Catalog of dessins d'enfants with \le 4 edges

TL;DR

This paper catalogs all dessins d'enfants with up to four edges, computes their Belyi pairs, automorphism groups, and provides matrix model representations, offering a comprehensive enumeration and analysis of these mathematical objects.

Contribution

It provides the first complete enumeration of dessins d'enfants with up to four edges, including their Belyi pairs, automorphism groups, and associated matrix models, using advanced computational techniques.

Findings

Total of 134 dessins enumerated

77 spherical, 53 genus 1, 4 genus 2 dessins identified

Matrix models constructed for each dessin based on their valencies

Abstract

In this work all the dessins d'enfant with no more than 4 edges are listed and their Belyi pairs are computed. In order to enumerate all dessins the technique of matrix model computations was used. The total number of dessins is 134; among them 77 are spherical, 53 of genus 1 and 4 of genus 2. The orders of automorphism groups of all the dessins are also found. Dessins are listed by the number of edges. Dessins with the same number of edges are ordered lexicographically by their lists of 0-valencies. The corresponding matrix model for any list of 0-valencies is given and computed. Complex matrix models for dessins with 1 -- 3 edges are used. For the dessins with 4 edges we use Hermitian matrix model, correlators for which are computed in [1].