{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:IOOS5PLCSRW43BAZ3H3RPDQZRS","short_pith_number":"pith:IOOS5PLC","schema_version":"1.0","canonical_sha256":"439d2ebd62946dcd8419d9f7178e198c992f68c9e76294f9e2dacb92080c1357","source":{"kind":"arxiv","id":"0901.0208","version":2},"attestation_state":"computed","paper":{"title":"Topological Index Theory for Surfaces in 3-Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"David Bachman","submitted_at":"2009-01-02T02:33:08Z","abstract_excerpt":"The disk complex of a surface in a 3-manifold is used to define its {\\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H \\setminus N(F)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. "},"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":"0901.0208","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-01-02T02:33:08Z","cross_cats_sorted":[],"title_canon_sha256":"4ece2ae42cbbefdb33e4de796a85614a51da96c228f0ef3027656e4d5c14df1a","abstract_canon_sha256":"8d534185fe424cbc2ca9e6344418bac2717d4f103a86651ad2fc654ad5eadedc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:18.941534Z","signature_b64":"LPrVYFK7n/gM1sjLgfQJ4sj1VIRqmISlIYNj8BwJRVj6eYgwObCACwyhgf4ogUmt+k90iVBTZAJ4jBSnN+wbCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"439d2ebd62946dcd8419d9f7178e198c992f68c9e76294f9e2dacb92080c1357","last_reissued_at":"2026-05-18T02:38:18.940947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:18.940947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topological Index Theory for Surfaces in 3-Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"David Bachman","submitted_at":"2009-01-02T02:33:08Z","abstract_excerpt":"The disk complex of a surface in a 3-manifold is used to define its {\\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H \\setminus N(F)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0208","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":"0901.0208","created_at":"2026-05-18T02:38:18.941029+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.0208v2","created_at":"2026-05-18T02:38:18.941029+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0208","created_at":"2026-05-18T02:38:18.941029+00:00"},{"alias_kind":"pith_short_12","alias_value":"IOOS5PLCSRW4","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"IOOS5PLCSRW43BAZ","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"IOOS5PLC","created_at":"2026-05-18T12:26:00.592388+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/IOOS5PLCSRW43BAZ3H3RPDQZRS","json":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS.json","graph_json":"https://pith.science/api/pith-number/IOOS5PLCSRW43BAZ3H3RPDQZRS/graph.json","events_json":"https://pith.science/api/pith-number/IOOS5PLCSRW43BAZ3H3RPDQZRS/events.json","paper":"https://pith.science/paper/IOOS5PLC"},"agent_actions":{"view_html":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS","download_json":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS.json","view_paper":"https://pith.science/paper/IOOS5PLC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.0208&json=true","fetch_graph":"https://pith.science/api/pith-number/IOOS5PLCSRW43BAZ3H3RPDQZRS/graph.json","fetch_events":"https://pith.science/api/pith-number/IOOS5PLCSRW43BAZ3H3RPDQZRS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS/action/storage_attestation","attest_author":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS/action/author_attestation","sign_citation":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS/action/citation_signature","submit_replication":"https://pith.science/pith/IOOS5PLCSRW43BAZ3H3RPDQZRS/action/replication_record"}},"created_at":"2026-05-18T02:38:18.941029+00:00","updated_at":"2026-05-18T02:38:18.941029+00:00"}