Stephen Bigelow

Braid groups are linear

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.

The Burau representation is not faithful for n = 5

The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton improved on Moodys techniques to bring this down to n>5. Their construction uses a simple closed curve on the 6-punctured disc with certain homological properties. In this paper we give such a curve on the 5-punctured disc, thus proving that the Burau representation is not faithful for n>4.

The mapping class group of a genus two surface is linear

In this paper we construct a faithful representation of the map- ping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer represen- tation of the braid group Bn , which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the close relationship between this group and Bn 1 . We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hildens result that this group is a Z2 central extension of the mapping class group of the 6-punctured sphere. The resulting representation has dimension sixty-four and will be described explicitly. In closing we will remark on subgroups of mapping class groups which can be shown to be linear using similar techniques. AMS Classication 20F36; 57M07, 20C15

DOES THE JONES POLYNOMIAL DETECT THE UNKNOT

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

A DIAGRAMMATIC ALEXANDER INVARIANT OF TANGLES

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

The pop-switch planar algebra and the Jones–Wenzl idempotents

The Jones-Wenzl idempotents are elements of the Temperley-Lieb planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra, which contains the Temperley-Lieb planar algebra. It is motivated by Jones idea of the graph planar algebra of type

A diagrammatic definition of Uq(𝔰𝔩2)

A_n

A homological definition of the Jones polynomial

. In the tensor category of idempotents of the pop-switch planar algebra, the

Representations of Braid Groups

n

The Lawrence-Krammer representation

th Jones-Wenzl idempotent is isomorphic to a direct sum of