On the Burau representation of B4

Bharathram, Vasudha; Birman, Joan

doi:10.2140/involve.2021.14.143

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On the Burau representation of $B_4$

Vasudha Bharathram and Joan Birman

Vol. 14 (2021), No. 1, 143–154

DOI: 10.2140/involve.2021.14.143

Abstract

In 1936 W. Burau discovered an interesting family of $n \times n$ matrices that give a linear representation of Artin’s classical braid group $B_{n}$ , $n = 1, 2, \dots$ . A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for $n \leq 3$ , nonfaithful for $n \geq 5$ , 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 $B_{4}$ 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

Vol. 14, No. 1, 2021