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. DAHA-Jones polynomials for $C^\vee C_1$ depend on $5$ 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 DAHA-superpolynomials for symmetric and wedge powers (and torus knots) conjecturally coincide with the Khovanov-Rozansky stable polynomials, those originated in the theory of BPS states and the superpolynomials defined via rational DAHA in connection with certain Hilbert schemes, though not much is known about such connections beyond the HOMFLYPT and Kauffman polynomials. We also define certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group instead of torus knots in the case of $C^\vee C_1$.