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Converse to Bigelow’s implication relating representation unfaithfulness and trivial HOMFLY-PT/Jones polynomials

Establish whether an inverse to Bigelow’s Proposition holds: if the Jones representation (equivalently, the Burau representation for B4) is faithful, determine whether this excludes the existence of non-trivial knots with trivial HOMFLY-PT or Jones polynomials.

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Background

Bigelow’s proposition states that if the Jones representation is unfaithful, then one can construct non-trivial knots with trivial HOMFLY-PT (and hence Jones) polynomials, connecting representation-theoretic properties to the Jones unknot problem.

The authors explicitly state they do not know the inverse theorem, i.e., whether faithfulness would preclude such constructions. Resolving this would clarify the logical relationship between representation faithfulness and the existence of knots with unit Jones polynomial.

References

We remark that we do not know the inverse theorem to the theorem \ref{Big1}, so the faithfulness of the Burau representation for ${\cal B}_4$ does not mean that the Jones problem is closed.

— 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