Belyi Function Decompositions for The Icosahedron of Genus 4

TL;DR

This paper investigates the decompositions of Belyi functions associated with the genus 4 icosahedron embedded in Bring's curve, extending known results from the regular icosahedron on the Riemann sphere to a more complex algebraic setting.

Contribution

It provides the first known decompositions of the Belyi function for the genus 4 icosahedron embedded in Bring's curve, linking it to the classical case on the Riemann sphere.

Findings

Decompositions of $eta_{I_4}$ are explicitly determined.

The decompositions relate algebraic curves of different types.

The structure of $eta_{I_4}$ mirrors known decompositions of $eta_{I_0}$.

Abstract

The icosahedron $I_4$ of genus 4 is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $\beta_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $\beta_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $\beta_{I_0}$ decompositions is already known. In the present paper we find $\beta_{I_4}$ decompositions. Note that $\beta_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $\beta_{I_4}$ we deal with maps between different algebraic curves.