On the Alexander polynomial of special alternating links

Abstract

The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any alternating link are trapezoidal -- was settled for special alternating links by the present authors (2023); K\'alm\'an, the second author, and Postnikov gave an alternative proof (2025). The present paper is a study of the special combinatorial and discrete geometric properties that Alexander polynomials of special alternating links possess along with a generalization to all Eulerian graphs, introduced by Murasugi and Stoimenow (2003). We prove that the Murasugi and Stoimenow generalized Alexander polynomials can be expressed in terms of volumes of root polytopes of unimodular matrices. The latter generalizes a result regarding the Alexander polynomials of special alternating links that follows by putting together the work of Li and Postnikov (2013) and K\'alm\'an, the second author, and Postnikov (2025). Furthermore, we conjecture a generalization of Fox's conjecture to the generalized Alexander polynomials of Murasugi and Stoimenow and bijectively relate two longstanding combinatorial models for the Alexander polynomials of special alternating links: Crowell's state model (1959) and Kauffman's state model (1982, 2006).