Vol. 14, No. 1, 2021

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On the Burau representation of $B_4$

Vasudha Bharathram and Joan Birman

Vol. 14 (2021), No. 1, 143–154
Abstract

In 1936 W. Burau discovered an interesting family of n × n matrices that give a linear representation of Artin’s classical braid group Bn, n = 1,2,. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n 3, nonfaithful for n 5, but the case of n = 4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

Keywords
Burau representation, braid group, free group, Heisenberg group, Klein 4-group
Mathematical Subject Classification
Primary: 20F36
Secondary: 20C99, 20E05
Milestones
Received: 17 July 2020
Revised: 1 October 2020
Accepted: 24 October 2020
Published: 4 March 2021

Communicated by Józef H. Przytycki
Authors
Vasudha Bharathram
Riverdale Country School
The Bronx, NY
United States
Joan Birman
Department of Mathematics
Columbia University
New York, NY
United States