Geometry & Topology, Vol. 3 (1999) Paper no. 16, pages 397--404.

The Burau representation is not faithful for n = 5

Stephen Bigelow

Abstract. 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.

Keywords. Braid group, Burau representation

AMS subject classification. Primary: 20F36. Secondary: 57M07, 20C99.

DOI: 10.2140/gt.1999.3.397

E-print: arXiv:math.GT/9904100

Submitted to GT on 21 July 1999. Paper accepted 23 November 1999. Paper published 30 November 1999.

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Stephen Bigelow
Department of Mathematics
UC Berkeley
Berkeley, CA 94720, USA

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