Jones polynomials of torus knots via DAHA

Details Jones polynomials of torus knots via DAHA Jones polynomials of torus knots via DAHA

2011-11-26

arxiv.org/abs/1111.6195v9

Mathematics

Quantum Algebra

v3: an extension toward super-polynomials; v4: the duality conjecture and some other modifications; v6: the special values con

Scientific paper

We suggest a construction for the QG-Jones colored polynomials of torus knots in terms of the PBW theorem of DAHA for any root systems and weights (proved only for type A). The main focus is on the corresponding 3-parametric super-extensions of this construction in type A. 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 q,t,a-polynomiality and duality conjectures for the DAHA super-polynomials are stated, essentially matching those 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.

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