Theoretical analysis reveals algebraic conditions linking HOMFLY-PT and Kauffman polynomials in twisted knots, confirming a conjectural correspondence for three-strand braids.
A bstract 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 braid index 4 or higher.
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Petrou et al. (2026) studied this question.
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