A filtration question on Belyi pairs and dessins

Abstract

A Bely\uı pair is a holomorphic map from a Riemann surface to $S^2$ with additional properties. A dessin d'enfants is a bipartite graph with additional structure. It is well know that there is a bijection between Belyĭ pairs and dessins d'enfants. Vassiliev has defined a filtration on formal sums of isotopy classes of knots. Motivated by this, we define a filtration on formal sums of Bely\uı pairs, and another on dessin d'enfants. We ask if the two definitions give the same filtration.

3 pages, no figures