Train track combinatorics and cluster algebras

The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this paper, we translate some concepts of train tracks into the language of cluster algebras using the Goncharov–Shen’s potential function [GS15]. Through this translation, we prove the sign stability [IK21] of the general pseudo-Anosov mapping classes.