{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DWZCYLGVTOQS334D7FFMY6ODWD","short_pith_number":"pith:DWZCYLGV","schema_version":"1.0","canonical_sha256":"1db22c2cd59ba12def83f94acc79c3b0ee840c22b35e13d5e8a205798d375373","source":{"kind":"arxiv","id":"1111.0609","version":3},"attestation_state":"computed","paper":{"title":"Virtual Khovanov homology using cobordisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Daniel Tubbenhauer","submitted_at":"2011-11-02T18:58:52Z","abstract_excerpt":"We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways.\n  Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathema"},"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":"1111.0609","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-11-02T18:58:52Z","cross_cats_sorted":[],"title_canon_sha256":"9f73387840307a3e784dff659739c5d0d6217bb73559d671aa68ed4bd0a1ba69","abstract_canon_sha256":"f613a57923faa2b3b7c8fdce431751b4a755c4ddcd3b22d5421f18325a2de65a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:39.879269Z","signature_b64":"2AQ9c+L2lTfrUFHYkrBKAr2X3Bjz3zssmbt118vtxGqK2yxkfz5LEZMiH9i+2SquBAj5eILl8VgGhWXDmj+ABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1db22c2cd59ba12def83f94acc79c3b0ee840c22b35e13d5e8a205798d375373","last_reissued_at":"2026-05-18T01:21:39.878737Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:39.878737Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Virtual Khovanov homology using cobordisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Daniel Tubbenhauer","submitted_at":"2011-11-02T18:58:52Z","abstract_excerpt":"We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways.\n  Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathema"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0609","kind":"arxiv","version":3},"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":"1111.0609","created_at":"2026-05-18T01:21:39.878836+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.0609v3","created_at":"2026-05-18T01:21:39.878836+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0609","created_at":"2026-05-18T01:21:39.878836+00:00"},{"alias_kind":"pith_short_12","alias_value":"DWZCYLGVTOQS","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DWZCYLGVTOQS334D","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DWZCYLGV","created_at":"2026-05-18T12:26:26.731475+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/DWZCYLGVTOQS334D7FFMY6ODWD","json":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD.json","graph_json":"https://pith.science/api/pith-number/DWZCYLGVTOQS334D7FFMY6ODWD/graph.json","events_json":"https://pith.science/api/pith-number/DWZCYLGVTOQS334D7FFMY6ODWD/events.json","paper":"https://pith.science/paper/DWZCYLGV"},"agent_actions":{"view_html":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD","download_json":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD.json","view_paper":"https://pith.science/paper/DWZCYLGV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.0609&json=true","fetch_graph":"https://pith.science/api/pith-number/DWZCYLGVTOQS334D7FFMY6ODWD/graph.json","fetch_events":"https://pith.science/api/pith-number/DWZCYLGVTOQS334D7FFMY6ODWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD/action/storage_attestation","attest_author":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD/action/author_attestation","sign_citation":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD/action/citation_signature","submit_replication":"https://pith.science/pith/DWZCYLGVTOQS334D7FFMY6ODWD/action/replication_record"}},"created_at":"2026-05-18T01:21:39.878836+00:00","updated_at":"2026-05-18T01:21:39.878836+00:00"}