Combinatorial description of jumps in spectral networks

TL;DR

This paper introduces a graph-based parametrization of rational quadratic differentials with simple poles, analyzing how their critical trajectory networks change topologically and relating these bifurcations to Stasheff polytopes.

Contribution

It provides a novel combinatorial framework for understanding topological jumps in spectral networks of quadratic differentials, linking bifurcation diagrams to well-known polytopes.

Findings

Bifurcation diagrams correspond to Stasheff polytopes.

Critical trajectories form networks depending on parameters.

Topological jumps in networks are characterized combinatorially.

Abstract

We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes.