Salter's question on the image of the Burau representation of $B_3$

TL;DR

This paper investigates Salter's question about the image of the Burau representation of the braid group $B_3$, providing an algorithmic counterexample that shows the central quotient does not match the expected subgroup.

Contribution

It offers the first negative answer to Salter's question for $n=3$ by explicitly constructing a counterexample using an algorithmic approach.

Findings

Counterexample constructed for $n=3$

Central quotient of Burau image differs from subgroup

Algorithmic method developed for the construction

Abstract

In 1974, Birman posed the question of under what conditions a matrix with Laurent polynomial entries is in the image of the Burau representation. In 1984, Squier observed that the matrices in the image are contained in a unitary group. In 2021, Salter formulated a specific question: whether the central quotient of the Burau image group is the central quotient of a certain subgroup of the unitary group. We solve this question negatively in the simplest nontrivial case, $n=3$, algorithmically constructing a counterexample.