A good way to understand the coefficients of a univariate polynomial with integer coefficients is to lift it to a “nice” multivariate polynomial with 0,1-coefficients. When the terms of the lift correspond to integer points of a magical polytope called a generalized permutahedron, a particularly nice story unfolds. I will illustrate the above by using it to prove a special case of Fox’s trapezoidal conjecture from 1962 that states that the absolute values of the coefficients of the Alexander polynomial of alternating links form a trapezoidal sequence. This talk is based on joint works with Hafner and Vidinas and with Kálmán and Postnikov.