The HZ character expansion and a hyperbolic extension of torus knots

TL;DR

The paper explores the structure of the HOMFLY-PT polynomial via the HZ character expansion, introduces hyperbolic extensions of torus knots, and discusses conditions for factorisability and decomposition of HZ functions.

Contribution

It constructs an infinite family of hyperbolic knots with HZ factorisability and conjectures a decomposition for non-factorisable HZ functions, supported by symmetry proofs.

Findings

Identified conditions for HZ factorisability involving hook-shaped Young diagrams.

Constructed a family of hyperbolic knots extending torus knots via braid operations.

Proved decomposition of HZ functions in the 3-strand case using Young diagram symmetries.

Abstract

The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY--PT polynomial into a rational function, can be applied directly to the characters, yielding hence the HZ character expansion. This illuminates the structure of the HZ functions and articulates conditions for their factorisability, including that non-vanishing contributions should come from hook-shaped Young diagrams. An infinite HZ-factorisable family of hyperbolic knots, that can be thought of as a hyperbolic extension of torus knots, is constructed by full twists, partial full twists and Jucys-Murphy twists, which are braid operations that preserve HZ factorisability. Among them, of interest is a family of pretzel links, which are the Coxeter links for E type Dynkin diagrams. Moreover, when the HZ function is non-factorisable, which occurs for the vast majority of knots and links, we conjecture that it can be decomposed into a sum of factorised terms. In the 3-strand case, this is proven using the symmetries of Young diagrams.