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Jones polynomials of torus knots via DAHA

Ivan Cherednik

TL;DR

The paper develops a unified DAHA framework to realize Quantum Group–Jones polynomials of torus knots across arbitrary root systems and colors, via the PBW theorem and DAHA coinvariants, with a detailed treatment of type $A$ and the DAHA super-polynomials. It introduces DAHA–Jones invariants, DAHA–super-polynomials, and hyper-polynomials, along with conjectures on stabilization, duality, and connections to Khovanov–Rozansky polynomials; it also examines the rational limit and rank-one cases. The work provides extensive numerical evidence across classical and exceptional root systems, including explicit formulas for many knots and weights, and discusses positivity constraints, physics connections (refined Chern–Simons/BPS states), and ties to Hilbert schemes and matrix models. It proposes a program to relate DAHA knot invariants to categorified knot homologies in a universal setting, shedding light on deep links among algebraic, geometric, and topological structures and offering a path toward computing colored invariants in broader root-system contexts. Overall, the approach yields a robust, algebraically founded route to q,t-deformations of Jones-type invariants for torus knots across Lie types and opens avenues for connecting knot invariants with geometric representation theory and mathematical physics.

Abstract

We suggest a new construction for the Quantum Groups - Jones polynomials of torus knots in terms of the PBW theorem of DAHA for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable 3-parametric 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, including small N (for some torus knots). The hyper-polynomials of types B and C are defined, generalizing the Kauffman invariants and containing an extra parameter vs. the super-polynomials. The special values and other features of the DAHA super and hyper-polynomials are discussed; there are many examples in the paper.

Jones polynomials of torus knots via DAHA

TL;DR

The paper develops a unified DAHA framework to realize Quantum Group–Jones polynomials of torus knots across arbitrary root systems and colors, via the PBW theorem and DAHA coinvariants, with a detailed treatment of type and the DAHA super-polynomials. It introduces DAHA–Jones invariants, DAHA–super-polynomials, and hyper-polynomials, along with conjectures on stabilization, duality, and connections to Khovanov–Rozansky polynomials; it also examines the rational limit and rank-one cases. The work provides extensive numerical evidence across classical and exceptional root systems, including explicit formulas for many knots and weights, and discusses positivity constraints, physics connections (refined Chern–Simons/BPS states), and ties to Hilbert schemes and matrix models. It proposes a program to relate DAHA knot invariants to categorified knot homologies in a universal setting, shedding light on deep links among algebraic, geometric, and topological structures and offering a path toward computing colored invariants in broader root-system contexts. Overall, the approach yields a robust, algebraically founded route to q,t-deformations of Jones-type invariants for torus knots across Lie types and opens avenues for connecting knot invariants with geometric representation theory and mathematical physics.

Abstract

We suggest a new construction for the Quantum Groups - Jones polynomials of torus knots in terms of the PBW theorem of DAHA for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable 3-parametric 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, including small N (for some torus knots). The hyper-polynomials of types B and C are defined, generalizing the Kauffman invariants and containing an extra parameter vs. the super-polynomials. The special values and other features of the DAHA super and hyper-polynomials are discussed; there are many examples in the paper.

Paper Structure

This paper contains 40 sections, 4 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Our approach
  3. Perspectives
  4. Acknowledgements
  5. Double Hecke algebras
  6. Affine root systems
  7. Affine Weyl groups
  8. Main definition
  9. Polynomial representation
  10. Automorphisms
  11. The evaluation map
  12. Super-polynomials via DAHA
  13. Jones polynomials
  14. Three super-conjectures
  15. Toward KhR polynomials
  16. ...and 25 more sections

Key Result

Lemma 2.2

Let $H^{(k)}$ be an algebraic expression with constant coefficients from ${\mathbb C}$ in terms of the standard $W$ --invariant polynomials (say, the symmetrizations of monomials) with respect to $\{Y_b\}$ and those with respect to $\{X_b\}$. Then $\widehat{H}^{(k)}$ can be considered as an operator $\qed$

Theorems & Definitions (13)

  • Definition 1.1
  • Conjecture 2.1: DAHA-Jones polynomials
  • Lemma 2.2
  • Proposition 2.3
  • Conjecture 2.4: Stabilization
  • Conjecture 2.5: Duality
  • Conjecture 2.6: Evaluations
  • Conjecture 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 3 more