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

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.

23 pages + Tables (51 pages)

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• Eigenvalue hypothesis for multi-strand braids
• Are there p-adic knot invariants?
• Kerov functions revisited
• Eigenvalue conjecture and colored Alexander polynomials
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• Harer-Zagier formulas for families of twisted hyperbolic knots
• Colored HOMFLY and Generalized Mandelbrot set