DAHA-Jones polynomials of torus knots

Abstract

DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author’s previous results for $$A_1$$ .