Denominator conjecture for some surface cluster algebras

TL;DR

This paper proves the denominator conjecture for a class of surface cluster algebras, showing that cluster monomials are uniquely identified by their denominator vectors in these cases.

Contribution

It establishes the denominator conjecture for surface cluster algebras with at least three boundary marked points, a significant step in understanding their structure.

Findings

Proves the denominator conjecture for specific surface cluster algebras

Demonstrates uniqueness of cluster monomials via denominator vectors

Focuses on algebras from marked surfaces with boundary points

Abstract

The denominator conjecture, proposed by Fomin and Zelevinsky, says that for a cluster algebra, the cluster monomials are uniquely determined by their denominator vectors with respect to an initial cluster. In this paper, for a cluster algebra from a marked surface with at least three boundary marked points, we establish this conjecture with respect to a given strong admissible tagged triangulation.