#### 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.

Notes on file formats
Stephen Bigelow

Department of Mathematics

UC Berkeley

Berkeley, CA 94720, USA

Email: bigelow@math.berkeley.edu

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