Zapponi-orientable dessins d'enfants

TL;DR

This paper extends Zapponi's concept of orientability for dessins d'enfants to a broader context, introduces twist orientability, and studies their invariance and dependence on classical invariants, especially in regular dessins.

Contribution

It generalizes Z-orientability to all dessins, introduces twist orientability, and analyzes their Galois invariance and relation to classical invariants.

Findings

Z-orientability and twist orientability are Galois-invariant.

These properties are generally independent of passport, monodromy, and automorphism groups.

In regular dessins, they are determined by the monodromy group.

Abstract

Almost two decades ago Zapponi introduced a notion of orientability of a clean dessin d'enfant, based on an orientation of the embedded bipartite graph. We extend this concept, which we call Z-orientability to distinguish it from the traditional topological definition, to the wider context of all dessins, and we use it to define a concept of twist orientability, which also takes account of the Z-orientability properties of those dessins obtained by permuting the roles of white and black vertices and face-centres. We observe that these properties are Galois-invariant, and we study the extent to which they are determined by the standard invariants such as the passport and the monodromy and automorphism groups. We find that in general they are independent of these invariants, but in the case of regular dessins they are determined by the monodromy group.