On the Belyi functions of planar circular maps

A map $(S,G)$ is a closed Riemann surface $S$ with an embedded graph $G$ such that $S\setminus G$ amounts to the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. The purpose of this article is to demonstrate a method of finding a Belyi function for planar circular maps and a way to plot a planar circular map by its Belyi function. Also we present a list of planar circular maps with the number of edges not exceeding five, their Belyi functions and their plots. We remark that the Belyi function for a planar circular map with $E$ edges obtained with the help of our method is a rational function of degree $E$.