A dimer view on Fox's trapezoidal conjecture

TL;DR

This paper provides a shorter, dimer model-based proof for a special case of Fox's trapezoidal conjecture related to Alexander polynomials of certain alternating links, and offers new insights into the polynomial's properties.

Contribution

It introduces a novel dimer model approach to prove a specific case of Fox's conjecture and derives new theorems about the Alexander polynomial from this perspective.

Findings

Shorter proof of Azarpendar, Juhász, and Kálmán's result

New theorems about the Alexander polynomial

Dimer model provides clear insights into polynomial properties

Abstract

Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been settled, some quite recently: Fox's conjecture was shown to hold for special alternating links by Hafner, M\'esz\'aros, and Vidinas (2023) and for certain diagrammatic Murasugi sums of special alternating links by Azarpendar, Juh\'asz, and K\'alm\'an (2024). In this paper, we give an alternative proof of Azarpendar, Juh\'asz, and K\'alm\'an's aforementioned beautiful result via a dimer model for the Alexander polynomial. In doing so, we not only obtain a significantly shorter proof of Azarpendar, Juh\'asz, and K\'alm\'an's result than the original, but we also obtain several theorems of independent interest regarding the Alexander polynomial, which are readily visible from the dimer point of view.