The HOMFLY-PT polynomial and HZ factorisation

TL;DR

This work studies the Harer--Zagier transform of the HOMFLY--PT polynomial, introducing $HZ$-factorisability as a condition where the transform becomes a rational function with numerator and denominator factors of the form $(1-\\lambda q^{\\beta})$. It shows that factorisability is preserved under full twists and Jucys--Murphy braids, enabling infinite families of $HZ$-factorisable knots and links, and provides a concrete inverse transform to recover the HOMFLY--PT polynomial and the Alexander polynomial from the $HZ$ function. A central theme is the connection between the $HZ$ data and topological-string physics through Kauffman polynomials and two-crosscap BPS invariants, including conjectural criteria linking $HZ$ factorisability to the vanishing of these invariants. The authors establish explicit relations between HOMFLY--PT and Kauffman polynomials for several factorisable families and demonstrate that, for knots, the $HZ$ exponents coincide with the indices in Khovanov homology, yielding a direct link to the Jones polynomial as a graded Euler characteristic. These results culminate in a framework that enumerates infinite families of hyperbolic knots/links with factorised $HZ$ transforms and ties knot invariants to homological and string-theoretic structures, with further work promised in Petrou II.

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.

Figures (5)

• Figure 1: Two operations that preserve the factorisability of the HZ transform.
• Figure 2: Relations between HZ-factorisable knots with braid index (a) $\iota=3$ and (b) $\iota=4$. They can be obtained from each other by introducing a full twist $F_m$ or a Jucys--Murphy braid $E_m$ as indicated by the arrows.
• Figure 3: The family $5_2^{3k}$ is the closure of the braid obtained by a braid representative for $5_2^+$ (left part) concatenated with $k$ full twists $F_3$ (one of such twists is depicted in the right part).
• Figure 4: The twisted torus knot $T(4,5,3,3k)$ is the closure of this braid.
• Figure 5: Hopf link $T(2,2)$ with parallel relative orientation on its components.

Theorems & Definitions (27)

• Remark 2.1
• Proposition 2.1
• proof
• Remark 2.2
• Theorem 3.1
• proof
• Corollary 3.1
• Theorem 3.2
• proof
• Remark 3.1