . . , xn, yn, . . . , y1 drawn in Figure 2.6.

Notice that the yi’s are oriented diﬀerently than the xi’s and are written in decreasing

²j order. Let Σuj be any word in the generators of F2n. Deﬁne a map φ : F2n → Z

²j by φ :Σuj 7→ Σ²j. Because of the opposite orientations for the generators of F2n this map is diﬀerent from the total winding number map as we deﬁned it above. Let ˜ D2n be the regular covering space corresponding to ker φ. Let β in Bn be induced

21 by h : Dn → Dn. Then the restriction of h to D2n will lift to a homeomorphism of ˜ ˜ D2n to itself. Passing to homology gives an automorphism of H1(D2n). In this way ˜ we obtain a homomorphism Bn → H1(D2n). We wish to show this homomorphism to be equivalent to the unreduced Burau representation. As we did for the proof of ˜ Proposition 6 we ﬁrst give a geometric realization of the space D2n.

Enlarge the punctures of D2n so that we are no longer removing points from D2n but rather small open disks. Call this space Y . Draw arcs A1,...,An in Y where th st the arc Ai joins the i and the (2n − i + 1) punctures. Cut Y along these arcs to ∗ ∗ + − ˜ obtain Y . The space Y then has two copies, Ai and Ai , of each arc Ai. Deﬁne Y ∗ ∗ + ∗ to be equal to the disjoint union of countably many copies Yj of Y with Ai ⊆ Yj − ∗ ˜ identiﬁed with Ai ⊆ Yj+1. Y is a regular covering of Y with a Z = hti action given by shifting one level upward. From the comments at the beginning of the proof of ˜ ∼ ˜ −1 Proposition 6 we have H1(D2n) = H1(Y ) as Z[t, t ]-modules.

Proposition 7. The ﬁrst homology group of the space Y˜ is a free Z[t, t−1]-module of −1 rank 2n − 1 generated by ν1, . . . , νn−1, ω1, . . . , ωn−1, and ρ where νi is a lift of xixi+1, −1 −1 ˜ ωi is a lift of yiyi−1, and ρ is a lift of xnyn (all relative to d0).

Proof. Divide Y down the middle letting L denote the left side and R the right side. Then both L and R are homeomorphic to the space X∗ deﬁned in section 3.1. Hence, ˜ ˜ −1 the proof of Proposition 6 tells us that H1(L) and H1(R) are both free Z[t, t ]- ^−1 ^−1 modules of rank n − 1 generated by νi = xixi+1 and ωi = yiyi−1, respectively. The intersection L˜ ∩ R˜ is homotopy equivalent to the countably inﬁnite discrete space ` j ˜ j∈Z{t d0}. We have the following Mayer-Vietoris sequence:

˜ ˜ ˜ δ ˜ ˜ η ˜ ˜ 0 −−−→ H1(L) ⊕ H1(R) −−−→ H1(Y ) −−−→ H0(L ∩ R) −−−→ H0(L) ⊕ H0(R)

˜ ∼ ˜ ˜ ˜ ˜ Hence, if K = ker η then we have H1(Y ) = H1(L) ⊕ H1(R) ⊕ K. Since H0(L ∩ R) = L j ˜ j∈Z Zaj, where we have set aj equal to t d0, we see that K is freely generated over

22 −1 Z[t, t ] by a0 − a1. But a quick examination of the deﬁnition of the connecting homomorphism δ shows that δρ = a0 − a1. This concludes the proof.

˜ With this result the action of Bn on H1(D2n) can be computed. Bn acts trivially on y1, . . . , yn so that Bn has trivial action on ω1, . . . , ωn−1. The action of Bn on the ˜ the elements ν1, . . . , νn−1 is induced from the braid action on H1(L). This is, by deﬁnition, the action of the reduced Burau representation given by (3.1). Lastly, by an explicit calculation we have σn−1 : ρ 7→ ρ + νn−1 and σi : ρ 7→ ρ for i < n − 1. 0 This exactly corresponds to the action of ψσn−1 on un as given by (3.7). Hence, forgetting about the trivial action on ω1, . . . , ωn−1, we ﬁnd that the matrices for this representation relative to {ν1, . . . , νn−1, ρ} are precisely those of the unreduced Burau representation given by (3.10).

3.3 The Krammer Representation

Still using the notation D, P = {p1, . . . , pn}, Dn = D − P introduced in section 3.1 we construct the space C of all unordered pairs of distinct points in Dn. That is,

C = J/S2 where J = {(x, y) ∈ Dn × Dn | x 6= y} and S2 acts on J by sending (x, y) to (y, x). Points in C are denoted by {x, y}. The projection J → C, taking (x, y) to

0 {x, y}, is evidently a two-fold covering map. Let d0 and d0 be points on the boundary 0 of Dn which are very close to each other. We take c0 = {d0, d0} to be the basepoint of C.

0 Let α : I → C be a path in C based at c0. Lift this path to J relative to (d0, d0).

This lifted path must be of the form (α1, α2) where α1, α2 : I → Dn are paths in Dn.

As a matter of notation we may then write α(s) = {α1(s), α2(s)}. Note that if α is a 0 loop then {α1(0), α2(0)} = {α1(1), α2(2)} = {d0, d0}. Hence, the paths α1 and α2 in

Dn are either both loops or they may be composed with one another. For example, consider the loops γ1 and γ2 in Figure 3.3.

If α1 and α2 are any paths in Dn with α1(s) 6= α2(s) for all s, then (α1, α2) is a

23 t = 1/2

p p j t = 1/2 i

' ' d0 d0 d0 d0

Figure 3.3: Two loops γ1 and γ2 in C. The point labeled s = 1/2 shows how γ1 is parametrized.

path in J. The image of this path under J → C is a path in C denoted by {α1, α2}.

If α1 is the path having constant value u, then we write {α1, α2} = {u, α2}.

Let α : I → C represent an element of π1(C). Define maps a and b from π1(C) to Z as follows: If α1 and α2 are both closed loops then a(α) is the sum of the winding numbers of α1 and α2 around the puncture points p1, . . . , pn where we specify clockwise as the positive orientation. If α1 and α2 are not both closed loops then a(α) is the winding number of the loop α1α2 around the puncture points. The map b is 1 defined by first composing the map I 3 s 7→ (α1(s)−α2(s))/|α1(s)−α2(s)| ∈ S with the projection S1 → RP 1 to obtain a loop in RP 1. The corresponding element of 1 ∼ H1(RP ) = Z is b(α). Hence, a measures how many times the loops α1 and α2 wind around the puncture points while b measures how many times they wind around each other. Let hq, ti denote the free Abelian group generated by q and t. Define a map

a(α) −b(α) 2 φ : π1(C) → hq, ti by φ : α → q t . For example, we have φγ1 = q and 2 φγ2 = tq where γ1 and γ2 are the loops in Figure 3.3. Let C˜ → C be the regular covering of C corresponding to the kernel of φ. Choose

24 ˜ ˜ a liftc ˜0 of c0 to C. The group hq, ti acts on C as a group of deck transformations. ˜ ±1 ±1 Hence, the homology group H2(C) becomes a Z[q , t ]-module.

Let h be a self-homeomorphism of Dn ﬁxing the boundary. Then h induces a homeomorphism C → C, also denoted by h, which is given by h : {x, y} 7→ {h(x), h(y)}. This map h lifts to a homeomorphism h˜ : C˜ → C˜ which commutes with the covering transformations q and t. Hence, h˜ induces a Z[q±1, t±1]-module isomorphism ˜ ˜ ˜ h∗ : H2(C) → H2(C).

˜ ˜ Deﬁnition 8. The representation κ : Bn → Aut(H2(C)) induced by h 7→ h∗ is the called the Krammer representation.

3.4 Forks

We would like to have a nice way of representing elements of the homology module ˜ H2(C). As we will see, one way of doing this is with forks.

A fork F in Dn is an embedded tree F ⊆ D formed by three edges and four vertices d0, pi, pj, and z such that F ∩ ∂D = {d0}, F ∩ P = {pi, pj}, and z is a common vertex for all three edges. The edge H connecting d0 to z is called the handle of the fork. The other two edges, which connect z to pi and pj, respectively, are called the tines of the fork. The union T of both of the tines of F is an arc in D with endpoints pi and pj. This arc T is usually called the tine edge of the fork F . In a small neighborhood of z the handle H lies on one side of T . We orient T so that this distinguished side lies on its right. The handle H also has a distinguished

0 side determined by d0. Simultaneously pushing T and H to their distinguished sides, 0 0 0 0 leaving pi and pj ﬁxed and pushing d0 to d0, gives a parallel copy F = T ∪ H of F 0 0 0 0 (see Figure 3.4). We have T ∩ T = {pi, pj}, H ∩ H = ∅, and F ∩ ∂D = {d0}. This parallel fork F 0 has an orientation which is induced from that of F . ˜ ˜ For any fork F we deﬁne a surface ΣF in C as follows. Set ΣF = {{x, y} ∈ C | 0 x ∈ T − P and y ∈ T − P }. Clearly, ΣF is homeomorphic to an open square. Let α1

25 p p p i j k

' d0 d0

Figure 3.4: A fork F and its parallel copy F 0

0 0 0 be a path from d0 to z along H and let α2 be a path from d0 to z along H . Consider 0 ˜ the path α = {α1, α2} in C joining c0 to {z, z }. Lift α to a pathα ˜ in C relative to ˜ ˜ c˜0. We deﬁne ΣF to be the lift of ΣF to C which containsα ˜(1). The surfaces ΣF and ˜ 0 ΣF have natural orientations which are determined by the orientations of T and T . th For each i = 1, . . . , n let Ui ⊆ Dn be an ²-neighborhood about the i puncture. Sn Let U be the set of points {x, y} ∈ C such that at least one of x or y lies in i=1 Ui. Let U˜ be the pre-image of U under the covering map C˜ → C. If F is any fork, then ˜ ˜ ˜ ˜ ΣF is an open square which has a closed subsquare SF ⊆ ΣF such that ΣF − SF ⊆ U. ˜ ˜ Hence, SF represents a relative homology class [SF ] ∈ H2(C, U). A direct calculation 2 in [3] shows that (q − 1) (qt + 1)[SF ] maps to 0 under the boundary homomorphism ˜ ˜ ˜ 2 ∂ : H2(C, U) → H1(U). Hence, (q − 1) (qt + 1)[SF ] = j∗vF where j∗ is the map ˜ ˜ ˜ ˜ H2(C) → H2(C, U) induced by inclusion and vF is a homology class is H2(C). Hence, associated with any fork F is a 2-dimensional homology class vF . This homology class 2 ˜ is represented by an immersed closed surface which is identical to (q − 1) (qt + 1)ΣF outside an ²-neighborhood of the puncture points. It follow by calculations performed in [3] that the homology class vF is well deﬁned by the fork F .

Let Fi,j be the fork lying entirely in the closed lower half of D such that its tine edge has endpoints pi and pj. Such a fork Fi,j for 1 ≤ i < j ≤ n is called a standard fork and is determined uniquely up to isotopy by the points pi and pj. Let vi,j be the

26 ˜ ˜ homology class in H2(C) determined by Fi,j. It is shown in [3] that H2(C) is a free ±1 ±1 Z[q , t ]-module with basis {vi,j}1≤i

3.5 Matrices for the Krammer Representation

˜ We now concentrate on giving explicit formulas for the action of Bn on H2(C). We accomplish this in a geometric way by viewing C˜ as a path space. Namely, C˜ is the

−1 space of all paths α : I → C with α(0) = c0 where we identify α with β if αβ is in ker φ. The map C˜ → C is given by α 7→ α(1). The base point of C˜ is the map ˜ having constant value c0. The group π1(C)/ ker φ = hq, ti acts on C by composition. That is, if γ is a loop in C, then the map α 7→ γα deﬁned by composition with ˜ ˜ γ is a self-homeomorphism of C. Moreover, this action of π1(C) on C descends to −1 π1(C)/ ker φ since γ ∈ ker φ ⇒ γαα ∈ ker φ ⇒ γα = α. Hence, the action of hq, ti on C˜ is given by γα = φ(γ)α.

Let F1, F2, and F3 be the forks pictured in Figure 3.5. Thinking of each fork F as standing for its associated 2-dimensional homology class vF , the following lemma shows how to write these forks in terms of the standard forks.

Lemma 9. The forks F1, F2, and F3 pictured in Figure 3.5 can be expressed in terms of standard forks as follows (cf. [13] pp. 463-464)

2 F1 = q Fi,i+1 (3.13)

2 F2 = −tq Fi,i+1 (3.14)

2 F3 = (1 − q)Fi−1,i + (q − q)Fi,i+1 + qFi−1,i+1 (3.15)

Proof. Notice that F1 and Fi,i+1 have the same tine edge so that ΣF1 = ΣFi,i+1 . Hence

F1 and Fi,i+1 represent the same homology class up to an application of a covering ˜ transformation. Recall that points in ΣF1 are paths in C joining c0 to ΣF1 along the

27 p p p j i i+1 F : F : F : 1 2 p p 3 p p p i i+1 i -1 i i+1

d0 d0 d0

Figure 3.5: The nonstandard forks F1, F2, and F3

˜ ˜ handle of the fork. With this observation it is clear that γ1ΣFi,i+1 = ΣF1 where γ1 is 2 pictured in Figure 3.3. Since φγ1 = q equation (3.13) follows.

In the case of F2, notice that F2 and Fi,i+1 also have the same tine edge so that

ΣF2 and ΣFi,i+1 are set-wise equal to each other. However, the handles of these forks ˜ approach the tines from diﬀerent sides. Hence ΣF2 and ΣFi,i+1 , and thereby ΣF2 and ˜ ΣFi,i+1 , have diﬀerent orientations. Thus, F2 and Fi,i+1 represent the same homology class up to an application of a covering transformation and a change of orientation. ˜ ˜ It is clear that γ2ΣFi,i+1 = ΣF2 as sets where γ2 is pictured in Figure 3.3. Hence, as 2 2 φγ2 = tq , we have F2 = −tq Fi,i+1 where the minus sign accounts for the change in orientation. This establishes equation (3.14). Next we prove equation (3.15).

As above, let U ⊆ C be an ²-neighborhood of the puncture points. Let SFi−1,i+1 ⊆ ˜ ˜ ΣFi−1,i+1 be a closed subsquare of the open square ΣFi−1,i+1 representing a relative ˜ ˜ homology class [SFi−1,i+1 ] ∈ H2(C, U). We may deform Fi−1,i+1 so that a small part 0 of its tine edge, and also the tine edge of its parallel copy Fi−1,i+1, lies in U. This allows us to write [SFi−1,i+1 ] = [S1] + [S2] + [S3] + [S4] where the Sj’s are subsquares of SFi−1,i+1 labeled by quadrant according to Figure 3.6. We do the same thing to F3. 0 0 0 0 Hence we obtain [SF3 ] = [S1] + [S2] + [S3] + [S4] as relative homology classes where ˜ ˜ ˜ SF3 ⊆ ΣF3 is a closed subsquare of ΣF3 with boundary in U. 0 ˜ The images of Sj and Sj under the covering map C → C are identical for each

i-1,i+1 i,i-1 i+1,i+1

S2 S

p p p 1

i -1 i i+1

i,i

Fi-1,i+1 : Si-1,i+1 : i-1,i i+1,i

3 S4

' d0 d0 i-1,i-1 i,i-1 i+1,i-1

Figure 3.6: Deforming Fi−1,i+1 so that a small part of its tine edge, and also the tine

edge of its parallel copy, lies in U. This allows us to write [SFi−1,i+1 ] = [S1] + [S2] + [S3] + [S4].

0 j = 1, 2, 3, 4. Hence, Sj is equal to Sj for each j = 1, 2, 3, 4 up to an application of a covering transformation. The precise relations between these surfaces are given by

0 2 S1 = q S1 (3.16)

0 S3 = S3 (3.17)

0 S2 = qS2 (3.18)

0 S4 = qS4 (3.19)

All one has to do to establish this is to ﬁnd the appropriate loop γ in C to multiply by. For the ﬁrst equation, the loop in question is the γ1-like loop (see Figure 3.3) which winds around the ith puncture. For the last two equations the loops are given

0 0 0 by {d0, δ0} and {δ0, d0}, respectively. Here δ0 and δ0 are loops in Dn based at d0 and 0 d0, respectively, which wind around pi in the clockwise direction. Hence, we have

0 0 0 0 [SF3 ] = [S1] + [S2] + [S3] + [S4] (3.20) 2 = q [S1] + q[S2] + [S3] + q[S4]

29 so that 2 [SF3 ] − q[SFi−1,i+1 ] = (q − q)[S1] + (1 − q)[S3]. (3.21)

Now, S1 = SFi,i+1 and S3 = SFi−1,i . Hence, multiplying equation (3.21) by (q − 1)2(qt + 1) we deduce (3.15).

There are obvious analogues of lemma 9 for forks with handles diﬀerent from those of F1, F2, and F3. For instance, we have

−2 −1 −2 = q Fi,i+1 and = −t q Fi,i+1. (3.22)

Now that we have proved Lemma 6 we may give the matrices for the Krammer representation. ˜ Theorem 10. Let {j, k} ∩ {i, i + 1} = ∅. The action of κσi on H1(C) is given as follows (cf. [22] page 14):

(κσi)vj,k = vj,k (3.23)

(κσi)vi+1,j = vi,j (3.24)

(κσi)vj,i+1 = vj,i (3.25)

(κσi)vi,j = −tq(q − 1)vi,i+1 + (1 − q)vi,j + qvi+1,j (3.26)

2 (κσi)vi,i+1 = −tq vi,i+1 (3.27)

(κσi)vj,i = (1 − q)vj,i + qvj,i+1 + q(q − 1)vi,i+1 (3.28)

Proof. Equations (3.23)–(3.25) are obvious. Equations (3.27) and (3.28) follow directly from (3.14) and (3.15), respectively. Equation (3.26) requires a calculation:

³ i i+1 j ´ . . . σiFi,j = σi (3.29)

³ i+1 j ´ = i . . . (3.30) µ ¶ i+1 j . . . = −tq2 i (3.31)

h ³ i i+1 j ´ ³ i i+1 j ´ ³ i i+1 j ´i = −tq2 (1 − q) . . . + (q2 − q) . . . + q . . . (3.32)

2 = (1 − q)Fi,j − t(q − q)Fi,i+1 + qFi+1,j (3.33)

30 The ﬁrst and second equalities are by deﬁnition; the third follows from (3.22); the fourth follows from (3.15); and the last equality follows from (3.22).

31 CHAPTER 4 THE ALEXANDER POLYNOMIAL

The Alexander polynomial is a link invariant ﬁrst discovered by J.W. Alexander in 1928. The Burau representation is closely related to the Alexander polynomial as the following theorem shows.

Theorem 11. Let β ∈ Bn be a braid on n strands. Then

n−1 (1 + t + ··· + t )∆βˆ(t) = det(ψrβ − I) (4.1)

ˆ where ∆βˆ(t) is the reduced Alexander polynomial of the link β.

The purpose of this section is to give a proof of Theorem 11 which does not require the use of the free differential calculus developed by Fox. Rather, we will give a proof which is more in line with the manner in which we obtained the Burau representation. In the first part we briefly give the definition of the Alexander polynomial. The second part then deals with the geometric aspect of Theorem 11. Finally, in the third part we finish off the theorem with a bit of algebra.

4.1 Deﬁnition of ∆L(t)

What follows is a quick deﬁnition of the reduced Alexander polynomial of a link. For a more thorough treatment see [16], [6], or just about any other book on knot theory. Let L be a link in S3. Let Σ be a Seifert Surface for the link L. That is, Σ is a compact connected orientable 2-manifold with ∂Σ = L. Let N be a regular neighborhood of the link L. That is, N is a disjoint union of solid tori D × S1

32 embedded in S3 so that the link L is the image under the embedding of the center meridians {0} × S1. Let C be the closure of S3 − N. Cut C along Σ so that we have two disjoint copies Σ+ and Σ− of the surface Σ. Call this space C∗. Now, as we did in our construction of the inﬁnite cyclic cover X˜, we take countably many copies

∗ ∗ + ∗ − ∗ Cj of C and identify Σ ⊆ Cj with Σ ⊆ Cj+1. We denote the resulting space by C∞. Again, as with X˜, there is a natural Z = hti action on C∞ which is deﬁned by “shifting one level upward.” In fact, our construction of C∞ parallels that of X˜ almost exactly; for, as is evident, C∞ is a regular covering space of C corresponding to the kernel of the map φ : π1(C) → Z which takes a curve in C to its algebraic intersection number with the surface Σ ⊆ C.

∞ −1 We will see in section 4.2 that H1(C ) is ﬁnitely presented as a Z[t, t ]-module. −1 That is, there are free Z[t, t ]-modules F0 = hu1, . . . , uni and F1 = hv1, . . . , vmi that ﬁt into an exact sequence of the form

∞ F0 → F1 → H1(C ) → 0.

m Suppose in this presentation the image of uj in F1 is given by Σi=1ai,jvi. The ∞ m × n matrix A = (ai,j) is called a presentation matrix for H1(C ). If m ≤ n then the reduced Alexander polynomial ∆L(t) of the link L is defined to be a generator of the smallest principle ideal that contains the ideal generated by all the m × m minors of A. Hence, if A is square then ∆L(t) = det A. If A is an (n − 1) × n matrix, as it will be for us in the next section, then the reduced Alexander polynomial of L is the g.c.d. of all the (n − 1) by (n − 1) minors of A. Notice that since ∆L(t) is defined to be any generator of a certain principle ideal of Z[t, t−1] it is well defined only up to multiplication by a unit in Z[t, t−1].

33 ∞ 4.2 Finding a Presentation Matrix for H1(C )

We use the Mayer-Vietoris long exact sequence to give a presentation of the Z[t, t−1]-

∞ module H1(C ). The reader will readily notice that what is being done in this section is quite similar to what was done in the proof of Theorem 5. Notice ﬁrst that in our construction above we could have taken C to be the closure of S − N, where S ∼= D × I is a cylinder containing the fattened link N. Making this

∞ ∞ change will not affect H1(C ). Hence, we shall think of C as having been defined in this manner. Looking at Figure 4.1 we see that the space C = S − N can be viewed as having three components. There are the caps A and B at either end, which are both homotopy equivalent to the space X defined in section 3.1; and there is the middle part T , which is the part in which all the braiding occurs. Let Y be the space defined in section 3.1 obtained from the 2-disk D by removing 2n disjoint open disks. The space T has a copy of Y at each end, call them Y0 and Y1. The Seifert surface we have drawn in Figure 4.1 intersects Y0 in n disjoint arcs A1,...,An where the arc th st Ai joins the i and the (2n − i + 1) punctures. In fact, any closed braid has such a Seifert surface. It therefore follows from our construction of C∞ that the induced ˜ ˜ ˜ coverings Y0 → Y0 and Y1 → Y1 are the same as the covering Y → Y constructed in section 3.1.

0 1 Let j , j : Y → C be homeomorphisms taking Y to Y0 and Y to Y1, respectively.

Let x1, . . . , xn, y1, . . . , yn be the loops in Y drawn in Figure 2.6. Let α : I → T be a 0 path from i0(d0) to i1(d0). We specify j (d0) to be the base point of C. We deﬁne 0 0 0 0 0 xi = j xi, and yi = j yi, which are loops in Y0 based at j d0. Analogously, we let 1 1 −1 1 1 −1 xi = α(j xi)α and yi = α(j yi)α .

Suppose the braid β ∈ Aut(Fn) acts on Fn by (2.6)-(2.7). From the discussion 1 0 1 −1 which follows Theorem 4 we have that xi is homotopic to wi xµi (wi ) inside T . It 0 1 should be clear from Figure 2.5 that yi is homotopic to yi inside T .

34 A T B

Figure 4.1: The complement in S3 of a closed braid with its Seifert Surface

Let U = T ∪B. We have C∞ = A˜∪U˜ where A˜ → A and U˜ → U are the coverings ∞ ˜ 0 ∞ induced by C → C. Choose once and for all a lift d0 of j (d0) to C . The retraction of A onto X induces a retraction of the induced covering A˜ onto X˜. Hence, the inclusion X,˜ → A˜ induces an isomorphism on homology. So by Proposition ˜ −1 6 we have that H1(A) is a free Z[t, t ]-module generated by v1, . . . , vn−1 where vi 0 0 −1 ˜ ˜ is the lift of the loop xi (xi+1) relative to the base point d0. Similarly, H1(U) is a −1 0 0 0 1 1 −1 free Z[t, t ]-module generated by v1, . . . , vn−1 where vi is a lift of the loop xi (xi+1) ˜ relative to d0.

The intersection A ∩ U is homotopy equivalent to Y0 by an obvious deformation retraction. In order to simplify notation let us regard A ∩ U as actually being equal to Y0. In fact, the retraction A∩U → Y0 lifts to a retraction on the induced coverings ˜ so that the inclusion Y0 ,→ A^∩ U induces an isomorphism on homology. Applying Mayer-Vietoris gives us the following exact sequence:

˜ ζ ˜ ˜ ∞ δ ˜ H1(Y0) −−−→ H1(A) ⊕ H1(U) −−−→ H1(C ) −−−→ H0(Y0) −−−→ · · · (4.2)

˜ ˜ ˜ The map H0(Y0) → H0(A)⊕H0(U) is an injection. Hence, we have δ = 0. We have

35 ˜ ˜ −1 already remarked that H1(A) and H1(U) are free over Z[t, t ]. Similarly, Proposition ˜ −1 7 shows that H1(Y0) is also free over Z[t, t ] generated by ν1, . . . , νn−1, ω1, . . . , ωn−1, ∞ and ρ. Hence, the exact sequence in (4.2) gives a presentation of H1(C ). The map ζ is given by   0 νi 7→ (vi, (ψrβ)vi)  ζ : 0 (4.3) ωi 7→ (vi, vi)   ρ 7→ (0, v(β))

∞ so that we have the following presentation matrix for H1(C ):   In−1 In−1 0   (4.4) ψrβ In−1 v(β)

By applying a sequence of matrix moves to (4.4) (see refLic91 Theorem 6.1) we

∞ obtain the following presentation matrix for H1(C ): h i ψrβ − I v(β) (4.5)

4.3 Proof of Theorem 11

−1 ∞ (4.5) gives us an (n − 1) × n presentation matrix of the Z[t, t ]-module H1(C ). th Let Dk(t) be the k minor of this matrix. That is, Dk(t) is the determinant of the th matrix obtained by deleting the k column of [ψrβ − I|v(β)]. The g.c.d. of these polynomials is ∆βˆ(t). j−1 n−1 Let v0 = (a0, . . . , an−1) where aj = (1 + t + ··· + t )/(1 + t + ··· + t ). We think of v0 as a column vector. We wish to show that for all braids β in Bn we have

(ψrβ − I)v0 = v(β) (4.6)

An easy calculation using (3.2) and (3.12) shows that (4.6) holds for σ1, . . . , σn−1.

36 Now, suppose β1 and β2 are two braids for which (4.6) holds. That is, (ψrβ1)v0 = vβ1 + v0 and (ψrβ2)v0 = vβ2 + v0. Then we have

(ψr(β1β2) − I)v0 = (ψrβ1)(ψrβ2)v0 − v0 (4.7)

= (ψrβ1)(vβ2 + v0) − v0 (4.8)

= (ψrβ1)(vβ2) + (ψrβ1)v0 − v0 (4.9)

= (ψrβ1)(vβ2) + vβ1 = v(β1β2) (4.10) where the last equality follows from (3.11). Hence, (4.6) does indeed hold for all

β ∈ Bn so that we may write [ψrβ − I|vβ] = [ψrβ − I|(ψrβ − I)v0]. Now, for k < n the determinant of the matrix obtained by deleting the kth column of [ψrβ − I|vβ] is (up to a sign) equal to the determinant of the matrix obtained by th replacing the k column of ψrβ −I with vβ. But as vβ = (ψrβ −I)v0, this is equal to th the determinant of (ψrβ − I)Tk where Tk is the matrix which has v0 in its k column and is equal to the identity everywhere else. Hence, we have (up to a sign)

Dk(t) = det((ψrβ − I)Tk) = det(ψrβ − I)ak. (4.11)

Setting k = 1 we have det(ψ β − I) D (t) = r . (4.12) 1 1 + t + ··· + tn−1 From these last two equations it follows that for all k = 1, . . . , n we have

k+1 Dk = (1 + t + ··· + t )D1(t). (4.13)

From this it follows that D1(t) is the g.c.d. of the polynomials D1(t),...,Dn(t). That is, D1(t) = ∆βˆ(t). Hence, Theorem 11 follows immediately from equation (4.12).

37 CHAPTER 5

REPRESENTATIONS OF Bn FROM Uq(sl2)

In this chapter we show how the Burau and Krammer representations of Bn can be obtained from the quantum algebra Uq(sl2). In particular, we ﬁrst develop the basic theory of braided bialgebras and show that these algebras give representations of Bn in a natural way. Then in sections 5.3 and 5.4 we show that in the case of Uq(sl2) the Burau and Krammer representations are summands of one of these natural representations.

5.1 Braided Bialgebras

Let k be a ﬁeld. An algebra over k is a k-vector space A with two k-linear maps

µ : A ⊗k A → A and η : k → A such that

µ(µ ⊗ 1A) = µ(1A ⊗ µ) (Associativity)

µ(η ⊗ 1A) = µ(1A ⊗ η) = 1A (Identity)

where in the second equation k ⊗k A and A ⊗k k are identified with A in the standard way. The first equality shows the multiplication µ to associative while the second shows the element η(1) ∈ A to be a left and right unit for µ. That is, A has a unitary ring structure defined by aa0 = µ(a ⊗ a0) and 1 = η(1). Moreover, the map η : k → A is a homomorphism of rings.

The algebra A is commutative if µ = µτA,A where τA,A : A ⊗k A → A ⊗k A is the

38 0 0 linear isomorphism deﬁned by τA,A : a ⊗ a 7→ a ⊗ a.A morphism of algebras is a k-linear map f : A → A0 satisfying

µ0(f ⊗ f) = fµ and η0 = fη (5.1)

Let A and A0 be k-algebras. There is an k-algebra structure on the tensor product

0 0 A ⊗k A . Multiplication is given by (µ ⊗ µ )(1 ⊗ τA,A0 ⊗ 1) and the unit is given by the composition of η ⊗ η0 with the canonical isomorphism k ∼= k ⊗ k.

A coalgebra over k is a k-vector space A with two linear maps ∆ : A → A ⊗k A and ² : A → k such that

(∆ ⊗ 1A)∆ = (1A ⊗ ∆)∆ (Coassociativity)

(² ⊗ 1A)∆ = (1A ⊗ ²)∆ = 1A (Coidentity)

The map ∆ is called the comultiplication of the coalgebra. The map ² is called the counit.A morphism of coalgebras is an k-linear map f : A → A0 satisfying

∆0f = (f ⊗ f)∆ and ²0f = ² (5.2)

Furthermore, the coalgebra A is said to be cocommutative if ∆ = τA,A∆. A coalgebra structure can be put on the tensor product of two coalgebras. Namely,

0 0 we define a comultiplication on A ⊗k A by (1 ⊗ τA,A0 ⊗ 1)(∆ ⊗ ∆ ). The counit is given as the composition of the canonical isomorphism k ⊗ k ∼= k with ² ⊗ ²0. Note that the field k itself is both an algebra and a coalgebra over itself in a trivial way. This leads us to our next definition.

Deﬁnition 12. A k-module A having both algebra and coalgebra structures is called a k-bialgebra if these structures are compatible with each other in the sense that the linear maps ∆ : A → A ⊗k A and ² : A → k are in fact morphisms of algebras. Or, what is equivalent, the linear maps µ : A ⊗k A → A and η : k → A are morphisms of coalgebras.

39 A linear map f : A → A0 from one bialgebra to another is a morphism of bialgebras if it is both a morphism of algebras and a morphism of coalgebras. Let (A, µ, η, ∆, ²) be a bialgebra. Since any algebra A is a ring via µ we may consider the category A-Mod of all left A-modules. Any A-module is also a vector space over k so we can take the tensor product U ⊗k V over k of any two A-modules U 0 0 and V . This product is an A⊗k A-module via (a⊗a )(u⊗v) = au⊗a v. The coproduct allows us to equip U ⊗k V with an A-module structure by a(u ⊗ v) = ∆(a)(u ⊗ v). Furthermore, the counit equips k with an A-module structure by ax = ²(a)x. For three A-modules U, V, W we have the canonical k-linear isomorphisms

∼ (U ⊗k V ) ⊗k W = U ⊗k (V ⊗k W ) (5.3) ∼ ∼ k ⊗k V = V = V ⊗k k (5.4) which are shown to be A-linear by the coassociativity and coidentity relations for δ and ², respectively. Furthermore, if f : V → V 0 and g : W → W 0 are two A-linear

0 0 homomorphisms, then the map f ⊗k g : V ⊗k V → W ⊗k W is also A-linear. Hence, the functor ⊗k makes A-Mod into a monoidal category (for the formal deﬁnition of a monoidal category see [12]). From here on out all tensor products will be taken with respect to k so we shall omit the k subscript from the tensor product symbol.

Deﬁnition 13. Let (A, µ, η, ∆, ²) be a k-bialgebra. A commutativity constraint c in the category A-Mod is a family of isomorphisms cV,W : V ⊗ W → W ⊗ V deﬁned for all pairs of A-modules V and W such that the following diagram commutes for all A-linear maps f, g: c V ⊗ W −−−→V,W W ⊗ V     f⊗gy yg⊗f (5.5)

c 0 0 V 0 ⊗ W 0 −−−−→V ,W W 0 ⊗ V 0

40 Deﬁnition 14. Let (A, µ, η, ∆, ²) be an k-bialgebra. A braiding in the category A- Mod is a commutativity constraint c which satisﬁes the following two relations for all A-modules U, V , W :

cU⊗V,W = (cU,W ⊗ 1V )(1U ⊗ cV,W ) (5.6)

cU,V ⊗W = (1V ⊗ cU,W )(cU,V ⊗ 1W ) (5.7)

A bialgebra (A, µ, η, ∆, ²) whose category of modules has a braiding will be called a braided bialgebra. Any cocommutative bialgebra is braided, the braiding being given by the ﬂip isomorphism τ. We will see an example of a non-cocommutative braided bialgebra in the next section. For now we have the following theorem which characterizes those bialgebras which admit braidings.

Theorem 15. A k-bialgebra (A, µ, η, ∆, ²) is braided if and only if there is an invertible element R of A ⊗ A such that

−1 τA,A∆(x) = R∆(x)R for all x ∈ A (5.8)

(∆ ⊗ 1A)(R) = R13R23 (5.9)

(1A ⊗ ∆)(R) = R13R12 (5.10)

where R12 = R ⊗ 1, R23 = 1 ⊗ R, and R13 = (τA,A ⊗ 1A)(1 ⊗ R).

Proof. We will give the proof of suﬃciency. For necessity refer to [12], page 19. Let R ∈ A ⊗ A be as in the statement of the theorem. Let V and W be left

R A-modules. Deﬁne cV,W : V ⊗ W → W ⊗ V by

R cV,W (v ⊗ w) = τV,W (R(v ⊗ w)) (5.11)

R for all v ∈ V and w ∈ W . We claim that the family {cV,W }V,W is a braiding in the category A-Mod .

41 Let x be in A, then using (5.8) we have

R cV,W (x(v ⊗ w)) = τV,W (R∆(x)(v ⊗ w))

= τV,W (τA,A∆(x)R(v ⊗ w))

= ∆(x)τV,W (R(v ⊗ w))

R = x(cV,W (v ⊗ w))

R so that cV,W is A-linear. R R −1 It is clear that cV,W satisﬁes (5.5) and that it has inverse given by (cV,W ) = −1 R τW,V . Hence, we need only verify conditions (5.6) and (5.7). Using (5.9) we have:

R R (cU,W ⊗ 1V )(1U ⊗ cV,W ) = (τU,W R ⊗ 1V )(1U ⊗ τV,W R)

= (τU,W ⊗ 1V )(R ⊗ 1)(1U ⊗ τV,W )(1 ⊗ R)

= (τU,W ⊗ 1V )(1U ⊗ τV,W )[(τA,A ⊗ 1A)(1 ⊗ R)](1 ⊗ R)

= (τU⊗V,W )(∆ ⊗ 1A)(R)

R = cU⊗V,W

Using (5.10) we have:

R R (1V ⊗ cU,W )(cU,V ⊗ 1W ) = (1V ⊗ τU,W R)(τU,V R ⊗ 1W )

= (1V ⊗ τU,W )(1 ⊗ R)(τU,V ⊗ 1W )(R ⊗ 1)

= (1V ⊗ τU,W )(τU,V ⊗ 1W )[(τA,A ⊗ 1A)(1 ⊗ R)](R ⊗ 1)

= (τU,V ⊗W )(1A ⊗ ∆)(R)

R = cU,V ⊗W

An invertible element R ∈ A ⊗ A is called a universal R-matrix.

The notion of a braiding will allow us us to construct representations of Bn in a systematic way. The following theorem provides the key result we need in order to achieve this.

42 Theorem 16. Let (A, µ, η, ∆, ²) be an R-bialgebra and let c be a braiding in the category A-Mod . Then for all A-modules U, V , and W we have:

(cV,W ⊗ 1U )(1V ⊗ cU,W )(cU,V ⊗ 1W ) = (1W ⊗ cU,V )(cU,W ⊗ 1V )(1U ⊗ cV,W ) (5.12)

Proof. We have

(cV,W ⊗ 1U )(1V ⊗ cU,W )(cU,V ⊗ 1W ) = (cV,W ⊗ 1U )cU,V ⊗W

= cU,W ⊗V (1U ⊗ cU,W ⊗V )

= (1W ⊗ cU,V )(cU,W ⊗ 1V )(1U ⊗ cV,W )

The ﬁrst and last equalities follow from (5.7) while the middle equality follows from

(5.5) with f = 1U and g = cV,W .

Now if A is any braided bialgebra with braiding c (for instance, c = cR as in (5.11) where R is a universal R-matrix for A) and V is any A-module, then we may deﬁne ci = 1 ⊗ · · · ⊗ cV,V ⊗ · · · ⊗ 1, an automorphism of the n-fold tensor product ⊗n th st V , where the cV,V term occupies the i and (i + 1) places. Theorem 16 then ⊗n implies that cici+1ci = ci+1cici+1. That is, the n−1 elements c1, . . . , cn−1 ∈ Aut(V ) satisfy the braid relations (2.1) - (2.2). Hence, for each A-module V we obtain a

⊗n representation ρV : Bn → GL(V ) given by σi 7→ ci.

In the next section we study the representations of Bn given by the universal

R-matrix of the braided bialgebra Uq(sl2). In particular, we will see that both the Burau and Krammer representations may be given in this way.

5.2 Uq(sl2)

2 Let k be a ﬁeld and let q ∈ k with q 6= 0, and q 6= 1. The quantum algebra Uq(sl2) is deﬁned to be the associative algebra with 1 generated by E, F , K, and K−1 with relations given by KK−1 = K−1K = 1 (5.13)

43 KEK−1 = q2E (5.14)

KFK−1 = q−2F (5.15) K − K−1 [E,F ] = (5.16) q − q−1

The idea is that Uq(sl2) is a quantum deformation of the Lie algebra sl2 of complex

2 × 2 matrices of trace 0. This algebra sl2 is generated by the elements E, F , and

H with relations [H,E] = 2E,[H,F ] = −2F , and [E,F ] = H. To deform sl2 into H Uq(sl2) we allow inﬁnite formal sums. Setting K = q as a formal power series the relations for sl2 will then give, modulo subtleties, the relations (5.13)–(5.16). For a more thorough discussion see [11], Chapters 7 and 17.

Deﬁne a comultiplication ∆ : Uq(sl2) → Uq(sl2) ⊗ Uq(sl2) by

∆(E) = E ⊗ 1 + K ⊗ E (5.17)

∆(F ) = F ⊗ K−1 + 1 ⊗ F (5.18)

∆(K) = K ⊗ K (5.19)

−1 Also, deﬁne a counit ² : Uq(sl2) → k by ² : K,K 7→ 1, and ² : E,F 7→ 0. It is easy to check that these co-operations give Uq(sl2) a bialgebra structure. In fact, Uq(sl2) is a braided bialgebra. qn−q−n Let [n]q denote the element q−q−1 in k and let [n]q! = [n]q[n − 1]q ... [1]q. We have the following universal R-matrix for Uq(sl2) (see for instance [7] or [10]):

³X∞ 2 n ´ n(n+1) (1 − q ) n n − 1 (H⊗H) R = q 2 E ⊗ F q 2 (5.20) [n] ! n=0 q

Hence, by the results of the previous section, for any Uq(sl2)-module V we obtain ⊗n R a representation ρ : Bn → GL(V ) deﬁned by ρ : σi 7→ ci where ci = 1 ⊗ · · · ⊗ cV,V ⊗ R · · · ⊗ 1 and cV,V is given by (5.11).

Let λ 6= 0 be in k and let Vλ = hv0, v1,... i be the k-vector space generated by the

44 distinct elements {v0, v1,... }. We make Vλ into a Uq(sl2)-module by letting Uq(sl2) act on Vλ as follows:

λ−2l Kvl = q vl (5.21)

F vl = vl+1 (5.22)

Evl = [l]q[λ + 1 − l]qvl−1 (5.23)

R Let c denote the automorphism cV,V where R is given by (5.20) and V = Vλ. A direct calculation gives

− 1 λ2 c(v0 ⊗ v0) = q 2 v0 ⊗ v0 (5.24)

− 1 λ(λ−2) c(v0 ⊗ v1) = q 2 v1 ⊗ v0 (5.25) h i − 1 λ(λ−2) −λ λ c(v1 ⊗ v0) = q 2 v0 ⊗ v1 + (q − q )v1 ⊗ v0 (5.26) h i − 1 (λ−2)2 −λ λ c(v1 ⊗ v1) = q 2 v1 ⊗ v1 + (q − q )v2 ⊗ v0 (5.27) h − 1 λ(λ−4) −1 −λ+1 λ−1 c(v2 ⊗ v0) = q 2 v0 ⊗ v2 + (q + q )(q − q )v1 ⊗ v1 i (5.28) −1 λ−1 −λ+1 λ −λ + q (q − q )(q − q )v2 ⊗ v0

− 1 λ(λ−4) c(v0 ⊗ v2) = q 2 (v2 ⊗ v0) (5.29)

5.3 The Burau Representation from Uq(sl2)

⊗n ⊗n Let (V )1 be the n-dimensional subspace of V generated by {uˆ1,..., uˆn} where th uˆi = v0 ⊗ · · · ⊗ v1 ⊗ · · · ⊗ v0, the vector v1 occurring in the i position. Equations (5.24)–(5.26) imply that

− 1 λ2 (ρσi)ˆuj = q 2 uˆj for j 6= i, i + 1 (5.30)

− 1 λ(λ−2) (ρσi)ûi+1 = q 2 uî (5.31) h i − 1 λ(λ−2) −λ λ (ρσi)ûi = q 2 uî+1 + (q − q )ûi (5.32)

⊗n so that the representation ρ : Bn → GL(V ) restricts to a representation ρ1 : Bn → ⊗n GL((V )1).

45 ⊗n 1 λ2 We normalize the representation ρ1 : Bn → GL((V )1) by setting ρ1 = q 2 ρ1. Then equations (5.30) - (5.32) become

(ρ1σi)ˆuj =u ˆj for j 6= i, i + 1 (5.33)

λ (ρ1σi)ˆui+1 = q uˆi (5.34)

λ 2λ (ρ1σi)ûi = q uî+1 + (1 − q )ûi. (5.35)

⊗n Let W1 ⊆ (V )1 be the k-vector subspace generated by {ui}1≤i≤n−1 where

λ ui = q uˆi − uˆi+1. (5.36)

Then equations (5.33) - (5.35) then give the following:

(ρ1σi)uj = uj for j 6= i − 1, i, i + 1 (5.37)

λ λ 2λ λ (ρ1σi)ui+1 = q q uî − uî+2 = q uî − quî+1 + quî+1 − uî+2 = q ui + ui+1 (5.38)

λ ¡ λ 2λ ¢ λ 2λ 3λ 2λ (ρ1σi)ui = q q uî+1 + (1 − q )ûi − q uî = q uî+1 − q uî = −q ui (5.39)

λ ¡ λ 2λ ¢ (ρ1σi)ui−1 = q uî−1 − q uî+1 + (1 − q )ûi (5.40) λ λ λ λ = q uî−1 − uî + q (q uî − uî+1) = ui−1 + q ui

⊗n Hence, ρ1 : Bn → GL((V )1) restricts to a representation ρ1 : Bn → GL(W1). If we −λi now rescale the basis {ui}1≤i≤n−1 of W1 by multiplying each ui by q , then in this new basis equations (5.37)–(5.40) reduce to

(ρ1σi)uj = uj for j 6= i − 1, i, i + 1 (5.41)

(ρ1σi)ui+1 = ui + ui+1 (5.42)

2λ (ρ1σi)ui = −q ui (5.43)

2λ (ρ1σi)ui−1 = ui−1 + q )ui (5.44)

Setting t = q2λ we have the reduced Burau representation as given by (3.1).

5.4 The Krammer Representation from Uq(sl2)

For 1 ≤ i < j ≤ n letw ˆi,j = v0 ⊗ · · · ⊗ v1 ⊗ · · · ⊗ v1 ⊗ · · · ⊗ v0, where the two th th v1 terms occur in the i and j positions, respectively. For 0 ≤ i ≤ n letw ˆi =

46 th ⊗n v0 ⊗ · · · ⊗ v2 ⊗ · · · ⊗ v0, the v2 term occurring in the i position.. Let (V )2 be the ¡ ¢ n ⊗n ( 2 + n)-dimensional subspace of V generated by {wˆ1,..., wˆn, wˆ1,2,..., wˆn−1,n}. Let {j, k} ∩ {i, i + 1} = ∅, then equations (5.24)–(5.29) imply that the representation

⊗n ⊗n ρ : Bn → GL(V ) restricts to a representation ρ2 : Bn → GL((V )2) given by

− 1 λ2 (ρ2σi)w ˆj,k = q 2 wˆj,k (5.45)

− 1 λ(λ−2) (ρ2σi)w î+1,j = q 2 wî,j (5.46) h i − 1 λ(λ−2) −λ λ (ρ2σi)w î,j = q 2 wî+1,j + (q − q )w î,j (5.47) h i − 1 (λ−2)2 −λ λ (ρ2σi)w î,i+1 = q 2 wî,i+1 + (q − q )w î (5.48)

− 1 λ2 (ρ2σi)w ˆj = q 2 wˆj (5.49) h − 1 λ(λ−4) −1 −λ+1 λ−1 (ρ2σi)w î = q 2 wî+1 + (q + q )(q − q )w î,i+1 i (5.50) −1 λ−1 −λ+1 λ −λ + q (q − q )(q − q )w î

− 1 λ(λ−4) (ρ2σi)w ˆi+1 = q 2 wˆi (5.51)

For each pair of integers {i, j} with 0 ≤ i < j ≤ n we deﬁne the following elements of k:

[λ] α = qλ(i−j) (5.52) i,j [2][λ − 1] [λ] β = qλ(j−i)−2 (5.53) i,j [2][λ − 1] ¡ ¢ ⊗n n Let W2 ⊆ (V )2 be the 2 -dimensional subspace generated by {wi,j}i

wi,j =w ˆi,j − αi,jwˆj − βi,jwˆi. (5.54)

⊗n We would like to show that ρ2 : Bn → GL((V )2) restricts to a representation ρ2 :

Bn → GL(W2), and that this restricted representation is equivalent to the Krammer representation.

⊗n 1 λ2 We normalize the representation ρ2 : Bn → GL((V )2) by setting ρ2 = q 2 ρ2. 1 λ2 Also, in order to simplify our equations let us set Ωi = q 2 (ρ2σi)w ˆi as given in (5.50).

47 Specifying {j, k} ∩ {i, i + 1} = ∅, we may then calculate the action of ρ2σi on the elements wi,j ∈ W2 directly from equations (5.54) and (5.45)–(5.51):

(ρ2σi)wj,k = wj,k (5.55)

λ 2λ (ρ2σi)wi+1,j = q wˆi,j − αi+1,jwˆj − βi+1,jq wˆi (5.56)

λ 2λ (ρ2σi)wj,i+1 = q wˆj,i − αj,i+1q wî − βj,i+1wˆj (5.57) h i λ −λ λ (ρ2σi)wi,j = q wî+1,j + (q − q )w î,j − αi,jwˆj − βi,jΩi (5.58) h i 2λ−2 −λ λ 2λ (ρ2σi)wi,i+1 = q wî,i+1 + (q − q )w î − αi,i+1q wî − βi,i+1Ωi (5.59) h i λ −λ λ (ρ2σi)wj,i = q wˆj,i+1 + (q − q )w ˆj,i − αj,iΩi − βj,iwˆj (5.60)

Proposition 17. Equations (5.55)–(5.60) reduce to the following:

(ρ2σi)wj,k = wj,k (5.61)

λ (ρ2σi)wi+1,j = q wi,j (5.62)

λ (ρ2σi)wj,i+1 = q wj,i (5.63)

λ(j−i) 2λ−2 λ −λ 2λ λ (ρ2σi)wi,j = q q (q − q )wi,i+1 + (1 − q )wi,j + q wi+1,j (5.64)

4λ−2 (ρ2σi)wi,i+1 = q wi,i+1 (5.65)

2λ λ λ(j−i) 2λ λ −λ (ρ2σi)wj,i = (1 − q )wj,i + q wj,i+1 + q q (q − q )wi,i+1 (5.66)

⊗n Hence, the representation ρ2 : Bn → GL((V )2) restricts to a representation ρ2 :

Bn → GL(W2). Moreover, this representation is equivalent to the Krammer representation.

Proof. The proof of the ﬁrst part of this proposition just involves simple algebraic manipulation of equations (5.55)–(5.60) using deﬁnitions (5.52)–(5.54) and the fact

48 1 λ2 that Ωi = q 2 (ρ2σi)w ˆi. For instance, to get equation (5.64) we calculate the following:

λ£ −λ λ ¤ (ρ2σi)wi,j = q wˆi+1,j + (q − q )w ˆi,j − αi,jwˆj − βi,jΩi

λ = q (wi+1,j + αi+1,jwˆj + βi+1,jwˆi+1)

2λ + (1 − q )(wi,j + αi,jwˆj + βi,jwˆi) − αi,jwˆj

2λ −1 −λ+1 λ−1 − βi,jq (q + q )(q − q )(wi,i+1 + αi,i+1wˆi+1 + βi,i+1wˆi)

2λ 2λ −1 λ−1 −λ+1 λ −λ − βi,jq wˆi+1 − βi,jq q (q − q )(q − q )w ˆi

λ 2λ λ(j−i) 2λ−2 λ −λ = q wi+1,j + (1 − q )wi,j + q q (q − q )wi,i+1 h i λ 2λ + q αi+1,j + (1 − q )αi,j − αi,j wˆj h i λ λ(j−i) 2λ−2 λ −λ 2λ + q βi+1,j + q q (q − q )αi,i+1 − βi,jq wˆi+1 h 2λ λ(j−i) 2λ−2 λ −λ + (1 − q )βi,j + q q (q − q )βi,i+1 i 2λ −1 λ−1 −λ+1 λ −λ − βi,jq q (q − q )(q − q ) wˆi

And the coefficients ofw ˆj,w î+1, andw î are readily shown to be zero:

λ 2λ λ λ q αi+1,j + (1 − q )αi,j − αi,j = q αi,j − q αi,j = 0 (5.67)

λ λ(j−i) 2λ−2 λ −λ 2λ q βi+1,j + q q (q − q )αi,i+1 − βi,jq (5.68)

λ −λ 2λ −1 2λ = q q βi,j − q (q − q)[2][λ − 1]αi,i+1βi,j − βi,jq (5.69) h i 2λ −λ −1 2λ = βi,j 1 − q q [λ](q − q) − q (5.70) h i λ λ −λ 2λ = βi,j 1 + q (q − q ) − q = 0 (5.71)

49 2λ λ(j−i) 2λ−2 λ −λ (1 − q )βi,j + q q (q − q )βi,i+1 (5.72) 2λ −1 λ−1 −λ+1 λ −λ − βi,jq q (q − q )(q − q )

2λ 2λ −1 = (1 − q )βi,j − q (q − q)[2][λ − 1]βi,i+1βi,j (5.73) 2λ −1 λ−1 −λ+1 λ −λ − βi,jq q (q − q )(q − q ) h i 2λ 2λ −1 λ−2 3λ−2 λ λ −λ = βi,j 1 − q − q (q − q)[λ]q − (q − q )(q − q ) (5.74) h i 2λ 3λ−2 λ −λ λ 3λ−2 λ −λ = βi,j 1 − q + q (q − q ) + (q − q )(q − q ) (5.75) h i 2λ λ −λ 3λ−2 λ 3λ−2 = βi,j 1 − q + (q − q )(q + q − q ) = 0 (5.76)

The remaining equations follow in similar fashion.

−λ(i+j) If we now rescale the basis {wi,j}i

(ρ2σi)wj,k = wj,k (5.77)

(ρ2σi)wi+1,j = wi,j (5.78)

(ρ2σi)wj,i+1 = wj,i (5.79)

−2 2λ 2λ 2λ 2λ (ρ2σi)wi,j = q q (q − 1)wi,i+1 + (1 − q )wi,j + q wi+1,j (5.80)

−2 4λ (ρ2σi)wi,i+1 = q q wi,i+1 (5.81)

2λ 2λ 2λ 2λ (ρ2σi)wj,i = (1 − q )wj,i + q wj,i+1 + q (q − 1)wi,i+1 (5.82)

Setting “q = q2λ” and t = −q−2 we obtain the Krammer representation (see Theorem 10).

5.5 The Verma Module

Certain parts of the above discussion deserve elaboration. Specifically, how we came to define the vector spaces Vλ, W1, and W2. As it stands now these definitions are rather obscure, their only justification being that they eventually work out to give the representations we want. In what follows we a somewhat better explanation of how these definitions arise.

50 −1 Let U+ be the subalgebra of Uq(sl2) generated by the elements E, K, and K .

Let hv+i be the one-dimensional k-module with basis {v+}. Give hv+i the structure of a U+-module by letting

λ Ev+ = 0 and Kv+ = q v+ (5.83)

The Verma module V is the induced module V = Uq(sl2) ⊗U+ hv+i. The relations for

Uq(sl2) given in eqrefqrel1–eqrefqrel4 allow us to calculate the action of Uq(sl2) on V . Speciﬁcally, for any lgeq0 we have

l −2l l λ−2l l KF ⊗ v+ = q F K ⊗ v+ = q F ⊗ v+ (5.84) so that K − K−1 K ⊗ v − K−1 ⊗ v EF ⊗ v = (FE + ) ⊗ v = + + = [λ] (1 ⊗ v ). (5.85) + q − q−1 + q − q−1 q + By induction on l we obtain K − K−1 K − K−1 EF l ⊗ v = (FEF l−1 + F l−1) ⊗ v = FEF l−1 ⊗ v + F l−1 ⊗ v + q − q−1 + + q − q−1 + (5.86)

¡ l−2 ¢ l−1 = F [l − 1]q[λ + 1 − (l − 1)]qF ⊗ v+ + [λ − 2(l − 1)]qF ⊗ v+ (5.87)

l−1 = ([l − 1]q[λ + 1 − (l − 1)]q + [λ − 2(i − 1)]q) F ⊗ v+ (5.88)

l−1 = [l]q[λ + 1 − l]qF ⊗ v+ (5.89)

l Hence, setting vl = F ⊗ v+ we see that the Verma module Vλ is precisely the module which we gave earlier in section 5.2.

Let us write V = Vλ. We may put a Uq(sl2)-module structure on the n-fold tensor product V ⊗n by xv = ∆(n)(x)v (5.90)

⊗n (n) (2) (3) where x ∈ Uq(sl2), v ∈ V , and ∆ is deﬁned by ∆ = ∆, ∆ = (∆ ⊗ 1)∆, etc. which, by cocommutativity, is well deﬁned no matter in which order the ∆’s are applied.

51 R Let R be the universal R-matrix for Uq(sl2) given by (5.20) and let cV,V be the R automorphism of V ⊗V deﬁned by (5.11). Let ci = 1⊗· · ·⊗cV,V ⊗· · ·⊗1 as in section R 5.2. Theorem 15 shows that cV,V is Uq(sl2)-linear. Hence, each ci is Uq(sl2)-linear as ⊗n a map on V , which is a Uq(sl2)-module by (5.90). (n) ⊗n For each x ∈ Uq(sl2) the element ∆ (x) acts on V as a k-linear transformation. Consider the elements

(n) n ∆ (E) = Σk=1(K ⊗ · · · ⊗ K ⊗ E ⊗ 1 ⊗ · · · ⊗ 1) (5.91) ∆(n)(K − qnλ−2l) = K ⊗ · · · ⊗ K − qnλ−2l1 ⊗ · · · ⊗ 1. (5.92)

⊗n For each l = 0, 1,... let Wl ⊆ V be the k-submodule given by

¡ (n) ¢ ¡ (n) nλ−2l ¢ Wl = ker ∆ (E) ∩ ker ∆ (K − q ) . (5.93) ¡ ¢ The vectors in ker ∆(n)(K − qnλ−2l) are said to have weight equal to l. This

n submodule is generated by all vectors vl1 ⊗ · · · ⊗ vln with Σi=1li = l. The vectors in Wl are called highest weight vectors and Wl itself is called a highest weight space.

The linearity of ci implies that Wl is invariant under ci. Hence, the representation ρ : ⊗n Bn → Aut(V ) deﬁned by σi 7→ ci will restrict for all l = 0, 1,... to a representation

ρl : Bn → Aut(Wl). These spaces Wl are finite dimensional over k. For the cases ¡n¢ l = 1, 2 we have dim(W1) = n − 1 and dim(W2) = 2 , which are the dimensions of the Burau and Krammer representations, respectively. Hence, this gives an indication that ρ1 and ρ2 might reduce to φr and κ, respectively. And this is in fact the case ¡ ¢ ¡ ¢ (n) (n) since a direct calculation shows that ∆ (E) (ui) = 0 and ∆ (E) (wi,j) = 0, λ where ui = q uî − uî+1 and wi,j =w î,j − αi,jwˆj − βi,jwî, as is given in (5.36) and

(5.54). That is to say, the spaces W1 deﬁned in section 5.3 and W2 deﬁned in section 5.4 are precisely the highest weight spaces of this section.

52 CHAPTER 6 TANGLE REPRESENTATIONS

In this chapter we discuss categorical representations of tangles. Braid group representations, as well as representations of the semigroup Stn of n-string links, can then be obtained by specialization. In particular we will obtain a representation of Stn which generalizes the Burau representation of Bn.

6.1 The Tangle Category

Recall that a geometric braid on n strands is, up to isotopy, a collection of n arcs

Ai disjointly embedded in D × I such that Ai(t) ∈ D × {t}, Ai(0) = (zi, 0), and

Ai(1) = (zφ(i), 1) where z1, . . . zn are n ﬁxed points in D and φ ∈ Sn. That is, a braid is made up of n disjoint paths Ai which move strictly upward and connect (zi, 0) to

(zφ(i), 1). More generally, an n-string link is the same as a geometric braid except that we eliminate the condition Ai(t) ∈ D × {t}. Tangles are similar to string links in that they are made up of disjoint arcs in D×I. For tangles, however, we do not require these arcs to join D × {0} to D × {1}, but stipulate only that the endpoints of the arcs are somewhere in {z1, . . . , zm} × {0, 1}. The deﬁnition which follows is more precise.

Let {z1, z2,... } be a discrete collection of points in D. Let ² = (²1, . . . , ²2n) be a sequence of n (+1)’s and n (−1)’s in any order. We think of ² as being equal to the set {z1, . . . , z2n} where each zi is oriented by ²i. Given any two oriented sets 0 0 0 0 ² = (²1, . . . , ²2n) and ² = (²1, . . . , ²2m) we can think of ² and ² as being contained in

53 _ _ + +

_ +

Figure 6.1: A generic tangle

D × I by ² ⊆ D × {0} and ²0 ⊆ D × {1}. Then ² and ²0 can be joined by a tangle, by which we mean any collection of maps ai : I → D × I, up to isotopy, such that exactly one of the following conditions holds (see Figure 6.1):

Condition 1. The point ai(0) is either a positive point of ² or a negative point of 0 0 ² ; and the point ai(1) is either a negative point of ² or a positive point of ² .

Condition 2. The map ai has ai(0) = ai(1) and lies completely in the interior of D × I.

The category of tangles is the category T having objects the oriented sets ² =

0 (²1, . . . , ²2n) and morphisms all tangles from ² and ² . The identity morphism in

HomT (², ²) is the tangle 1² which consists of vertical lines. Note that if ² and ²0 have the same number of points then they are isomorphic as objects of T , an isomorphism between them being any “braid-like” tangle joining ² with ²0. A representation of T is deﬁned to be any functor F from T to the category k-Vect of vector spaces over the ﬁeld k.

For any n ∈ N we deﬁne ²n = (+1,..., +1, −1,..., −1) in Obj(T ). We have

Bn ⊆ AutT (²n) as shown in Figure 6.2 (a). Hence, any representation F : T → k-Vect

54 ______+ + + + + +

______+ + + + + + (a) (b)

Figure 6.2: (a) Showing how braids are viewed in the tangle category. (b) Showing how string links are viewed in the tangle category.

of the tangle category restricts to an “honest” representation Bn → Autk(F (²n)) of the braid groups.

Furthermore, the semigroup of n-string links Stn is contained in EndT (²n) as shown in Figure 6.2 (b). Hence, as for braids, a representation F : T → k-Vect of the tangle category will restrict to a representation Stn → Endk(F (²n)) of the of the semigroup of n-string links.

6.2 An Algebra of Tangle Diagrams

Let t ∈ C be a nonzero complex number. For each ² = (²1, . . . , ²2n) we deﬁne a complex vector space V² as follows. We think of ² as sitting in D ⊆ R3. Deﬁne a tangle diagram s to be any collection

3 of maps ai : I → D × (−∞, 0] ⊆ R , up to isotopy, such that exactly one of the following conditions holds (see Figure 6.3 (a)):

Condition 1’. The point ai(0) is a negative point of ² and the point ai(1) is a positive point of ².

Condition 2’. The map ai has ai(0) = ai(1) and lies completely in D × (−∞, 0).

55 ______+ + + + + +

(a) (b)

Figure 6.3: (a) A generic tangle diagram. Diagrams such as this one generate the space V². (b) A tangle diagram in V3.

Let W² be the free complex vector space generated by all tangle diagrams s. Let

R² ⊆ W² be the subspace generated by all elements of the form

− − z . (6.1)

We deﬁne V² = W²/R² as a vector space over C. Equation (6.1) relates three tangle diagrams which are identical outside a small neighborhood of a crossing. That is, if s+ is a tangle with a speciﬁed positive crossing, and if s− and s0 are the same tangle with the crossing changed to a negative and a

+ − 0 smoothed crossing, respectively, then we have s = s + zs in V².

Notice that tangle diagrams with a free component are zero in V² since by (6.1) any such diagram may be reduced to a combination of diagrams with a free circle and such diagrams are zero by

= = + z . (6.2)

0 In this context, any tangle T in HomT (², ² ) will naturally give a homomorphism

T∗ : V² → V²0 which takes any tangle diagram s in V² to the diagram T · s in V²0 obtained by stacking T on top of s. The homomorphism T∗ is well deﬁned since it preserves the relation s+ = s− + zs0.

56 With these deﬁnitions in hand we now deﬁne a functor F : T → k-Vect by

F (²) = V² and F (T ) = T∗.

We would like to know more about these vector spaces V². But as noted above, ∼ ∼ for any ² we have ² = ²n for some n. Hence, applying F we see that V² = V²n .

Denote the space V²n by Vn (an example of a tangle diagram in Vn is given in

Figure 6.3 (b)). Suppose a tangle diagram s is given by maps ai : I → D × (∞, 0] and we label the ﬁrst n maps so that ai(0) = z2n−i+1. There is then an associated element π(s) = (a1(1), . . . , an(1)) of the symmetric group Sn. Let si denote the tangle diagram with π(si) = (i i + 1) and which has no crossings other than a single positive crossing. Also, let 1 denote the diagram that has no crossings.

We deﬁne a multiplication on Vn by concatenation. For example, Ã ! Ã ! Ã ! ______+ + + _ + + + + + + _ _ · = (6.3)

This multiplication is well deﬁned since it preserves the relation s+ = s− + zs0.

Hence, Vn becomes a C-algebra with this multiplication. Moreover, it is clear that

Vn is ﬁnitely generated as an algebra over C by the elements s1, . . . , sn−1 (and their inverses).

The diagrams si satisfy the braid relations

sisj = sjsi |i − j| ≥ 2 (6.4)

sisi+1si = si+1sisi+1 1 ≤ i ≤ n − 2 (6.5) as well as the relation 2 1/2 −1/2 si = (t − t )si + 1. (6.6)

1/2 By setting gi = t si the relation in (6.6) becomes

2 gi = (t − 1)gi + t. (6.7) ∼ Hence, as a C-algebra we have Vn = Hn(t) where Hn(t) is the Hecke algebra of type An−1. That is, Hn(t) is the free complex algebra with generators g1, . . . , gn−1 satisfying the braid relations and the relation given in (6.7).

57 The Hecke algebra Hn(t) has been well studied. If t is not a root of unity it is isomorphic to CSn, the complex group algebra of the symmetric group, and its irreducible representations are indexed by Young diagrams (see [21] Theorem 2.2 and [9] Theorem 4.5; also, Cole Giller uses skein theory in [8] to give a direct proof that ∼ Vn = CSn).

Bn has a representation inside Hn(t) by σi 7→ gi. Hence, any representation of

Hn(t) will restrict to a representation of Bn. Jones [9] initiated a detailed study of the representations of Bn which arise in this way. In particular he points out that by sending gi to the matrix −ψrσi given in (3.2) one obtains an irreducible representation 2 of Hn(t). This is just a matter of verifying that (−ψrσi) = (t−1)(−ψrσi)+t. Hence, by semisimplicity there is a submodule W of Hn(t) such that the left regular action of Hn(t) on W is given by the representation gi 7→ −ψrσi.

Now the functor F : T → k-Vect deﬁned by F (²) = V² and F (T ) = T∗ is an extension to tangles of the left regular action of Hn(t) on itself. That is, any tangle ∼ diagram s ∈ Vn = Hn(t) can be straightend out and paired with n vertical lines, as in Figure 6.2, to give a tangle, i.e., a morphism in the category T . Let T be this tangle.

Then the action of T∗ on Vn is precisely the left regular action of s on Vn. Hence, by restricting to braids the functor deﬁnes a representation of the braid group on the n − 1-dimensional module W by σi 7→ (σi)∗|W . The action of (σi)∗ on −1/2 Vn is the same as the left regular action of si. We have si = t gi. Hence, the rep- −1/2 resentation is given by σi 7→ −t ψrσi. Of course, this is the Burau representation up to normalization by a constant.

Furthermore, restricting to string links gives a representation of Stn on W which generalzes the Burau representation. Matrices for this representation can be computed by using the decomposition rule s+ = s− + zs0 to express string links as linear combinations of braids. We give an outline of why this should be the same as the generalized Burau representation of string links given by Lin [17].

The map gi 7→ −1 deﬁnes a one dimensional, hence irreducible, representaion of

58 the Hecke algebra. So there exists some submodule ε of Hn(t) upon which each gi acts by −1. We can then consider the the direct sum of these representations, W ⊕ ε.

It is given by   −ψrσi 0 gi 7→   (6.8) 0 -1 so that the induced representation on Bn is given by    

−1/2 −1/2 −ψrσi 0 −1/2 ψrσi 0 σi 7→ si = t gi 7→ t   = −t   . (6.9) 0 -1 0 1

It is apparent from the equations in (3.2) that the unreduced Burau matrix, as given in (3.9), has eigenvector (1/(t+1), 1,..., 1)T with eigenvalue 1. Hence, applying a basis transformation we see that this matrix is identical to the matrix in (6.9). Hence, the matrix in (6.9) can be put into the form that the underduced Burau matrix takes in equation (3.4). We may then take the transpose of this representation which is again a representation of the Hecke algebra. The matrices of this representation are stochastic, meaning that they have eigenvector (1,..., 1)T with eigenvalue 1. We denote this representation by ψ. We may then consider the representation ψ/ε, where by ε we now mean the map

−1/2 which takes gi to −1, and therefore takes σi to −t . This is well deﬁned as a representation of string links. Moreover, the resulting matrix for a string link, say l, is a stochastic matrix since ψ(l) has eigenvector (1,..., 1)T with eigenvalue ε(l). This is similar to what occurs in Lin’s paper. As an example, consider the string link l given by

l = . (6.10)

This link decomposes as l = 1 − zσ1. Hence, the matrix for l is given as follows

59 (cf. [17], page 295):     1 0 1 − t t ψ(l) 1   −1/2   = −1/2 − z(−t ) (6.11) ε(l) 1 − z(−t ) 0 1 1 0   3 − t − t−1 t − 1 1   = −1 . (6.12) 2 − t 1 − t−1 1

Further, we note that this phenomena of extending the Burau representation to string links is not limited to just this particular representation. Indeed, any representation of the braid group which arises from an irreducible representation of the

Hecke algebra will have an extension to Stn. The matrices for these representations are, again, computable by using the relation s+ = s− + zs0.

6.3 The Conway Polynomial

The Conway polynomial is really just the Alexander polynomial normalized in such a way that we no longer have the abiguity concerning multiplication by a unit in Z[t, t−1]. The Conway polynomial is easy to compute, there being a skein relation by which we can reduce a link to a combination of unknots. We end this chapter by noting that the Conway polynomial arises out of the tangle theoretic context described in the previous section.

Proposition 18. The Conway-normalized Alexander polynomial ∇L(t) of a link L is a polynomial in Z[t1/2, t−1/2] which is characterized by

∇unknot(t) = 1 (6.13) and

∇L+ = ∇L− + z∇L0 (6.14) where z = t1/2 − t−1/2 and the links L+, L−, and L0 are identical outside a neighborhood of a crossing and diﬀer inside the neighborhood according to equation (6.1).

60 Proof. See Lickorish [16], Theorem 8.6.

+ −

Deﬁne tangles Ti : ²i → ²i+1 and Ti : ²i+1 → ²i by

+ + _

+ + + _ _ _ _

... + ...... − ...

Ti = Ti = (6.15)

+ + + _ _

+ + _ _ _

Let β be a braid in Bn. Consider β to belong to AutT (²n) as in Figure 6.2 (a). The space V1 is one-dimensional over C with basis given by the tangle diagram consisting of a single unknotted loop from z2 to z1. Hence, any linear map from V1 to itself is multiplication by a constant in C. Consider the map (∇β)∗ from V1 to itself induced by the tangle + + − − ∇β = T1 ...Tn−1βTn−1 ...T1 . (6.16)

+ − 0 Since the relation s = s +zs inside V1 is identical to the skein relation in (6.14) we see easily the following identity:

(∇β)∗ = ∇βˆ(t) (6.17)

This should come as no surprise, though, given the similarity of the relations used de-

ﬁne the Conway polynomial and the vector space Vn. Indeed, the Conway polynomial was virtually built into our deﬁniton of the space Vn.

61 BIBLIOGRAPHY

[1] Emil Artin, Theorie der Z¨opfe, Hamburg Abh., 4 (1925), 47-72. [2] Stephen Bigelow, The Burau representation is not faithfull for n = 5, Geom. Topol., 3 (1999), 397–404. [3] Stephen Bigelow, Braid groups are linear, J. Amer. Math. Soc., 14 (2001) no. 2, 471–486. [4] Joan S. Birman, Briads, links, and mapping class groups, Princeton University Press (1974), Annals of Mathematics Studies, no. 82. [5] W. Burau, Uber¨ Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Semin. Hamburg. Univ., 11 (1935), 179–186. [6] Gerhard Burde and Heiner Zieschang, Knots, de Gruyter (1985), de Gruyter Studies in Mathematics, no. 5. [7] G. Felder and C. Wieczerkowski, Topological Representations of the Quantum Group Uq(sl2), Commun. Math. Phys., 138 (1991), 583–605. [8] Cole A. Giller, A Family of Links and the Conway Calculus, Trans. Amer. Math. Soc., 270 (1982), no. 1, 75–109. [9] V.F.R. Jones, Hecke Algebra Representations of Braid Groups and Link Polyno- mials, Ann. of Math., 126 (1987), no. 2, 335–388. [10] Jurge Frolich and Thomas Kerler, Quantum Groups, Quantum Categories and Quantum Field Theory, Springer-Verlag, Berlin (1993), Lecture Notes in Math- ematics, no. 1542. [11] Christian Kassel, Quantum Groups, Springer-Verlag (1995). [12] Christian Kassel, Marc Rosso, Vladimir Turaev, Quantum Groups and Knot Invariants, Societe Mathematique de France (1997), Panoramas et Syntheses, no. 5.

[13] Daan Krammer, The braid group B4 is linear, Invent. Math., 132 (2000), no. 3, 451–486.

62 [14] Daan Krammer, Braid Groups are linear, preprint, Basel (2000).

[15] R.J. Lawrence, Homological Representations of the Hecke Algebra, Commun. Math. Phys., 135 (1990), 141–191.

[16] W.B. Lickorish, An Introduction to Knot Theory, Springer-Verlag (1997).

[17] Xiao-Song Lin, Feng Tian, and Zhenghan Wang, Burau representation and ren- dom walks on string links, Paciﬁc J. Math., 182 (1998), no. 2, 289–302.

[18] John Moody, The Burau representation of the braid group Bn is unfaithful for large n, Bull. Amer. Math. Soc. (N.S.), 25 (1991), no. 2, 379–384.

[19] Daniel S. Silver and Susan G. Williams, A generalized Burau representation for string links, Paciﬁc J. Math., 197 (2001), no. 1, 241–255.

[20] Vladimir Turaev, Faithful Linear Representations of the Braid Groups, arXiv: math.GT/0006202 (2000).

[21] Hanz Wenzl, Hecke algebras of type An and subfactors, Invent. Math., 92 (1988), no. 2, 349–383.

[22] Matthew Zinno, On Krammer’s Representation of the Braid Group, arXiv:math. RT/0002136 (2000).