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Faithfulness of the Burau representation for the four-strand braid group

Determine whether the reduced Burau representation of the four-strand braid group B4 is faithful; that is, prove or disprove that distinct elements of B4 are mapped to distinct matrices under the reduced Burau representation.

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Background

The Burau representation’s faithfulness is known for B_n with n≤3 and known to be unfaithful for n≥5, leaving B4 as the sole unresolved case. Bigelow’s results connect faithfulness of the Burau representation for B4 with faithfulness of the Jones representation, tying this question to the Jones unknot problem.

Recent progress by Datta shows that the Burau representation of B4 is faithful almost everywhere, but unresolved ‘tricky cases’ remain. Resolving faithfulness for B4 would have major implications for constructing or ruling out knots with trivial Jones or HOMFLY-PT polynomials.

References

That said, the question of the faithfulness of the Burau representation is currently open only for ${\cal B}_4$.

— Closed 4-braids and the Jones unknot conjecture (2402.02553 - Korzun et al., 4 Feb 2024) in Subsection: Connection of Jones and Burau representations for B4 group