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 $n \geq 9$ and Long and Paton improved on Moody’s techniques to bring this down to $n \geq 6$ . 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 $n \geq 5$ .

Keywords

braid group, Burau representation

Mathematical Subject Classification

Primary: 20F36

Secondary: 57M07, 20C99

References

Forward citations

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

Volume 3, issue 1 (1999)

Stephen Bigelow