Rectangular superpolynomials for the figure-eight knot

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.