A cluster theory approach from mutation invariants to Diophantine equations

TL;DR

This paper introduces a cluster theory framework to analyze mutation invariants and Diophantine equations, providing classifications and explanations for differences in cluster algebra types and their associated Diophantine structures.

Contribution

It offers a novel classification of sign-equivalent exchange matrices and connects cluster algebra invariants with Diophantine equations, expanding understanding of their interplay.

Findings

Classification of sign-equivalent exchange matrices

Diophantine explanation for cluster algebra type differences

Identification of Diophantine equations with cluster structures

Abstract

In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive integer points of the Markov mutation invariant and its variant. As an application, several classes of Diophantine equations with cluster algebraic structures are exhibited.