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

We construct a generalization of the C^ C₁-type double affine Hecke algebra for the skein algebra on the twice-punctured torus 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 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.