Fox’s unimodality conjecture for the Alexander polynomial of alternating links

Establish 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_{-n}|, |a_{-n+1}|, …, |a_n| is unimodal.

Background

Fox formulated a classical conjecture asserting unimodality of the absolute values of the Alexander polynomial coefficients for alternating links. Numerous partial results are known for special families, but the general case remains unresolved.

References

Moreover, Fox conjectured that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a unimodal sequence , i.e. The general conjecture, however, remains unsolved to this day.

— 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