{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:SBPOKNZZLALPOF3WIMKUXMIK5J","short_pith_number":"pith:SBPOKNZZ","schema_version":"1.0","canonical_sha256":"905ee537395816f7177643154bb10aea6130b7495f6beefe55462f4452d05b17","source":{"kind":"arxiv","id":"2602.21659","version":2},"attestation_state":"computed","paper":{"title":"Crossing Numbers of Knots on Closed Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Makoto Ozawa","submitted_at":"2026-02-25T07:37:40Z","abstract_excerpt":"Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. The proof combines a surface ascending-number estimate, a bridge-numb"},"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":"2602.21659","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-02-25T07:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"b60d3564372a26560fa1ae296d773200347266c39c72ba866b43e4f8579bfde1","abstract_canon_sha256":"e7e5cab9a74d8dc6b1f05ef7754e3b36f90e02c182852f9dac04b8683370c1ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T02:04:40.157974Z","signature_b64":"tALNB1NsAAzCKp9j2hgwhZDVAC6G/VK21Qxhg9dVuxMDBtuFhMsxBYZDdlDorPQ+tuUBqoXsc1KK9/GTd+YZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"905ee537395816f7177643154bb10aea6130b7495f6beefe55462f4452d05b17","last_reissued_at":"2026-05-22T02:04:40.156893Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T02:04:40.156893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Crossing Numbers of Knots on Closed Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Makoto Ozawa","submitted_at":"2026-02-25T07:37:40Z","abstract_excerpt":"Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. 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