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

We construct a generalization of the $C^{\vee }{C}_1$-type double affine Hecke algebra for the skein algebra on the twice-punctured torus ${\mathrm{\Sigma }}_{1,2}$ 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 ${\mathrm{\Sigma }}_{1,2}$. 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.