Recovering the topology of surfaces from cluster algebras

TL;DR

This paper introduces a new method to determine the topology of bordered surfaces with marked points using their cluster algebras, leveraging maximal triangulations and exchange quivers.

Contribution

It provides a novel approach to recover surface topology from cluster algebras, offering new proofs for automorphism and isomorphism problems and clarifying exceptions.

Findings

New proof of automorphism and isomorphism results

Method identifies surface topology from cluster algebra data

Explains exceptions due to pathological triangulations

Abstract

We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange quivers with maximal number of arrows in the mutation class. The method gives new proofs of the automorphism and isomorphism problems for the surface cluster algebras, as well as the uniqueness of the Fomin-Shapiro-Thurston block decompositions of the exchange quivers of the surface cluster algebras. The previous proofs of these results followed a different approach based on Gu's direct proof of the last result. The method also explains the exceptions to these results due to pathological problems with the maximal triangulations of several surfaces.