A note on double affine Hecke algebra for skein algebra on twice-punctured torus

Abstract

We construct a generalization of the $C^{\vee }{C}_1$-type double affine Hecke algebra for the skein algebra on the twice-punctured torus ${\Sigma }_{1,2}$ using the Heegaard dual of the Iwahori--Hecke operator recently introduced in our previous article. We show that the automorphisms of our algebra correspond to the Dehn twists about the curves on ${\Sigma }_{1,2}$. We also give the cluster algebraic construction of the the classical limit of the skein algebra, where the Dehn twists are given in terms of the cluster mutations.

15 pages, 4 figures