Rectangular superpolynomials for the figure-eight knot

TL;DR

This paper presents a new, simplified formula for rectangular HOMFLY polynomials of the figure-eight knot, extending previous work to arbitrary rectangular representations and introducing positive, $eta$-deformed superpolynomials.

Contribution

The authors derive a simple sum-over-Young-diagrams formula for rectangular HOMFLY polynomials, incorporating $eta$-deformation to Macdonald dimensions, and extend differential expansion to all rectangular representations.

Findings

Coefficients are quantum dimensions of symmetric groups, restricted by diagram size.

The formula produces positive Laurent polynomials as candidates for superpolynomials.

Extension of differential expansion to arbitrary rectangular representations with preserved factorization.

Abstract

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple coefficients $D_{\lambda^{tr}}(r)\cdot D_\lambda(s)$ in front of the $Z$-factors. Somewhat miraculously, these coefficients are made from quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict summation to diagrams with no more than $s$ rows and $r$ columns. They possess a natural $\beta$-deformation to Macdonald dimensions and produces positive Laurent polynomials, which can be considered as plausible candidates for the role of the rectangular superpolynomials. Both polynomiality and positivity are non-evident properties of arising expressions, still they are true. This extends the previous suggestions for symmetric and antisymmetric representations (when $s=1$ or $r=1$ respectively) to arbitrary rectangular representations. As usual for differential expansion, there are additional gradings. In the only example, available for comparison -- that of the trefoil knot $3_1$, to which our results for $4_1$ are straightforwardly extended, -- one of them reproduces the "fourth grading" for hyperpolynomials. Factorization properties are nicely preserved even in the 5-graded case.