Rational Exceptional Belyi Coverings

TL;DR

This paper classifies rational exceptional Belyi coverings of genus 0 surfaces, identifying all such coverings up to degree 15 and computing their Belyi functions, including new infinite series.

Contribution

It provides a complete classification of rational exceptional Belyi coverings up to degree 15 and introduces new infinite series, expanding the understanding of these coverings.

Findings

All rational exceptional Belyi coverings up to degree 15 identified.

Belyi functions computed for degrees up to 6 and some for degree 7.

New infinite series of rational exceptional Belyi coverings discovered.

Abstract

Exceptional Belyi covering is a connected Belyi covering uniquely determined by its ramification scheme or the respective dessin d'enfant. We focus on rational exceptional Belyi coverings of compact Riemann surfaces of genus 0. Well known examples are cyclic, dihedral, and Chebyshev coverings. Using Maple, we identified all rational exceptional Belyi coverings up to degree 15. Their Belyi functions were calculated for degrees up to 6 along with some for degree 7. We also found new infinite series.