Jones unknot problem: does the Jones polynomial detect the unknot?

Determine whether the Jones polynomial J(q) detects the unknot; equivalently, prove or disprove the existence of a non-trivial knot K whose Jones polynomial satisfies J^K(q)=1 for all q.

Background

The paper reviews the status of polynomial knot invariants and emphasizes that, while the Jones polynomial is powerful, it does not distinguish all knots and takes the same value on certain mutants. The central question is whether the Jones polynomial detects the unknot, i.e., whether any knot with unit Jones polynomial must be trivial.

The authors note extensive prior efforts: computational checks up to 24 crossings found no counterexamples, and related invariants such as Khovanov do detect the unknot. They position their work within this context, focusing on braid presentations and especially four-strand braids as a promising setting to investigate the Jones unknot problem.