Torsor Structures on Spanning Trees

TL;DR

This paper proves a conjecture relating two actions of the Picard group on spanning trees of ribbon graphs, showing they differ in nonplanar cases and providing explicit examples of their relationship.

Contribution

It proves Baker and Wang's conjecture for nonplanar ribbon graphs without multiple edges and extends it to graphs with multiple edges.

Findings

Confirmed the conjecture for nonplanar ribbon graphs without multiple edges.

Extended the conjecture to certain graphs with multiple edges.

Provided explicit examples illustrating the relationship between the torsor structures.

Abstract

We study two actions of the (degree 0) Picard group on the set of the spanning trees of a finite ribbon graph. It is known that these two actions, denoted $\beta_q$ and $\rho_q$ respectively, are independent of the base vertex $q$ if and only if the ribbon graph is planar. Baker and Wang conjectured that in a nonplanar ribbon graph without multiple edges there always exists a vertex $q$ for which $\rho_q\neq\beta_q$. We prove the conjecture and extend it to a class of ribbon graphs with multiple edges. We also give explicit examples exploring the relationship between the two torsor structures in the nonplanar case.