We define composite DAHA-superpolynomials of torus knots, depending on pairs
of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the
theory of the skein 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; such a connection is a
challenge in the theory of the annulus. At the end, we construct two
DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E and
closely related to the exceptional Deligne-Gross series of root systems; this
theme is of experimental nature.Comment: v2: 3 references were added and minor editin
