Special colored Superpolynomials and their representation-dependence

TL;DR

The paper introduces special superpolynomials by setting q=1, revealing a simplified representation dependence similar to special HOMFLY polynomials, with findings confirmed for certain knots and representations.

Contribution

It generalizes special HOMFLY polynomials to superpolynomials with an extra parameter t, analyzing their representation dependence for specific knots and representations.

Findings

Dependence on representation R as |R|th power for symmetric cases

Dependence for antisymmetric representations at t=1

Potential interpolation between these relations remains unexplored

Abstract

We introduce the notion of "special superpolynomials" by putting q=1 in the formulas for reduced superpolynomials. In this way we obtain a generalization of special HOMFLY polynomials depending on one extra parameter t. Special HOMFLY are known to depend on representation R in especially simple way: as |R|-th power of the fundamental ones. We show that the same dependence persists for our special superpolynomials in the case of symmetric representations, at least for the 2-strand torus and figure-eight knots. For antisymmetric representations the same is true, but for t=1 and arbitrary q. It would be interesting to find an interpolation between these two relations for arbitrary representations, but no superpolynomails are yet available in this case.