A filtration question on Belyi pairs and dessins

TL;DR

This paper explores a new filtration concept on Belyi pairs and dessins d'enfants, investigating whether these filtrations coincide, inspired by Vassiliev's knot theory filtration.

Contribution

It introduces a novel filtration on Belyi pairs and dessins d'enfants, and examines their potential equivalence, bridging knot theory and algebraic geometry.

Findings

Defined filtrations on Belyi pairs and dessins d'enfants

Proposed the question of their equivalence

Connected concepts from knot theory to algebraic geometry

Abstract

A Bely\u{\i} 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 Bely\u{i} 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\u{\i} pairs, and another on dessin d'enfants. We ask if the two definitions give the same filtration.