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

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

TL;DR

This work presents a streamlined, MacDonald-polynomial–based evolution for torus knot superpolynomials, derived by deforming the split W-representation of HOMFLY polynomials via a cut-and-join operator. By replacing Schur characters with MacDonald dimensions and refining expansion coefficients, the authors obtain explicit superpolynomials for arbitrary representations and many torus knots, including new results like P_{[1]}^{[m,km±1]}. The approach extends to non-torus knots and offers multiple reductions, such as Alexander and Heegaard–Floer polynomials, with Catalan-number connections in special limits. The framework emphasizes positivity of coefficients in the MacDonald-parameterization, though acknowledges subtleties around unknot choices and non-fundamental representations, suggesting a deep link to Khovanov–Rozhansky homologies and related algebraic structures. Overall, the paper provides a practical recursive scheme for computing torus-knot superpolynomials and highlights rich connections to matrix models, Hurwitz theory, and combinatorial path models.

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\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

TL;DR

This work presents a streamlined, MacDonald-polynomial–based evolution for torus knot superpolynomials, derived by deforming the split W-representation of HOMFLY polynomials via a cut-and-join operator. By replacing Schur characters with MacDonald dimensions and refining expansion coefficients, the authors obtain explicit superpolynomials for arbitrary representations and many torus knots, including new results like P_{[1]}^{[m,km±1]}. The approach extends to non-torus knots and offers multiple reductions, such as Alexander and Heegaard–Floer polynomials, with Catalan-number connections in special limits. The framework emphasizes positivity of coefficients in the MacDonald-parameterization, though acknowledges subtleties around unknot choices and non-fundamental representations, suggesting a deep link to Khovanov–Rozhansky homologies and related algebraic structures. Overall, the paper provides a practical recursive scheme for computing torus-knot superpolynomials and highlights rich connections to matrix models, Hurwitz theory, and combinatorial path models.

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\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.

Paper Structure

This paper contains 66 sections, 330 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations.
  3. HOMFLY polynomials for torus knots
  4. General construction chi
  5. HOMFLY polynomials from $R$-matrix
  6. Fundamental representation
  7. Other representations
  8. Non-torus knots
  9. Deformation to MacDonald characters.
  10. General construction
  11. MacDonald dimensions
  12. From HOMFLY to the superpolynomial: an example
  13. The rules for $\zeta$ evaluation
  14. Comments
  15. Non-fundamental representations. The problem of unknot
  16. ...and 51 more sections

Figures (2)

  • Figure 1: The figure which illustrates the notation in the generalization of the standard hook formula to the MacDonald dimensions (\ref{['McDdh']}).
  • Figure 2: The Schröder path $\pi$ (on the left) and the corresponding Dyck path $T(\pi)$ (on the right) drawn in red, with steps from the sets $T^{-1}(V(T(\pi)))$ and $V(T(\pi))$ drawn in thick blue, and the bounce path corresponding to $T(\pi)$ drawn in dashdotted black. $S(\pi)=12$, $b(\pi)=5$