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Superpolynomials for torus knots from evolution induced by cut-and-join operators. (English) Zbl 1342.57004
Summary: 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 $$\mathcal P^{[m,km\pm 1]}_{[1]}$$ for arbitrary $$m$$ and $$k$$. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81T45 Topological field theories in quantum mechanics
##### Keywords:
Chern-Simons theories; quantum groups
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