Homology in combinatorial refraction billiards
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40687_2025_524_ReferencePDF.pdf
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3.32 MB
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Author(s)
Defant, Colin
Liu, Derek
Date Issued
May 14, 2025
Journal
Research in the Mathematical Sciences
Publisher
Springer International Publishing
Citation
Defant, C., Liu, D. Homology in combinatorial refraction billiards. Res Math Sci 12, 36 (2025).
Version
Author's final manuscript
Abstract
Given a graph G with vertex set { 1 , … , 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s40687-025-00524-8
