{"paper":{"title":"Minimal charts of type (3,3)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Akiko Shima, Teruo Nagase","submitted_at":"2016-09-27T04:30:05Z","abstract_excerpt":"Let $\\Gamma$ be a chart. For each label $m$, we denote by $\\Gamma_m$ the \"subgraph\" of $\\Gamma$ consisting of all the edges of label $m$ and their vertices. Let $\\Gamma$ be a minimal chart of type $(m;3,3)$. That is, a minimal chart $\\Gamma$ has six white vertices, and both of $\\Gamma_m\\cap\\Gamma_{m+1}$ and $\\Gamma_{m+1}\\cap\\Gamma_{m+2}$ consist of three white vertices. Then $\\Gamma$ is C-move equivalent to a minimal chart containing a \"subchart\" representing a 2-twist spun trefoil or its \"reflection\"."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08257","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"}