Special colored superpolynomials and their representation-dependence

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.