{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:QKEFQOTSHXZUQJJP5PWBLKIXBT","short_pith_number":"pith:QKEFQOTS","schema_version":"1.0","canonical_sha256":"8288583a723df348252febec15a9170ce0d615981fdf8d1a95b07474d8dd4240","source":{"kind":"arxiv","id":"1102.5081","version":1},"attestation_state":"computed","paper":{"title":"Virtual Knot Invariants Arising From Parities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Denis Petrovich Ilyutko, Igor Mikhailovich Nikonov, Vassily Olegovich Manturov","submitted_at":"2011-02-24T20:31:54Z","abstract_excerpt":"In \\cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\\em parity}, i.e.\\ a function on crossings valued in $\\{0,1\\}$ and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than $\\mathbb{Z}_2$ and investigate them for different knot theories. For some knot theories we show that there i"},"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":"1102.5081","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-02-24T20:31:54Z","cross_cats_sorted":[],"title_canon_sha256":"01400922aa415b4ff9228b51028ee79bfb597e5ca6cb2b4142aef43e25e039df","abstract_canon_sha256":"8f44b566e56ca21dd364890200d505595e40f4e076b1d2c158922cd19e241df2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:57.450602Z","signature_b64":"Z4VPtBWjhVedglLSm4EMnkkTG8Jsikv+fWILF1Ie5/PfojsFfxol42RbjMmWzY1umgJwSLa/dosEmQ30O6dKDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8288583a723df348252febec15a9170ce0d615981fdf8d1a95b07474d8dd4240","last_reissued_at":"2026-05-18T04:27:57.449987Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:57.449987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Virtual Knot Invariants Arising From Parities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Denis Petrovich Ilyutko, Igor Mikhailovich Nikonov, Vassily Olegovich Manturov","submitted_at":"2011-02-24T20:31:54Z","abstract_excerpt":"In \\cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\\em parity}, i.e.\\ a function on crossings valued in $\\{0,1\\}$ and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than $\\mathbb{Z}_2$ and investigate them for different knot theories. For some knot theories we show that there i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5081","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":"1102.5081","created_at":"2026-05-18T04:27:57.450048+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.5081v1","created_at":"2026-05-18T04:27:57.450048+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5081","created_at":"2026-05-18T04:27:57.450048+00:00"},{"alias_kind":"pith_short_12","alias_value":"QKEFQOTSHXZU","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"QKEFQOTSHXZUQJJP","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"QKEFQOTS","created_at":"2026-05-18T12:26:39.201973+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/QKEFQOTSHXZUQJJP5PWBLKIXBT","json":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT.json","graph_json":"https://pith.science/api/pith-number/QKEFQOTSHXZUQJJP5PWBLKIXBT/graph.json","events_json":"https://pith.science/api/pith-number/QKEFQOTSHXZUQJJP5PWBLKIXBT/events.json","paper":"https://pith.science/paper/QKEFQOTS"},"agent_actions":{"view_html":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT","download_json":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT.json","view_paper":"https://pith.science/paper/QKEFQOTS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.5081&json=true","fetch_graph":"https://pith.science/api/pith-number/QKEFQOTSHXZUQJJP5PWBLKIXBT/graph.json","fetch_events":"https://pith.science/api/pith-number/QKEFQOTSHXZUQJJP5PWBLKIXBT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT/action/storage_attestation","attest_author":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT/action/author_attestation","sign_citation":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT/action/citation_signature","submit_replication":"https://pith.science/pith/QKEFQOTSHXZUQJJP5PWBLKIXBT/action/replication_record"}},"created_at":"2026-05-18T04:27:57.450048+00:00","updated_at":"2026-05-18T04:27:57.450048+00:00"}