The rotor-routing torsor and the Bernardi torsor disagree for every non-planar ribbon graph

TL;DR

This paper proves that the rotor-routing torsor and Bernardi torsor, two structures on spanning trees related to the Picard group, differ for all non-planar ribbon graphs, confirming a conjecture.

Contribution

It establishes that the two torsors do not coincide for non-planar ribbon graphs, resolving a conjecture and extending understanding of their differences.

Findings

The torsors agree for planar graphs.

The torsors disagree for all non-planar graphs.

Confirmed the conjecture by Baker and Wang.

Abstract

Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar.