{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:7YNOKWKTC3IEVOSM4XBXC2IL34","short_pith_number":"pith:7YNOKWKT","schema_version":"1.0","canonical_sha256":"fe1ae5595316d04aba4ce5c371690bdf22a5c926146da2dc4107b46b5d279dbb","source":{"kind":"arxiv","id":"1410.3755","version":2},"attestation_state":"computed","paper":{"title":"3-manifolds Modulo Surgery Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Lucas Culler","submitted_at":"2014-10-14T16:23:18Z","abstract_excerpt":"Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $\\Sigma$, we consider the abelian group $K(\\Sigma)$ generated by bordered 3-manifolds with boundary $\\Sigma$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that $K(\\Sigma)$ is a finitely generated free abelian group and compute its rank. We also construct an explicit basis and show that it generates all bordered 3-manifolds in a certain stronger sense. Our basis is strictly contained in another finite generating set which was construct"},"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":"1410.3755","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-14T16:23:18Z","cross_cats_sorted":[],"title_canon_sha256":"cb52b9e1ff0e96bd4bfbf75d71c423ad5d9d1492ef9ba2efc42bea6c455a9653","abstract_canon_sha256":"dcaa0eac791e82fc5cbbc792ca420a8ef23e52a781c61a7cd795c5d671536c41"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:00.625810Z","signature_b64":"kZcE485aXe2P3qJGamgYnGAeG5G5rr6O6SkbJxNt5P7GYoKyZ6degdZSxZV/LeUTGixOjYGM9GCgzrhQ64irAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe1ae5595316d04aba4ce5c371690bdf22a5c926146da2dc4107b46b5d279dbb","last_reissued_at":"2026-05-18T02:39:00.625335Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:00.625335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"3-manifolds Modulo Surgery Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Lucas Culler","submitted_at":"2014-10-14T16:23:18Z","abstract_excerpt":"Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $\\Sigma$, we consider the abelian group $K(\\Sigma)$ generated by bordered 3-manifolds with boundary $\\Sigma$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that $K(\\Sigma)$ is a finitely generated free abelian group and compute its rank. We also construct an explicit basis and show that it generates all bordered 3-manifolds in a certain stronger sense. Our basis is strictly contained in another finite generating set which was construct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3755","kind":"arxiv","version":2},"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":"1410.3755","created_at":"2026-05-18T02:39:00.625414+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.3755v2","created_at":"2026-05-18T02:39:00.625414+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3755","created_at":"2026-05-18T02:39:00.625414+00:00"},{"alias_kind":"pith_short_12","alias_value":"7YNOKWKTC3IE","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"7YNOKWKTC3IEVOSM","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"7YNOKWKT","created_at":"2026-05-18T12:28:19.803747+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/7YNOKWKTC3IEVOSM4XBXC2IL34","json":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34.json","graph_json":"https://pith.science/api/pith-number/7YNOKWKTC3IEVOSM4XBXC2IL34/graph.json","events_json":"https://pith.science/api/pith-number/7YNOKWKTC3IEVOSM4XBXC2IL34/events.json","paper":"https://pith.science/paper/7YNOKWKT"},"agent_actions":{"view_html":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34","download_json":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34.json","view_paper":"https://pith.science/paper/7YNOKWKT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.3755&json=true","fetch_graph":"https://pith.science/api/pith-number/7YNOKWKTC3IEVOSM4XBXC2IL34/graph.json","fetch_events":"https://pith.science/api/pith-number/7YNOKWKTC3IEVOSM4XBXC2IL34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34/action/storage_attestation","attest_author":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34/action/author_attestation","sign_citation":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34/action/citation_signature","submit_replication":"https://pith.science/pith/7YNOKWKTC3IEVOSM4XBXC2IL34/action/replication_record"}},"created_at":"2026-05-18T02:39:00.625414+00:00","updated_at":"2026-05-18T02:39:00.625414+00:00"}