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 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.

Link polynomial calculus and the AENV conjecture

“…The answers are represented either in the differential hierarchy (Z-expansion) form of [25,36], which is convenient to control the representation dependence, or in the evolution based form of [29,43], convenient to control the dependence on the shape of the knot. These two representations look very different, but in every particular case one can easily convert between them.…”

“…In this section we provide results about the link family, to which the Hopf and Whitehead links naturally belong and study the evolution a la [29,43] along this family (see figure 2).…”

“…In fact, one can consider these formulas from the point of view of the evolution method of [29] and [43] This time it is clear that the powers of A are limited, however, instead the differential structure of (5.3) is obscure. An alternative formulation, where the both properties are transparent is as follows: and the coefficients f remain to be determined.…”

Checks of integrality properties in topological strings

“…where the sum runs over all Young diagrams R ("colors"), χ R are the Schur functions of time variablesp k and the Adams (plethysm) operation Ad d : p k −→ p kd acts at the topological locus [73][74][75][76]87] by raising the power of parameters:…”

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…In particular, the knot superpolynomials of [49][50][51], constructed with the help of double-affine Hecke algebras (DAHA) [52], still lack a clear R-matrix realization within the Reshetikhin-Turaev (RT) formalism, either original [53][54][55] or modern [56][57][58]. On the other hand, the DIM algebra is naturally related with DAHA by a kind of Schur duality (see [59] for a degenerate version of this correspondence).…”