Analytic and Topological Nets

TL;DR

This paper characterizes planar graphs that are pullbacks of Jordan curves under rational maps and shows their density among certain inverse images of branched covers, linking topology and complex dynamics.

Contribution

It provides a characterization of planar graphs as pullbacks of Jordan curves under rational maps and proves their density among inverse images of branched covers.

Findings

Characterization of planar graphs as pullbacks of Jordan curves.

Density of such pullbacks within inverse images of branched covers.

Connection between graph topology and complex rational maps.

Abstract

We characterize which planar graphs arise as the pullback, under a rational map $r$, of an analytic Jordan curve passing through the critical values of $r$. We also prove that such pullbacks are dense within the collection of $f^{-1}(\Sigma)$, where $f$ is a branched cover of the sphere and $\Sigma$ is a Jordan curve passing through the branched values of $f$.