High Energy Physics - Theory
[Submitted on 21 Jun 2011 (v1), last revised 2 Dec 2016 (this version, v4)]
Title:Superpolynomials for toric knots from evolution induced by cut-and-join operators
View PDFAbstract: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.
Submission history
From: Andrei Mironov [view email][v1] Tue, 21 Jun 2011 19:55:57 UTC (69 KB)
[v2] Mon, 18 Jun 2012 16:54:56 UTC (85 KB)
[v3] Mon, 20 Aug 2012 10:58:37 UTC (85 KB)
[v4] Fri, 2 Dec 2016 18:30:44 UTC (84 KB)
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