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.
Categories: math.NT