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\pm 1] 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.

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

We suggest a construction for the Quantum Groups–Jones polynomials of torus knots in terms of the PBW theorem of double affine Hecke algebra (DAHA) for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable type A variant of this construction. A connection is expected with the approach to super-polynomials due to Aganagic and Shakirov via the Macdonald polynomials at roots of unity and the Verlinde algebra. The duality conjecture for the DAHA super-polynomials is stated, essentially matching that due to Gukov and Stosic. A link to Khovanov–Rozansky polynomials is provided. The hyper-polynomials of types B and C are defined, generalizing the Kauffman polynomials. The special values and other features of the DAHA super- and hyper-polynomials are discussed.

Jones polynomials of torus knots via DAHA

We study the framing dependence of the Wilson loop observable of U(N) Chern-Simons gauge theory at large N. Using proposed geometrical large N dual, this leads to a direct computation of certain topological string amplitudes in a closed form. This yields new formulae for intersection numbers of cohomology classes on moduli of Riemann surfaces with punctures (including all the amplitudes of pure topological gravity in two dimensions). The reinterpretation of these computations in terms of BPS degeneracies of domain walls leads to novel integrality predictions for these amplitudes. Moreover we find evidence that large N dualities are more naturally formulated in the context of U(N) gauge theories rather than SU(N).

Framed knots at large N

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.

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

Super-A-polynomial for knots and BPS states

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g + 1 two strand braids, parallel or antiparallel, and depend on g + 1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g + 1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU
 q
 (N ) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

/pdf/colored-homfly-polynomials-for-the-pretzel-knots-and-links-4rksllq2zn.pdf

Colored HOMFLY polynomials for the pretzel knots and links

Colored knot polynomials for arbitrary pretzel knots and links

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is described in terms of the Vassiliev invariants and Kontsevich integral.

Alexey Sleptsov

Papers

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

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

Colored HOMFLY polynomials for the pretzel knots and links

Colored knot polynomials for arbitrary pretzel knots and links

Genus expansion of HOMFLY polynomials