  We study the skein algebra of the genus 2 surface and its action on the skein
module of the genus 2 handlebody. We compute this action explicitly, and we
describe how the module decomposes over certain subalgebras in terms of
polynomial representations of double affine Hecke algebras. Finally, we show
that this algebra is isomorphic to the $t=q$ specialisation of the genus two
spherical double affine Hecke algebra recently defined by Arthamonov and
Shakirov.
