{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:6UKTYYHNN3UDTHNCAQ7JMPQYJF","short_pith_number":"pith:6UKTYYHN","schema_version":"1.0","canonical_sha256":"f5153c60ed6ee8399da2043e963e184953048ba94330d52c1f8b3de8b9bf97b3","source":{"kind":"arxiv","id":"1507.08370","version":1},"attestation_state":"computed","paper":{"title":"Bridge trisections of knotted surfaces in $S^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Alexander Zupan, Jeffrey Meier","submitted_at":"2015-07-30T04:08:31Z","abstract_excerpt":"We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and "},"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":"1507.08370","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-30T04:08:31Z","cross_cats_sorted":[],"title_canon_sha256":"1748d943a319549012f8eab88bb8d5bd24fbc83fde45a584dda85102a6b5327b","abstract_canon_sha256":"3d37db9bb7629d808a495a7b17233b9002c1a19232b61f59de953a6ba5cd9662"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:23.646239Z","signature_b64":"DBML3wG4lK6BgD2OmUitK7N4ULWbGujD5kYTU9MzFCLBen9TFMXYeNSLTX1pgQTxvL75b3xw9+Qrg7x7BYaFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5153c60ed6ee8399da2043e963e184953048ba94330d52c1f8b3de8b9bf97b3","last_reissued_at":"2026-05-18T00:38:23.645512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:23.645512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bridge trisections of knotted surfaces in $S^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Alexander Zupan, Jeffrey Meier","submitted_at":"2015-07-30T04:08:31Z","abstract_excerpt":"We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08370","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":"1507.08370","created_at":"2026-05-18T00:38:23.645644+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08370v1","created_at":"2026-05-18T00:38:23.645644+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08370","created_at":"2026-05-18T00:38:23.645644+00:00"},{"alias_kind":"pith_short_12","alias_value":"6UKTYYHNN3UD","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"6UKTYYHNN3UDTHNC","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"6UKTYYHN","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/6UKTYYHNN3UDTHNCAQ7JMPQYJF","json":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF.json","graph_json":"https://pith.science/api/pith-number/6UKTYYHNN3UDTHNCAQ7JMPQYJF/graph.json","events_json":"https://pith.science/api/pith-number/6UKTYYHNN3UDTHNCAQ7JMPQYJF/events.json","paper":"https://pith.science/paper/6UKTYYHN"},"agent_actions":{"view_html":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF","download_json":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF.json","view_paper":"https://pith.science/paper/6UKTYYHN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08370&json=true","fetch_graph":"https://pith.science/api/pith-number/6UKTYYHNN3UDTHNCAQ7JMPQYJF/graph.json","fetch_events":"https://pith.science/api/pith-number/6UKTYYHNN3UDTHNCAQ7JMPQYJF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF/action/storage_attestation","attest_author":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF/action/author_attestation","sign_citation":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF/action/citation_signature","submit_replication":"https://pith.science/pith/6UKTYYHNN3UDTHNCAQ7JMPQYJF/action/replication_record"}},"created_at":"2026-05-18T00:38:23.645644+00:00","updated_at":"2026-05-18T00:38:23.645644+00:00"}