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{
    "pk": 64880,
    "title": "Triangular-grid billiards and plabic graphs",
    "subtitle": null,
    "abstract": "Given a polygon \\(P\\) in the triangular grid, we obtain a permutation \\(\\pi_P\\) via a natural billiards system in which beams of light bounce around inside of \\(P\\). The different cycles in \\(\\pi_P\\) correspond to the different trajectories of light beams. We prove that \\[\\operatorname{area}(P)\\geq 6\\operatorname{cyc}(P)-6\\quad\\text{and}\\quad\\operatorname{perim}(P)\\geq\\frac{7}{2}\\operatorname{cyc}(P)-\\frac{3}{2},\\] where \\(\\operatorname{area}(P)\\) and \\(\\operatorname{perim}(P)\\) are the (appropriately normalized) area and perimeter of \\(P\\), respectively, and \\(\\operatorname{cyc}(P)\\) is the number of cycles in \\(\\pi_P\\). The inequality concerning \\(\\operatorname{area}(P)\\) is tight, and we characterize the polygons \\(P\\) satisfying \\(\\operatorname{area}(P)=6\\operatorname{cyc}(P)-6\\). These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let \\(G\\) be a connected reduced plabic graph with essential dimension \\(2\\). Suppose \\(G\\) has \\(n\\) marked boundary points and \\(v\\) (internal) vertices, and let \\(c\\) be the number of cycles in the trip permutation of \\(G\\). Then we have \\[v\\geq 6c-6\\quad\\text{and}\\quad n\\geq\\frac{7}{2}c-\\frac{3}{2}.\\]\n \nMathematics Subject Classifications: 05D99, 51M04\n \nKeywords: Triangular grid, billiards, plabic graph, membrane",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Triangular grid"
        },
        {
            "word": "billiards"
        },
        {
            "word": "plabic graph"
        },
        {
            "word": "membrane"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/5tg1p6sx",
    "frozenauthors": [
        {
            "first_name": "Colin",
            "middle_name": "",
            "last_name": "Defant",
            "name_suffix": "",
            "institution": "Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Pakawut",
            "middle_name": "",
            "last_name": "Jiradilok",
            "name_suffix": "",
            "institution": "Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2023-03-14T15:39:51Z",
    "date_accepted": "2023-03-14T15:39:51Z",
    "date_published": "2023-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64880/galley/49690/download/"
        }
    ]
}