{"paper":{"title":"Splitting a contraction of a simple curve traversed $m$ times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Gregory R. Chambers, Yevgeny Liokumovich","submitted_at":"2015-10-12T20:16:16Z","abstract_excerpt":"Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $\\gamma$ be a simple closed curve on $M$. Let $m \\gamma$ denote the curve formed by tracing $\\gamma$ $m$ times. We prove that if $m \\gamma$ is contractible through curves of length less than $L$, then $\\gamma$ is contractible through curves of length less than $L$. In the last section we state several open questions about controlling length and the number of self-intersections in homotopies of curves on Riemannian surfaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03445","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"}