Character varieties on a four-holed sphere

For each $\mathbf{k}\in {\mathbb{C}}^4$, let $V_{\mathbf{k}}$ be the character variety on a four-holed sphere and ${\mathrm{\Gamma }}_{\mathbf{k}}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for ${\mathrm{\Gamma }}_{\mathbf{k}}$-action on $V_{\mathbf{k}}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for $V_{\mathbf{k}}$ are in the same ${\mathrm{\Gamma }}_{\mathbf{k}}$-orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the ${\mathrm{\Gamma }}_{\mathbf{k}}$-orbits and the mapping class group orbits, and classify their restricted global shapes by analyzing the limited local edge configurations at each vertex, using a descent argument.