#### 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 ${B}_{n}$, $n=1,2,\dots \phantom{\rule{0.3em}{0ex}}$. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for $n\le 3$, nonfaithful for $n\ge 5$, 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