A relation between the HOMFLY-PT and Kauffman polynomials via characters
Abstract
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. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.
Cite
@article{arxiv.2603.03628,
title = {A relation between the HOMFLY-PT and Kauffman polynomials via characters},
author = {Andreani Petrou and Shinobu Hikami},
journal= {arXiv preprint arXiv:2603.03628},
year = {2026}
}