ÍâÍøÌìÌÃ

Fox’s trapezoidal conjecture from 1962 states that the absolute values of the coefficients of the Alexander polynomial of alternating links form a trapezoidal sequence. Stoimenow strengthened Fox’s conjecture to log-concavity (without internal zeros) in 2005. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of algebraic alternating links, and Ozsváth and Szabó (2003) for genus 2 alternating knots, among others. We will show how to prove Fox’s conjecture for special alternating links by lifting the Alexander polynomials of these links to "nice" multivariate polynomials with 0,1 coefficients. The terms of the lift correspond to integer points of a generalized permutahedron, allowing for an application of the theory of Lorentzian polynomials developed by Brändén and Huh (2019). This talk is based on joint works with Hafner and Vidinas, and, Kálmán and Postnikov.
Ìý