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

Abstract

The Burau representation is a natural action of the braid group $B_n$ on the free $\mathbb{Z}\left[t,t^{-1}\right]$–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 $n\ge 5$.