Unresolved ‘tricky cases’ in the faithfulness analysis of Burau(B4)

Ascertain the faithfulness status of the Burau representation of the four-strand braid group in the remaining unresolved cases identified by Datta (2022), completing the ‘almost everywhere’ analysis.

Background

Datta (2022) provided strong evidence that the Burau representation of B4 is faithful ‘almost everywhere,’ but the authors note that certain tricky cases were not settled. These residual cases prevent a definitive resolution of the faithfulness question for B4.

Clarifying these tricky cases is essential both for the representation theory of braid groups and for the implications discussed by Bigelow that tie faithfulness to the existence of knots with trivial polynomial invariants.