The HOMFLY-PT polynomial and HZ factorisation
arXiv:2412.04933
The paper studies how the Harer‑Zagier transform converts the HOMFLY‑PT polynomial of certain knots into a rational function that factorises in a simple way, and shows that this HZ factorisation is preserved under specific twist operations, linking the factorisation to Khovanov homology gradings and to relations with the Kauffman polynomial.
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 the Jucys-Murphy twists, which are hence used to generate infinite HZ-factorisable families of hyperbolic knots. 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 double-grading in Khovanov homology. Moreover, a relation between the HOMFLY-PT and Kauffman polynomials, which was only known to hold for torus knots, is now proven for several of these hyperbolic families. Such a relation has a peculiar implication in topological string theory, namely, it is equivalent to the vanishing of the two-crosscap BPS invariants. It is conjectured that the HOMFLY-PT/Kauffman relation provides a criterion for HZ factorisability.