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The Burau representation is not faithful for n = 5

Stephen Bigelow

Geometry & Topology 3 (1999) 397–404

DOI: 10.2140/gt.1999.3.397

arXiv: math.GT/9904100

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.

Keywords

braid group, Burau representation

Mathematical Subject Classification

Primary: 20F36

Secondary: 20C99, 57M07

References
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Publication

Received: 21 July 1999
Accepted: 23 November 1999
Published: 30 November 1999
Proposed: Joan Birman
Seconded: Shigeyuki Morita, Dieter Kotschick

Authors
Stephen Bigelow
Department of Mathematics
University of California at Berkeley
Berkeley
California 94720
USA