Presentations of NET maps

TL;DR

This paper introduces a normal form for nearly Euclidean Thurston (NET) maps using affine data, enabling systematic computation and tabulation of their fundamental invariants.

Contribution

It provides a normal form for NET maps based on affine data, facilitating algorithmic computation of their invariants.

Findings

Normal form for NET maps established

Algorithmic methods for invariant computation developed

Large census of NET maps and their invariants created

Abstract

A branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of simple affine data. This data can then be used as input for algorithms developed for the computation of fundamental invariants, now systematically tabulated in a large census.