Train track combinatorics and cluster algebras
Shunsuke Kano
TL;DR
This paper connects train track concepts with cluster algebras, demonstrating sign stability of pseudo-Anosov mapping classes through a novel translation of ideas.
Contribution
It introduces a new translation of train track concepts into cluster algebra language, proving sign stability for pseudo-Anosov classes.
Findings
Proves sign stability of pseudo-Anosov mapping classes
Translates train track concepts into cluster algebra framework
Establishes a new link between topology and algebra
Abstract
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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics