{"paper":{"title":"Properties of minimal charts and their applications V: charts of type $(3,2,2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Akiko Shima, Teruo Nagase","submitted_at":"2019-01-31T02:44:28Z","abstract_excerpt":"Let $\\Gamma$ be a chart, and we denote by $\\Gamma_m$ the union of all the edges of label $m$. A chart $\\Gamma$ is of type $(3,2,2)$ if there exists a label $m$ such that $w(\\Gamma)=7$, $w(\\Gamma_m\\cap\\Gamma_{m+1})=3$, $w(\\Gamma_{m+1}\\cap\\Gamma_{m+2})=2$, and $w(\\Gamma_{m+2}\\cap\\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00007","kind":"arxiv","version":1},"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"}