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A skein-theoretic model for the double affine Hecke algebras Morton, Hugh


We consider oriented braids in the thickened torus $T^2 \times I$, together with a single fixed base string. The based skein $H_n(T^2,*)$ is defined to be $\mathbb Z [s^{\pm 1}, q^{\pm 1}]$-linear combinations of $n$-braids subject to the Homflypt skein relation $X_{+}- X_{-} = (s-s^{-1})X_0$. In addition a braid string is allowed to cross through the base string at the expense of multiplying by the parameter $q$. Composition of braids induces an algebra structure on $H_n(T^2,*)$. We show that this algebra satisfies the relations of the double affine Hecke algebra ${\tilde H}_n$, as defined by Cherednik. We discuss how to include closed curves in the thickened torus in the model in an attempt to incorporate earlier work with Peter Samuelson on the Homflypt skein of $T^2$ into the setting of the algebras ${\tilde H}_n$, with an eye on the elliptic Hall algebra and the work of Schiffman and Vasserot.

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