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We propose a skein model for the quantum cluster algebras of surface typewith coefficients. We introduce a skein algebra$\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface$(\Sigma,\mathbb{W})$, and prove that it has a quantum cluster structure. Thewalled surfaces naturally generalize the marked surfaces withmulti-laminations, which have been used to describe the quantum clusteralgebras of geometric type for marked surfaces by Fomin--Thurston [FT18].Moreover, we give skein theoretic interpretation for some ofquasi-homomorphisms [Fra16] between these quantum cluster algebras.

Skein and cluster algebras with coefficients for unpunctured surfaces