Georgia Institute of Technology College of Sciences

The Fox trapezoidal conjecture is a longstanding open problem about the coefficients of the Alexander polynomial of alternating links. In this talk, we will discuss recent work which settled this conjecture for “special alternating links”. The first tool is a graph theoretic model of the Alexander polynomial of an alternating link discovered by Crowell in 1959. The second is the theory of Lorentzian polynomials, developed by Brändén and Huh in 2019 and a key part of Huh’s Fields medal work. We will show how a version of Crowell’s model produces a refinement of the Alexander polynomial of special alternating links that is Lorentzian, from which the result follows quickly.