Belyi Function Decompositions for The Icosahedron of Genus 4

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 ${\beta }_{I_0}:{\mathbb{CP}}^1\to {\mathbb{CP}}^1$ and ${\beta }_{I_4}:\mathcal{B}\to {\mathbb{CP}}^1$ 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.

22 pages, 15 figures