Shabat Polynomials and Monodromy Groups of Trees Uniquely Determined by their Passport

TL;DR

This paper explores the relationship between Shabat polynomials and monodromy groups for planar acyclic dessins, focusing on those uniquely identified by their passports, linking algebraic and combinatorial properties.

Contribution

It provides a detailed analysis of Shabat polynomials and monodromy groups for dessins uniquely determined by their passports, bridging algebraic and combinatorial perspectives.

Findings

Characterization of dessins uniquely determined by passports

Connection between Shabat polynomials and monodromy groups

Insights into the algebraic and combinatorial structure of dessins

Abstract

A dessin d'enfant, or dessin, is a bicolored graph embedded into a Riemann surface. Acyclic dessins can be described analytically by pre-images of certain polynomials, called Shabat polynomials, and also algebraically by their monodromy groups, that is, the group generated by rotations of edges about black and white vertices. In this paper we investigate the Shabat polynomials and monodromy groups of planar acyclic dessins that are uniquely determined by their passports.