Refined composite invariants of torus knots via DAHA

Abstract

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the skein theory of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E closely related to the Deligne-Gross approach to the exceptional root systems; this theme is of experimental nature.