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 \ 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 having no more than five edges, 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. © 2015 Springer Science+Business Media New York.