Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

TL;DR

This paper extends the framework of Laurent phenomenon algebras to include punctured surfaces with laminations, demonstrating that all such surfaces can be associated with LP structures, broadening the scope of cluster algebra models.

Contribution

It introduces a method to incorporate punctured surfaces and laminations into Laurent phenomenon algebra structures, expanding the class of surfaces that admit LP structures.

Findings

All punctured and unpunctured surfaces admit LP structures.

Laminations enable the extension of LP structures to more complex surfaces.

The framework unifies cluster algebra models for various surface types.

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.