The HOMFLY-PT polynomial and HZ factorisation

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.