The Burau representation is not faithful for n = 5
Geometric Topology
2014-11-11 v2 Group Theory
Abstract
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 Moody's 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.
Keywords
Cite
@article{arxiv.math/9904100,
title = {The Burau representation is not faithful for n = 5},
author = {Stephen Bigelow},
journal= {arXiv preprint arXiv:math/9904100},
year = {2014}
}
Comments
8 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper16.abs.html