arXiv subjects

Andreani Petrou

Publications and source records attributed to Andreani Petrou.

4 recordsLinked to original sources

arXiv2026

A relation between the HOMFLY-PT and Kauffman polynomials via characters

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N+1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. The conjecture remains open for knots with 4 and higher strands.

hep-th↗

arXiv2025

Braid twists and HZ factorisation via character expansion

The HOMFLY-PT polynomial of a closed braid admits a character expansion, expressed as a sum of SU(N) characters over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY-PT polynomial into a rational function, is applied directly to the characters, yielding the HZ character expansion. We use this expansion to illumine the hidden structure of the HZ function that enables its decomposition into a sum of factorised terms. We further articulate explicit conditions for full HZ factorisation, which include that non-vanishing contributions come solely from hook-shaped Young diagrams. We show that these conditions remain invariant under three mutually commuting braid twists: full twists, partial full twists and Jucys-Murphy twists. We, hence, employ such twists to construct infinite, HZ-factorisable families of knots and links, which can often be thought of as a hyperbolic extension of torus knots. Remarkably, these families encompass the Coxeter links corresponding to E-type Dynkin diagrams.

math-ph↗

arXiv2024

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.

math-ph↗

arXiv2023

Harer-Zagier formulas for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived formulas for the Harer-Zagier transform of the HOMFLY-PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of Pretzel knots for which the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zeros have a remarkable structure as the modulus of their product always equals unity.

math-ph↗