Jones Polynomials of Torus Knots via DAHA

Abstract

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.