Rectangular superpolynomials for the figure-eight knot $4_1$

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in an arbitrary rectangular representation $R=[r^s]$ as a sum over all Young subdiagrams $\lambda$ of $R$ with surprisingly simple coefficients of the $Z$ factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict the summation to diagrams with no more than $s$ rows and $r$ columns. Moreover, the $\beta$-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot $3_1$, to which our results for the knot $4_1$ are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.