Homology in Combinatorial Refraction Billiards

TL;DR

This paper explores the topological properties of billiard trajectories in a toric hyperplane arrangement derived from a graph, classifying graphs based on whether all trajectories are contractible or not.

Contribution

It characterizes graphs as expelling or ensnaring based on the contractibility of billiard trajectories, providing necessary and sufficient conditions for these classifications.

Findings

Expelling graphs are exactly bipartite graphs.

Complement of an ensnaring graph cannot have a clique as a connected component.

Gluing ensnaring graphs at a vertex yields another ensnaring graph.

Abstract

Given a graph $G$ with vertex set $\{1,\ldots,n\}$, we can project the graphical arrangement of $G$ to an $(n-1)$-dimensional torus to obtain a toric hyperplane arrangement. Adams, Defant, and Striker constructed a toric combinatorial refraction billiard system in which beams of light travel in the torus, refracting (with refraction coefficient $-1$) whenever they hit one of the toric hyperplanes in this toric arrangement. Each billiard trajectory in this system is periodic. We adopt a topological perspective and view the billiard trajectories as closed loops in the torus. We say $G$ is ensnaring if all of the billiard trajectories are contractible, and we say $G$ is expelling if none of the billiard trajectories is contractible. Our first main result states that a graph is expelling if and only if it is bipartite. We then provide several necessary conditions and several sufficient conditions for a graph to be ensnaring. For example, we show that the complement of an ensnaring graph cannot have a clique as a connected component. We also discuss ways to construct ensnaring graphs from other ensnaring graphs. For example, gluing two ensnaring graphs at a single vertex always yields another ensnaring graph.