Character varieties on a four-holed sphere

TL;DR

This paper investigates the structure of character varieties on a four-holed sphere, providing a method to determine if two integral solutions are related by the mapping class group, using graph-based techniques.

Contribution

It introduces a novel approach using delta and Vieta maps to analyze orbits on character varieties, establishing decidability of orbit equivalence.

Findings

Decidability of orbit relation for integral solutions

Introduction of graph models for orbit analysis

Application of delta and Vieta maps in character variety study

Abstract

For each $k\in\mathbb{A}^4(\mathbb{C})$, consider the character variety $X_k$ on a four-holed sphere. We prove that it is decidable whether or not any two integral solutions of $X_k$ are in the same mapping class group orbit. For this, using a delta map and Vieta maps, we will introduce graphs corresponding to the orbits and observe the properties of vertices on the graph.