Pairs of tree dessins, their Shabat polynomials, and monodromy groups
Authors
Benjamin Dupont
Revekka Kyriakoglou
Vassilis Metaftsis
Efstratios Prassidis
Alexandros Singh
Abstract
Coverings of the Riemann sphere by itself, ramified over two points, are given by so-called Shabat polynomials. The correspondence between Grothendieck's dessins d'enfants and Belyi maps then implies a bijection between Shabat polynomials and tree dessins (bicolored plane trees). Dessins can be assigned a combinatorial invariant known as their passport, which records the degrees of their vertices. We consider all possible passports determining a pair of tree dessins, determining the associated Shabat polynomials and monodromy groups.
