Iterated torus knots and double affine Hecke algebras

We give a topological realization of the (spherical) double affine Hecke algebra SH_q,t of type sl₂, and we use this to construct a module over SH_q,t for any knot K⊂S³ . As an application, we give a purely topological interpretation of Cherednik's 2-variable polynomials P_n (r,s;q,t) of type sl₂  from [Che13] (where r,s ∈ Z are relatively prime), and we give a new proof that these specialize to the colored Jones polynomials of the r,s torus knot.
We then generalize Cherednik's construction (for sl₂ ) to all iterated cables of the unknot and prove the corresponding specialization property. Finally, in the appendix we compare our polynomials associated to iterated torus knots to the ones recently defined in [CD14], in the specialization t=−q ² .