Abstract:
A Belyǐ map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ is a rational function with at most three critical values; we may assume these are $\{ 0, \, 1, \, \infty \}$. A Dessin d'Enfant is a planar bipartite graph on the sphere obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$. This project sought to expand on a database of such Belyǐ pairs, their corresponding Dessins d'Enfant, and their monodromy groups. We did so for up to degree $N = 5$ in the hopes of generating an algorithm to generate Dessins from monodromy triples. No prior knowledge is necessary.