Four-strand braid case of the Jones unknot problem

Determine whether there exists a non-trivial knot arising from a closed four-strand braid whose Jones polynomial is equal to 1; equivalently, resolve the Jones unknot problem in the specific case of 4-strand braids.

Background

The authors focus on knots presented as closures of four-strand braids because of their connection to the faithfulness of the Burau representation and the Jones (Reshetikhin–Turaev) representation. Bigelow’s work links the existence of knots with trivial HOMFLY-PT (and thus Jones) polynomials to unfaithfulness of these representations in the four-strand case.

While two- and three-strand cases are understood (no non-trivial knots with unit Jones polynomial), the four-strand case remains unresolved, making it a central target of this paper’s analysis and proposed search methods.