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

• DOI: 10.1007/jhep03(2013)021
• Publication Date: 2013-03-05T11:42:58Z
• Authors: P. Dunin-Barkowski; A. Mironov; A. Morozov; A. Sleptsov; A. Smirnov
• Keywords: Antisymmetric relation; Representation; Matrix (chemical analysis); Wilson loop; Substitution (logic)

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.<br/>23 pages + Tables (51 pages)<br/>