{"paper":{"title":"The number of framings of a knot in a 3-manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Patricia Cahn, Rustam Sadykov, Vladimir Chernov","submitted_at":"2014-04-23T15:02:42Z","abstract_excerpt":"In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of $S^1\\times S^2$; the knot $K$ need not be zero-homologous and the manifold is not required to be compact.\n  We show that when $M$ is orientable, the number $|K|$ is infinite unless $K$ intersects a non-separating sphere at exactly one point, in which case $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5851","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"}