{"paper":{"title":"The structure of a minimal $n$-chart with two crossings II: Neighbourhoods 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":"2017-09-26T04:46:28Z","abstract_excerpt":"Given a 2-crossing minimal chart $\\Gamma$, a minimal chart with two crossings, set $\\alpha=\\min\\{~i~|~$there exists an edge of label $i$ containing a white vertex$\\}$, and $\\beta=\\max\\{~i~|~$there exists an edge of label $i$ containing a white vertex$\\}$. In this paper we study the structure of a neighbourhood of $\\Gamma_\\alpha\\cup\\Gamma_\\beta$, and propose a normal form for 2-crossing minimal $n$-charts, here $\\Gamma_\\alpha$ and $\\Gamma_\\beta$ mean the union of all the edges of label $\\alpha$ and $\\beta$ respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08827","kind":"arxiv","version":2},"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"}