On Fox’s trapezoidal conjecture

Fox’s trapezoidal conjecture, proposed in 1962, states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot form a trapezoidal sequence: they strictly increase, possibly plateau, and then strictly decrease. In this talk, I will discuss recent progress on the conjecture, based on joint work with András Juhász and Tamás Kálmán. In particular, we prove the conjecture for diagrammatic plumbings of special alternating links and obtain partial results for alternating three-braid closures