Harer–Zagier transform of the HOMFLY-PT polynomial for families of twisted hyperbolic knots

Abstract

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform -a discrete Laplace transform- of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure and, for all knots, they have the property that the modulus of the product of all the zeros equals unity.