Jones polynomial unknot detection problem

Determine whether the Jones polynomial is an unknot detector; specifically, ascertain whether any knot in S^3 that has the same Jones polynomial as the unknot must be isotopic to the unknot, or provide a counterexample showing that the Jones polynomial fails to detect the unknot.

Background

The notes compare the detection capabilities of various knot invariants. Khovanov and Heegaard Floer homologies are known to detect the unknot, while the Alexander polynomial does not. In contrast, whether the Jones polynomial detects the unknot remains unresolved. This longstanding problem asks if equality of a knot’s Jones polynomial with that of the unknot forces the knot to be the unknot.

The authors highlight this gap in knowledge while motivating the development of refined homotopical invariants that may offer stronger detection properties than polynomial invariants alone.