Superpolynomials for toric knots from evolution induced by cut-and-join operators

Understand

The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory.

• Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients.
• A lot of explicit expressions are presented for different representations (Young diagrams), many of them new.
• In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k.