On dessins d'enfants with equal supports

Abstract

For a Belyi function ${\hat{I}}^2:\mathbb{C}{\mathbb{P}}^1\to \mathbb{C}{\mathbb{P}}^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' ${\mathcal{D}}_{\hat{I}}^2$ is defined as the set $β^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $β^{-1}\{-1\}$ and $β^{-1}\{1\}$ correspondingly. Merely the set $β^{-1}([-1,1])$ without a graph structure is called a support of ${\mathcal{D}}_{\hat{I}}^2$. In this note, we solve the following problem: under what conditions different dessins ${\mathcal{D}}_{{\hat{I}}_1^2}$ and ${\mathcal{D}}_{{\hat{I}}_2^2}$ have equal supports?