Volume 3, issue 1 (1999)

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The Burau representation is not faithful for $n = 5$

Stephen Bigelow

Geometry & Topology 3 (1999) 397–404
 arXiv: math.GT/9904100
Abstract

The Burau representation is a natural action of the braid group ${B}_{n}$ on the free $ℤ\left[t,{t}^{-1}\right]$–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 $n\ge 9$ and Long and Paton improved on Moody’s techniques to bring this down to $n\ge 6$. 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 $n\ge 5$.

Keywords
braid group, Burau representation
Mathematical Subject Classification
Primary: 20F36
Secondary: 57M07, 20C99