The HOMFLY-PT polynomial and HZ factorisation

TL;DR

This paper explores the properties of the HOMFLY-PT polynomial under the Harer-Zagier transform, identifying conditions for factorisation, and relating it to Khovanov homology, Kauffman polynomials, and topological string theory invariants.

Contribution

It introduces the concept of HZ factorisation for specific knot families and links it to algebraic and topological invariants, expanding understanding of polynomial relations in knot theory.

Findings

HZ factorisation is preserved under full twists and Jucys-Murphy braids.

HOMFLY-PT polynomial can be encoded by two integer sets related to Khovanov homology.

A proven relation between HOMFLY-PT and Kauffman polynomials for certain families.

Abstract

The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under full twists and concatenation with the Jucys-Murphy braid, which are hence used to generate infinite HZ-factorisable families. For such families, the HOMFLY-PT polynomial can be fully encoded in two sets of integers, corresponding to the numerator and denominator exponents, which turn out to be related to the Khovanov homology and its Euler characteristics. Moreover, a relation between the HOMFLY-PT and Kauffman polynomials, which was originally found for torus knots, is now proven for several such families. Interestingly, this relation is equivalent to the vanishing of the two-crosscap BPS invariants in topological string theory. It is conjectured that the HOMFLY-PT-Kauffman relation provides a criterion for HZ factorisability.