The spherical Double Affine Hecke Algebra (DAHA) can be viewed as a noncommutative
$(q,t)$-deformation of the $ {\rm SL}(N, {\mathbb C}) $-character variety of the fundamental group
of a torus. This deformation inherits a major topological property from its
commutative counterpart: namely, the mapping class group of a torus, ${\rm SL}(2, {\mathbb Z}) $,
acts by atomorphisms on DAHA.

In my talk, I will review the theory of symmetric polynomials from a
topological viewpoint. I will then define a genus two analogue of the
spherical DAHA of type $A_1$ and show that the mapping class group
of a closed genus two surface acts by automorphisms on such an algebra.
I will conclude with a discussion of potential applications to TQFT and knot theory.

Based on arXiv:1704.02947 (joint with Sh. Shakirov)