Faithfulness of the Burau representation of B4

Determine whether the (unreduced) Burau representation ρ: B4 → GL4(Z[t, t−1]) is faithful, i.e., whether ρ is injective on the braid group B4. This is the remaining undecided case in the classical faithfulness question for the Burau representation, which is known to be faithful for n ≤ 3 and unfaithful for n ≥ 5.

Background

The Burau representation is a foundational linear representation of braid groups, originally defined via matrices assigned to Artin’s generators. It plays a central role in low-dimensional topology and in the paper of the linearity of braid groups. Extensive work has established that the Burau representation is faithful for Bn when n ≤ 3 and is not faithful for n ≥ 5 (with results by Moody, Long–Paton, and Bigelow).

The only remaining case is n = 4, for which the faithfulness of the Burau representation is unresolved. The paper references recent advances aimed at closing this problem, underscoring its ongoing status in the literature.