A skein-theoretic model for the double affine Hecke algebras

We will illustrate pictorially the use of ${\mathbb Z}[s^{\pm 1}, q^{\pm 1}]$-linear combinations of braids in the thickened torus $T^{2}\times I$ to construct an algebra induced by composing $n$-string braids.

We will show, with the help of pictures, that this algebra satisfies the relations of the double affine Hecke algebra $\ddot{H}_{n}$, which will be introduced algebraically.

We will finish with a rather speculative plan to include closed curves in our model in an attempt to incorporate earlier work with Peter Samuelson on the Homfly skein of $T^{2}$ into the setting of the algebras $\ddot{H}_{n}$. This is done with an eye on the elliptic Hall algebra and the work of Schiffman and Vasserot, which we will discuss very briefly.