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.
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
Fox's conjecture states:
For any alternating link $K$, the sequence of coefficients of $|\Delta|K$ is {\em trapezoidal}. In other words, there exist indices $0 \leq i_1 \leq i_2 \leq n$ such that $$ |a_0| < |a_1| < \dots < |a{i_1}| = |a_{i_1 + 1}| = \cdots = |a_{i_2}| > \cdots > |a_{n - 1}| > |a_n|. $$
— 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 1.2 (labelled conjecture:fox)