{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:67FDT62VPRJ54Y66FYNXZN3EVE","short_pith_number":"pith:67FDT62V","schema_version":"1.0","canonical_sha256":"f7ca39fb557c53de63de2e1b7cb764a91651f9cf80bc9b7e891f9623e4a2191a","source":{"kind":"arxiv","id":"1502.06352","version":1},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1502.06352","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-02-23T09:05:02Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6ec4d0dcccd8241ffb3a02d684cf6f8bca35233602b79f50eca8faa87ab8a9a8","abstract_canon_sha256":"751131a62b7defec4ab6dc9c5f3c9c45bc3495a771e832ef3010bca919081bfb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:49.564613Z","signature_b64":"QbH5qFB81Jw4ZAVIZWinBkmWoZ0eA+O2/Rh5uPmlqyEJFMKIU6mQOSoR6Or3/W8w2vMhVlqpp7Go2Gl/Oe7dDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7ca39fb557c53de63de2e1b7cb764a91651f9cf80bc9b7e891f9623e4a2191a","last_reissued_at":"2026-05-18T00:30:49.564052Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:49.564052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1502.06352","created_at":"2026-05-18T00:30:49.564130+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.06352v1","created_at":"2026-05-18T00:30:49.564130+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.06352","created_at":"2026-05-18T00:30:49.564130+00:00"},{"alias_kind":"pith_short_12","alias_value":"67FDT62VPRJ5","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"67FDT62VPRJ54Y66","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"67FDT62V","created_at":"2026-05-18T12:29:07.941421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE","json":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE.json","graph_json":"https://pith.science/api/pith-number/67FDT62VPRJ54Y66FYNXZN3EVE/graph.json","events_json":"https://pith.science/api/pith-number/67FDT62VPRJ54Y66FYNXZN3EVE/events.json","paper":"https://pith.science/paper/67FDT62V"},"agent_actions":{"view_html":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE","download_json":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE.json","view_paper":"https://pith.science/paper/67FDT62V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.06352&json=true","fetch_graph":"https://pith.science/api/pith-number/67FDT62VPRJ54Y66FYNXZN3EVE/graph.json","fetch_events":"https://pith.science/api/pith-number/67FDT62VPRJ54Y66FYNXZN3EVE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE/action/storage_attestation","attest_author":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE/action/author_attestation","sign_citation":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE/action/citation_signature","submit_replication":"https://pith.science/pith/67FDT62VPRJ54Y66FYNXZN3EVE/action/replication_record"}},"created_at":"2026-05-18T00:30:49.564130+00:00","updated_at":"2026-05-18T00:30:49.564130+00:00"}