Homology in Combinatorial Refraction Billiards

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.