{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2002:QWNV67H5KNB5S2PVOTF2L6AZ3T","short_pith_number":"pith:QWNV67H5","schema_version":"1.0","canonical_sha256":"859b5f7cfd5343d969f574cba5f819dcd431c0dfee4d0733259050b9e1fb5bc0","source":{"kind":"arxiv","id":"math/0203042","version":1},"attestation_state":"computed","paper":{"title":"A norm for the cohomology of 2-complexes","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Vladimir Turaev","submitted_at":"2002-03-05T14:14:01Z","abstract_excerpt":"We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from \\pi_1(X)."},"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":"math/0203042","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2002-03-05T14:14:01Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"d833b38be0792f24c4a65f81c1f2ebe8658e5e89b977655fea518ae52fceb951","abstract_canon_sha256":"df3ef8721f09009655eb87bf73113122529dd5257acdcf655a55d6bb89a4fbe0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:33.092861Z","signature_b64":"XmVLZdRWe9rA8Vt+9CxLdDc5V+7ot34tTusncJc9AWT317BVumSckP2fIEQAm7w0m0SUU/Y9rpHEn0hE9uIJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"859b5f7cfd5343d969f574cba5f819dcd431c0dfee4d0733259050b9e1fb5bc0","last_reissued_at":"2026-05-18T02:41:33.092439Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:33.092439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A norm for the cohomology of 2-complexes","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Vladimir Turaev","submitted_at":"2002-03-05T14:14:01Z","abstract_excerpt":"We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from \\pi_1(X)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0203042","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":"math/0203042","created_at":"2026-05-18T02:41:33.092502+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0203042v1","created_at":"2026-05-18T02:41:33.092502+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0203042","created_at":"2026-05-18T02:41:33.092502+00:00"},{"alias_kind":"pith_short_12","alias_value":"QWNV67H5KNB5","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_16","alias_value":"QWNV67H5KNB5S2PV","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_8","alias_value":"QWNV67H5","created_at":"2026-05-18T12:25:51.375804+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/QWNV67H5KNB5S2PVOTF2L6AZ3T","json":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T.json","graph_json":"https://pith.science/api/pith-number/QWNV67H5KNB5S2PVOTF2L6AZ3T/graph.json","events_json":"https://pith.science/api/pith-number/QWNV67H5KNB5S2PVOTF2L6AZ3T/events.json","paper":"https://pith.science/paper/QWNV67H5"},"agent_actions":{"view_html":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T","download_json":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T.json","view_paper":"https://pith.science/paper/QWNV67H5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0203042&json=true","fetch_graph":"https://pith.science/api/pith-number/QWNV67H5KNB5S2PVOTF2L6AZ3T/graph.json","fetch_events":"https://pith.science/api/pith-number/QWNV67H5KNB5S2PVOTF2L6AZ3T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T/action/storage_attestation","attest_author":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T/action/author_attestation","sign_citation":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T/action/citation_signature","submit_replication":"https://pith.science/pith/QWNV67H5KNB5S2PVOTF2L6AZ3T/action/replication_record"}},"created_at":"2026-05-18T02:41:33.092502+00:00","updated_at":"2026-05-18T02:41:33.092502+00:00"}