On some log-concavity properties of the Alexander-Conway and Links-Gould invariants

Abstract

The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $Δ_L(t)$ and has been shown to share some of its most geometric features in several recent works. Here we suggest that $\mathrm{LG}$ likely shares another of the Alexander polynomial's most distinctive - and mysterious - properties: for alternating links, the coefficients of the Links-Gould polynomial alternate and appear to form a log-concave two-indexed sequence with no internal zeros. The former was observed by Ishii for knots with up to 10 crossings. We further conjecture that they satisfy a bidimensional property of unimodality, thereby replicating a long-standing conjecture of Fox (1962) regarding the Alexander polynomial, and a subsequent refinement by Stoimenow. We also point out that the Stoimenow conjecture reflects a more structural phenomenon: after a suitable normalization, the Alexander polynomial of an alternating link appears to be a Lorentzian polynomial. We give compelling experimental and computational evidence for these different properties.