Stoimenow’s log-concavity-without-gaps conjecture for the Alexander polynomial of alternating links

Prove that for any alternating link L with Conway-normalized Alexander polynomial Δ_L(t) = Σ_{k=-n}^{n} a_k t^k, the sequence of absolute values |a_k| is log-concave and has no internal zeros.

Background

Stoimenow strengthened Fox’s unimodality conjecture by proposing that the absolute values of the Alexander polynomial’s coefficients not only be unimodal but satisfy the stronger property of log-concavity and contain no internal zeros. This would imply the trapezoidal shape and unimodality of the sequence.

References

Stoimenow refined the statement by conjecturing that the sequence should be log-concave with no internal zeros .

— On some log-concavity properties of the Alexander-Conway and Links-Gould invariants  (2509.16868 - Harper et al., 21 Sep 2025) in Section 1.2

A strengthening of Conjecture \ref{conj:fox}, due to Stoimenow, suggests the sequence is actually {\em log-concave}:

For any alternating link $K$, the sequence of coefficients of $|\Delta|K$ satisfies $$ |a_i|2 \geq |a{i - 1}||a_{i + 1}| $$ for all $i = 1, \dots, n - 1$. Furthermore, none of the $a_i$ are zero. $$

— An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties  (2608.28484 - Boninger, 28 Aug 2026) in Section 1, subsection “Fox's trapezoidal conjecture”; Conjecture 1.3 (labelled conjecture:lc)