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

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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 \( \mathcal{P}_{{\left[ 1 \right]}}^{{\left[ {m,km\pm 1} \right]}} \) for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots.

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  1. ITEP, Bol. Cheremushinskaya, 25, Moscow, Russia

    P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov & A. Smirnov

  2. MIPT, Institutsky per., 9, Dolgoprudny, Russia

    P. Dunin-Barkowski, A. Sleptsov & A. Smirnov

  3. Theory Department, Lebedev Physics Institute, Leninsky pr., 53, Moscow, Russia

    A. Mironov

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  1. P. Dunin-Barkowski
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ArXiv ePrint: 1106.4305

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Dunin-Barkowski, P., Mironov, A., Morozov, A. et al. Superpolynomials for torus knots from evolution induced by cut-and-join operators. J. High Energ. Phys. 2013, 21 (2013). https://doi.org/10.1007/JHEP03(2013)021

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