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

TL;DR

This paper explores the mathematical relationship between HOMFLY-PT and Kauffman polynomials for specific knots, using algebraic characters, and discusses implications for knot theory conjectures.

Contribution

It establishes conditions linking HOMFLY-PT and Kauffman polynomials via algebraic characters and examines their relation to the Harer-Zagier factorisability conjecture.

Findings

Relation holds for knots with full twists and Jucys-Murphy twists

Proves a conjectural 1-1 correspondence for a large family of 3-strand knots

Counterexamples show the relation does not extend to all 4-strand knots

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.