{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:LAN4SAUJLJZ73LYMCUDXKFSJON","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"aad6595ffa34443f00413ffa5e5c4df9614a85286b6b3d66c60b6ae8281ce62d","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-03-02T09:04:22Z","title_canon_sha256":"56922954c98c1c975221eaa7e352204366796ba4143dcceee6dab42b7883119c"},"schema_version":"1.0","source":{"id":"0903.0238","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.0238","created_at":"2026-05-18T02:16:48Z"},{"alias_kind":"arxiv_version","alias_value":"0903.0238v3","created_at":"2026-05-18T02:16:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.0238","created_at":"2026-05-18T02:16:48Z"},{"alias_kind":"pith_short_12","alias_value":"LAN4SAUJLJZ7","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"LAN4SAUJLJZ73LYM","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"LAN4SAUJ","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:8d525bf4380cfac40af6d890de9224feb27c5ea9ddc9757d96c9cb82862fa150","target":"graph","created_at":"2026-05-18T02:16:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n.\n  As an application, we determine ","authors_text":"Masamichi Takase, Tobias Ekholm","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-03-02T09:04:22Z","title":"Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.0238","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2a86503c12ad29e3f41695782e728e3d7d591204dd1db81b24d03c2e6bab8422","target":"record","created_at":"2026-05-18T02:16:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"aad6595ffa34443f00413ffa5e5c4df9614a85286b6b3d66c60b6ae8281ce62d","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-03-02T09:04:22Z","title_canon_sha256":"56922954c98c1c975221eaa7e352204366796ba4143dcceee6dab42b7883119c"},"schema_version":"1.0","source":{"id":"0903.0238","kind":"arxiv","version":3}},"canonical_sha256":"581bc902895a73fdaf0c15077516497347def4cea62c374d00997ea454ffe696","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"581bc902895a73fdaf0c15077516497347def4cea62c374d00997ea454ffe696","first_computed_at":"2026-05-18T02:16:48.335444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:16:48.335444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T7Qj3JT+AsLV2cfMLQNxdz4JQlWHCgVtWE0aXXTAPXYEW+wHinj9W9wACauxkM6ZPa6dtVOsoKXzVZ87hjuODA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:16:48.336176Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.0238","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a86503c12ad29e3f41695782e728e3d7d591204dd1db81b24d03c2e6bab8422","sha256:8d525bf4380cfac40af6d890de9224feb27c5ea9ddc9757d96c9cb82862fa150"],"state_sha256":"f4f34ce338b95964e36be61da9d0858ba5d6986fe965d9ff0d8642857f855cc6"}