The Burau representation is not faithful for n=5

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 >= 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.