The Burau representation is not faithful for n = 5

The Burau Representation is Not Faithful for n = 5 Stephen

A presentation for the image of Burau(4)| Z2

Let Bn denote the n-strand braid group. We recall that this admits a representation fin : Bn ---+ GLnl (Z[ t , t l ] ) the (reduced) Burau representation [1]. Although open for a long time, it is now known that this representation is not faithful for n > 6, (See [8, 5]) and it is an old result that it is faithful for n = 3. (See [7]) Despite these counterexamples, there is no understanding of t...

Braid Groups Are Linear

The braid groups Bn were originally introduced by Emil Artin in 1926. They have many interpretations, for example, as the group of geometric braids in R, as the Artin group of type An, as the fundamental group of a certain hyperplane arrangement, as a subgroup of the automorphism group of a free group, and so on. In this paper we will use the interpretation of Bn as the mapping class group of a...

Burau Representation and Random Walk on String Links

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the trans...

2 Sang

Given two pseudo-Anosov homeomorphisms with distinct invariant measured foliations, some powers of their isotopy classes generate a rank two free subgroup of the mapping class group of the surface [6]. This construction gives an example of all pseudo-Anosov subgroup of the mapping class group. Whittlesey [13] gives a positive answer to the natural question of the existence of all pseudo-Anosov ...