Homology in Combinatorial Refraction Billiards
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913809707302912 |
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| author | Defant, Colin Liu, Derek |
| author_facet | Defant, Colin Liu, Derek |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homology in Combinatorial Refraction Billiards Defant, Colin Liu, Derek Combinatorics 05E18, 37E99, 20F34 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. |
| title | Homology in Combinatorial Refraction Billiards |
| topic | Combinatorics 05E18, 37E99, 20F34 |
| url | https://arxiv.org/abs/2502.06013 |