{"paper":{"title":"Local geometry of the k-curve graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Tarik Aougab","submitted_at":"2015-08-03T17:42:29Z","abstract_excerpt":"Let $S$ be an orientable surface with negative Euler characteristic. For $k \\in \\mathbb{N}$, let $\\mathcal{C}_{k}(S)$ denote the $\\textit{k-curve graph}$, whose vertices are isotopy classes of essential simple closed curves on $S$, and whose edges correspond to pairs of curves that can be realized to intersect at most $k$ times. The theme of this paper is that the geometry of Teichm\\\"uller space and of the mapping class group captures local combinatorial properties of $\\mathcal{C}_{k}(S)$. Using techniques for measuring distance in Teichm\\\"uller space, we obtain upper bounds on the following t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00502","kind":"arxiv","version":4},"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"}