Dessins d'enfants for analysts

TL;DR

This paper introduces an algorithmic approach for exactly computing Belyi functions and dessins d'enfants for hypermaps and triangulations in genus 0 or 1, enabling analysis of larger graphs with minimal algebraic geometry.

Contribution

It presents a novel numerical iterative method combined with lattice reduction for computing Belyi functions, expanding applicability to larger graphs compared to previous methods.

Findings

Applicable to larger graphs than previous methods

Uses minimal algebraic geometry, making it more accessible

Provides a self-contained approach for computing dessins d'enfants

Abstract

We present an algorithmic way of exactly computing Belyi functions for hypermaps and triangulations in genus 0 or 1, and the associated dessins, based on a numerical iterative approach initialized from a circle packing combined with subsequent lattice reduction. The main advantage compared to previous methods is that it is applicable to much larger graphs; we use very little algebraic geometry, and aim for this paper to be as self-contained as possible.