Superpolynomials for torus 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± 1} ]}} for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots.
- Publication:
-
Journal of High Energy Physics
- Pub Date:
- March 2013
- DOI:
- 10.1007/JHEP03(2013)021
- arXiv:
- arXiv:1106.4305
- Bibcode:
- 2013JHEP...03..021D
- Keywords:
-
- Chern-Simons Theories;
- Quantum Groups;
- High Energy Physics - Theory;
- Mathematics - Geometric Topology
- E-Print:
- 23 pages + Tables (51 pages)