T-Path Formula and Atomic Bases for Cluster Algebras of Type D

TL;DR

This paper extends a combinatorial expansion formula for arcs on surfaces to punctured polygons and proves that cluster monomials form the atomic basis in type D cluster algebras.

Contribution

It introduces a T-path expansion formula for punctured polygons and provides a combinatorial proof of the atomic basis property for type D cluster algebras.

Findings

Extended T-path formula to once-punctured polygons

Proved cluster monomials form the atomic basis in type D

Provided combinatorial proof for the basis property

Abstract

We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D.