Jones polynomials of torus knots via DAHA

TL;DR

This paper introduces a new DAHA-based approach to compute Jones polynomials and super-polynomials of torus knots, linking quantum groups, Macdonald polynomials, and Khovanov-Rozansky invariants.

Contribution

It presents a novel DAHA construction for Jones polynomials and super-polynomials, connecting various knot invariants and extending to hyper-polynomials of types B and C.

Findings

DAHA super-polynomials are constructed and connected to Macdonald polynomials.

Duality conjecture for DAHA super-polynomials is formulated.

Hyper-polynomials generalize Kauffman invariants with additional parameters.

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.