Mutation invariance of the Alexander polynomial refinement

Determine whether the polynomial P_K is invariant under arbitrary link mutations, beyond the orientation-preserving mutations for which invariance is established.

Background

The paper proves that P is invariant under orientation-preserving mutations by an argument analogous to its flype-invariance proof. General mutation can change the orientation or isotopy type of a link, and the author does not establish invariance in that broader setting. The unresolved issue is explicitly identified as a natural direction for further study.

References

An argument much like the proof of Lemma \ref{lem:flype_invar} can be used to show $P$ is invariant under orientation-preserving mutations, but we do not know if $P$ is mutation invariant in general. This is a natural question for further study.

— An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties  (2608.28484 - Boninger, 28 Aug 2026) in Remark following Theorem 5.3, Section 5, “Diagram Independence”