On dessins d'enfants with equal supports

Abstract

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