Braid twists and HZ factorisation via character expansion

Abstract

The HOMFLY-PT polynomial of a closed braid admits a character expansion, expressed as a sum of SU(N) characters over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY-PT polynomial into a rational function, is applied directly to the characters, yielding the HZ character expansion. We use this expansion to illumine the hidden structure of the HZ function that enables its decomposition into a sum of factorised terms. We further articulate explicit conditions for full HZ factorisation, which include that non-vanishing contributions come solely from hook-shaped Young diagrams. We show that these conditions remain invariant under three mutually commuting braid twists: full twists, partial full twists and Jucys-Murphy twists. We, hence, employ such twists to construct infinite, HZ-factorisable families of knots and links, which can often be thought of as a hyperbolic extension of torus knots. Remarkably, these families encompass the Coxeter links corresponding to E-type Dynkin diagrams.