Homology in Combinatorial Refraction Billiards

TL;DR

This work develops a topological lens on toric combinatorial refraction billiards by projecting graph-associated hyperplane arrangements to the $(n-1)$-torus and encoding dynamics via the discrete map $\Theta_G$. It proves a sharp expelling criterion: a graph is expelling if and only if it is bipartite, and it offers a collection of necessary and sufficient conditions for ensnaring graphs, including complete graphs, odd cycles, wedges, unions, and complement-based constructions. By introducing winding vectors and a homology-based framework, the paper translates geometric contractibility into combinatorial invariants, enabling precise criteria and constructive graph operations that preserve ensnaring. The results connect graph structure with toric homology, yield dense families of ensnaring graphs through complements, and open multiple directions for generalization, including mirrors and other affine Weyl groups, with potential implications for higher-dimensional combinatorial dynamical systems. Overall, the work advances a robust methodology for classifying graph-driven toric billiards via homological properties and discrete dynamical systems.

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.

Figures (7)

• Figure 1: A beam of light refracts through a horizontal line.
• Figure 2: On the top is an unbounded combinatorial refraction billiard trajectory in $\widetilde{\mathfrak{S}}_3$. On the bottom is the corresponding toric combinatorial refraction billiard trajectory, with each state represented both as an arrow in the torus and as a stone diagram.
• Figure 3: On the top is a bounded combinatorial refraction billiard trajectory in $\widetilde{\mathfrak{S}}_3$. On the bottom is the corresponding toric combinatorial refraction billiard trajectory, with each state represented both as an arrow in the torus and as a stone diagram.
• Figure 4: Four applications of the map $\Theta_{G}$. At each step, we represent a triple in $\Xi_5$ as a stone diagram with the associated coin diagram drawn below.
• Figure 5: A sequence transforming one pleasant stone diagram into another.

Theorems & Definitions (50)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 2.1
• Example 2.2
• Example 2.3
• Proposition 2.4