Unistructurality of cluster algebras from surfaces without punctures

TL;DR

This paper proves that cluster algebras from surfaces without punctures are unistructural, meaning their cluster variables uniquely determine their clusters and seeds, using the bracelet basis and skein relations.

Contribution

It establishes unistructurality for cluster algebras from unpunctured surfaces and characterizes unistructurality for disjoint unions of quivers.

Findings

Cluster algebras from unpunctured surfaces are unistructural.

Unistructurality for disjoint unions of quivers depends on their connected components.

The bracelet basis and skein relations are key tools in the proof.

Abstract

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis, and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural.