Abstract. We give a generators and relations presentation of the Homflypt skein algebra H of the torus T 2 , and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that H is isomorphic to the t = q specialization of the elliptic Hall algebra of Burban and Schiffmann [BS12].As an application, for an iterated cable K of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the λ-colored Homflypt polynomial of K. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the gl N double affine Hecke algebra. This proves one of the Connection Conjectures in [CD14].

Abstract. In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot K ⊂ S 3 . We prove this in a number of nontrivial cases, including all (2, 2p + 1) torus knots, the figure eight knot, and all 2-bridge knots (when q = ±1). As the main application of the conjecture, we construct 3-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev in [RT90].We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies the colored Jones polynomials of K satisfy an inhomogeneous recursion relation.

It is known that the fundamental group homomorphism π 1 (T 2 ) → π 1 (S 3 \ K) induced by the inclusion of the boundary torus into the complement of a knot K in S 3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL 2 -character varieties of the corresponding fundamental groups. In our earlier work [BS16], we proposed a conjecture that the classical restriction map admits a canonical 2parameter deformation into a smooth cubic surface. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [BH95] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel-Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel-Hilden conjecture holds is closed under taking connect sums and certain knot coverings.To Efim Zelmanov on the occasion of his 60th birthday Contents

Abstract. We give a topological realization of the (spherical) double affine Hecke algebra SHq,t of type sl 2 , and we use this to construct a module over SHq,t for any knot K ⊂ S 3 . As an application, we give a purely topological interpretation of Cherednik's 2-variable polynomials Pn(r, s; q, t) of type sl 2 from [Che13] (where r, s ∈ Z are relatively prime).We then generalize the construction of these polynomials (for sl 2 ) from torus knots to all iterated cables of the unknot and prove they specialize to the colored Jones polynomials of the knot. Finally, in the Appendix we compare this construction to a later construction of Cherednik and Danilenko.

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