On the Burau representation of $B_3$ modulo $p$

TL;DR

This paper introduces an algorithm to determine the faithfulness of the Burau representation of the 3-strand braid group over finite fields, proves faithfulness for primes up to 13, and resolves a related open question for all primes.

Contribution

It provides a new algorithm for testing faithfulness and completely solves Salter's question on the Burau representation over finite fields.

Findings

Faithfulness confirmed for primes p ≤ 13

Algorithm effectively determines faithfulness for any prime p

Salter's question answered for all primes p

Abstract

We present an algorithm that, given a prime $p$ as input, determines whether or not the Burau representation of the 3-strand braid group modulo $p$ is faithful. We also prove that the representation is indeed faithful when $p\le 13$. Additionally, we re-pose Salter's question on the Burau representation of $B_3$ over finite fields $\mathbb{F}_p$, and solve it for every $p$.