Character Expansion and the Harer-Zagier Transform of Knot Polynomials

It is known that statistical mechanical solvable models give knot polynomials. The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young tableaux. The Harer-Zagier transform (HZ), which converts the HOMFLY-PT polynomial into a rational polynomial in $q$ and $\lambda$, can also be expanded in terms of the HZ of characters. When the HZ of a knot is fully factorisable, such an expansion consists of Young diagrams of hook shape. However, for the vast majority of knots and links the HZ is not factorisable. We show that, in non-factorised cases with up to 7-strands, the HZ transform can be decomposed into a sum of factorised terms. Furthermore, by the substitution $\lambda= q^\mu$ for a magic number $\mu$, the HZ can be reduced to a simple rational function in $q$ consisting of products of cyclotomic polynomials.