Generalized double affine Hecke algebra for double torus

We propose a generalization of the double affine Hecke algebra of type- \(C^\vee C_1\) $C^{\vee }{C}_1$ at specific parameters by introducing a “Heegaard dual” of the Hecke operators. Shown is a relationship with the skein algebra on double torus. We give automorphisms of the algebra associated with the Dehn twists on the double torus.