The braid group B n B_n can be defined as the mapping class group of the n n -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over R \mathbf R . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case n = 4 n=4 . In this paper, we show that it is faithful for all n n .

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range $(4,3+\sqrt{3})$, which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.Comment: 45 pages (final version; improved introduction

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 ≥ 9 and Long and Paton improved on Moody's techniques to bring this down to n ≥ 6. 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 ≥ 5. AMS Classification numbers Primary: 20F36Secondary: 57M07, 20C99

We give a new definition of the Jones polynomial. Let L be an oriented knot or link obtained as the plat closure of a braid β ∈ B 2n . We define a covering spaceC of the space of unordered n-tuples of distinct points in the 2n-punctured disk. We then describe two n-manifoldsS and T inC , and show that the Jones polynomial of L can be defined as an intersection pairing betweenS and βT . Our construction is similar to one given by Lawrence, but more concrete.

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, 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 B_{n-1}. We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hilden's result that this group is a Z_2 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.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-34.abs.htm