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

By P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, A. Smirnov. 2011

175 citations recorded by the source corpus.

A bstractThe 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a097723072229c878440ba8f060ae5aaa22e59eb

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Citations in the corpus

Homological algebra of knots and BPS states (2011, background): “ of this paper the Sr-colored superpolynomials are related (by a simple change of variables) to the Λr-colored invariants that one would ﬁnd by following the same steps in grading conventions of e.g. [3, 58, 59]: (4.2) PSr here = P Λr elsewhere Note, that the Sr-colored invariant is related to the Λ”
Jones polynomials of torus knots via DAHA (2011, methodology): “14) and up to a (general) proportionality coefficient; see also formulas (169) and (176) from [12].”
Torus knots and the rational DAHA (2012, methodology): “−t Hm,n(x;q,t). This equation can be compared with (6.10), what allows one to recast the formulas for Hm,n in terms of the standard Macdonald polynomials. Such formulas were previously conjectured in [DMMSS11], where ∇ was denoted by eWˆ . 7. DIFFERENTIALS In this section, we construct a family of d”
Refined knot invariants and Hilbert schemes (2013, methodology): “sources: for m = 1they are remarkably simple and were computed ﬁrst in [23] and later rediscovered in [57, 62]. For general m and small n some of these coefﬁcients were computed in [57, Section 5.3], [15] and [62]. Although the individual terms cn,m(T) have a nice factorized form, their sums cn,m(λ)”