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

TL;DR

This paper introduces a simplified method for deriving superpolynomials of toric knots using evolution operators and MacDonald polynomials, providing explicit formulas and new results for various representations.

Contribution

It presents a novel, simplified approach to compute superpolynomials for toric knots via evolution induced by cut-and-join operators, expanding explicit formulas to new representations.

Findings

Explicit superpolynomial formulas for various Young diagram representations.

New superpolynomial expressions for torus knots with specific parameters.

Simplified derivation method avoiding complex Littlewood-Richardson coefficients.

Abstract

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. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist.