Geodesic
On the Belyi functions of planar circular maps
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 111-133.

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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 $E$. 
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M. A. Deryagina; A. D. Mednykh. On the Belyi functions of planar circular maps. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 111-133. http://geodesic.mathdoc.fr/item/FPM_2013_18_6_a6/

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