Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

Abstract

It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. Previously we linked this framework to Lam and Pylyavskyy's Laurent phenomenon algebras, showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures.