On dessins d'enfants with equal supports

TL;DR

This paper investigates the conditions under which different dessins d'enfants, associated with Belyi functions ramified over three points, share the same support set on the Riemann sphere.

Contribution

It provides a characterization of when two dessins d'enfants have identical supports, clarifying the relationship between Belyi functions and their underlying support sets.

Findings

Identifies conditions for equal supports of dessins d'enfants

Characterizes the structure of Belyi functions with shared supports

Enhances understanding of dessins d'enfants and their supports

Abstract

For a Belyi function $\beta:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ 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?