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Paper detail

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>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.

preprint1999arXivOpen access
Stephen Bigelow
math.GRmath.GT
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Stephen Bigelow

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