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

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, like for torus knots, 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, which mostly lies on the unit circle and deviates from it only in pairs.