{"paper":{"title":"The structure of a minimal $n$-chart with two crossings I: Complementary domains of $\\Gamma_1\\cup\\Gamma_{n-1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Akiko Shima, Teruo Nagase","submitted_at":"2017-04-05T01:05:35Z","abstract_excerpt":"This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $\\Gamma$ we denote by $\\Gamma_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $\\Gamma$ with exactly two crossings, we can show that the two crossings are contained in $\\Gamma_\\alpha\\cap\\Gamma_\\beta$ for some labels $\\alpha<\\beta$. In this paper, we study the structure of a disk $D$ not containing any crossing but satisfying $\\Gamma\\cap \\partial D\\subset\\Gamma_{\\alpha+1}\\cup \\Gamma_{\\beta-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01232","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}