Trapezodial property of the generalized Alexander polynomial

Abstract

Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this date. Recently Fox's conjecture was settled for all special alternating links. In this paper we take a broad view of the Alexander polynomials of special alternating links, showing that they are a generating function for a statistic on certain vector configurations. We study three types of vector configurations: (1) vectors arising from cographic matroids, (2) vectors arising from graphic matroids, (3) vectors arising from totally positive matrices. We prove that Alexander polynomials of special alternating links belong to both classes (1) and (2), and prove log-concavity, respectively trapezoidal, properties for classes (2) and (3). As a special case of our results, we obtain a new proof of Fox's conjecture for special alternating links.