Orders of the canonical vector bundles over configuration spaces of finite graphs

TL;DR

This paper determines the order of the canonical vector bundle over configuration spaces of finite graphs, showing it is 2 for planar graphs and 4 for nonplanar graphs, revealing a topological distinction based on planarity.

Contribution

It establishes the exact order of the canonical vector bundle over configuration spaces for finite graphs, distinguishing planar and nonplanar cases.

Findings

Order is 2 for planar graphs

Order is 4 for nonplanar graphs

Topological properties depend on graph planarity

Abstract

We prove that the order of the canonical vector bundle over the configuration space is $2$ for a general planar graph, and is $4$ for a nonplanar graph.