The Burau representation is not faithful for n = 5

Abstract

The Burau representation is a natural action of the braid group Bn 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.