{"paper":{"title":"On the Morse-Novikov number for 2-knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Andrei Pajitnov, Hisaaki Endo","submitted_at":"2015-02-23T09:05:02Z","abstract_excerpt":"Let $K\\subset S^4$ be a 2-knot, that is, a smoothly embedded 2-sphere in $S^4$. The Morse-Novikov number $\\mathcal M\\mathcal N(K)$ is the minimal possible number of critical points of a Morse map $S^4\\setminus K\\to S^1$ belonging to the canonical class in $H^1(S^4\\setminus K)$. We prove that for a classical knot $K\\subset S^3$ the Morse-Novikov number of the spun knot $S(K)$ is $\\leq 2\\mathcal M\\mathcal N(K)$. This enables us to compute $\\mathcal M\\mathcal N(S(K))$ for every classical knot $K$ with tunnel number 1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.06352","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"}