{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:J655WWLSPWAXUOUO5KPVCV6TOQ","short_pith_number":"pith:J655WWLS","canonical_record":{"source":{"id":"1007.1693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-10T03:51:42Z","cross_cats_sorted":[],"title_canon_sha256":"2fb389c227503033a0426a16c03bf21703b5f82e4c824451a048c84897ac9dfa","abstract_canon_sha256":"2287e07ec105ed5a10cf898e501a007a3e1dbc40f770d087f7c39300f89508a6"},"schema_version":"1.0"},"canonical_sha256":"4fbbdb59727d817a3a8eea9f5157d3743d78f221d32841040321200bcb0c9908","source":{"kind":"arxiv","id":"1007.1693","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1693","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1693v2","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1693","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"pith_short_12","alias_value":"J655WWLSPWAX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"J655WWLSPWAXUOUO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"J655WWLS","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:J655WWLSPWAXUOUO5KPVCV6TOQ","target":"record","payload":{"canonical_record":{"source":{"id":"1007.1693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-10T03:51:42Z","cross_cats_sorted":[],"title_canon_sha256":"2fb389c227503033a0426a16c03bf21703b5f82e4c824451a048c84897ac9dfa","abstract_canon_sha256":"2287e07ec105ed5a10cf898e501a007a3e1dbc40f770d087f7c39300f89508a6"},"schema_version":"1.0"},"canonical_sha256":"4fbbdb59727d817a3a8eea9f5157d3743d78f221d32841040321200bcb0c9908","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:59.105881Z","signature_b64":"NQZ4hzH5+9pJotabnT8Hg/XkLUKVW+vpX//k1Pwa5UfWS3CC2QSalx5tVDik8TsIqZppxuLwHTGyv4w5L2xPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fbbdb59727d817a3a8eea9f5157d3743d78f221d32841040321200bcb0c9908","last_reissued_at":"2026-05-18T04:39:59.105232Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:59.105232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.1693","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GAk2ViZ6OuStdla/LEqfASXQBEIuE7PODOubw//lvpmiEr9YJfgkb6BUWsOJsMcuZWOcTrzOjTwo6SGb7yrUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:04:38.121465Z"},"content_sha256":"4f11f6b90f25c339cf635f50e7ebd43c29b8fdb675fac57916399ab6826a4a4e","schema_version":"1.0","event_id":"sha256:4f11f6b90f25c339cf635f50e7ebd43c29b8fdb675fac57916399ab6826a4a4e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:J655WWLSPWAXUOUO5KPVCV6TOQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite type invariants of nanowords and nanophrases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Andrew Gibson, Noboru Ito","submitted_at":"2010-07-10T03:51:42Z","abstract_excerpt":"Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree one and degree two respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1693","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DdwT5aICssy5XnVDMNl2CLatC5MbGrhXwLKp6gKtC1QTLI+yYDSjq2L7b8KHLDi6vNN4snSHsgt8elNMlxlJBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:04:38.122246Z"},"content_sha256":"93dcb0214f0a82be07a7bbfa9c15dac7bee0f2ae24ff28afbaf1086550e225a7","schema_version":"1.0","event_id":"sha256:93dcb0214f0a82be07a7bbfa9c15dac7bee0f2ae24ff28afbaf1086550e225a7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/bundle.json","state_url":"https://pith.science/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-12T00:04:38Z","links":{"resolver":"https://pith.science/pith/J655WWLSPWAXUOUO5KPVCV6TOQ","bundle":"https://pith.science/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/bundle.json","state":"https://pith.science/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J655WWLSPWAXUOUO5KPVCV6TOQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:J655WWLSPWAXUOUO5KPVCV6TOQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2287e07ec105ed5a10cf898e501a007a3e1dbc40f770d087f7c39300f89508a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-10T03:51:42Z","title_canon_sha256":"2fb389c227503033a0426a16c03bf21703b5f82e4c824451a048c84897ac9dfa"},"schema_version":"1.0","source":{"id":"1007.1693","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1693","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1693v2","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1693","created_at":"2026-05-18T04:39:59Z"},{"alias_kind":"pith_short_12","alias_value":"J655WWLSPWAX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"J655WWLSPWAXUOUO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"J655WWLS","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:93dcb0214f0a82be07a7bbfa9c15dac7bee0f2ae24ff28afbaf1086550e225a7","target":"graph","created_at":"2026-05-18T04:39:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree one and degree two respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.","authors_text":"Andrew Gibson, Noboru Ito","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-10T03:51:42Z","title":"Finite type invariants of nanowords and nanophrases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1693","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4f11f6b90f25c339cf635f50e7ebd43c29b8fdb675fac57916399ab6826a4a4e","target":"record","created_at":"2026-05-18T04:39:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2287e07ec105ed5a10cf898e501a007a3e1dbc40f770d087f7c39300f89508a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-10T03:51:42Z","title_canon_sha256":"2fb389c227503033a0426a16c03bf21703b5f82e4c824451a048c84897ac9dfa"},"schema_version":"1.0","source":{"id":"1007.1693","kind":"arxiv","version":2}},"canonical_sha256":"4fbbdb59727d817a3a8eea9f5157d3743d78f221d32841040321200bcb0c9908","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4fbbdb59727d817a3a8eea9f5157d3743d78f221d32841040321200bcb0c9908","first_computed_at":"2026-05-18T04:39:59.105232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:59.105232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NQZ4hzH5+9pJotabnT8Hg/XkLUKVW+vpX//k1Pwa5UfWS3CC2QSalx5tVDik8TsIqZppxuLwHTGyv4w5L2xPDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:59.105881Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.1693","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f11f6b90f25c339cf635f50e7ebd43c29b8fdb675fac57916399ab6826a4a4e","sha256:93dcb0214f0a82be07a7bbfa9c15dac7bee0f2ae24ff28afbaf1086550e225a7"],"state_sha256":"3e0640bc02302beff8069896b9644ef4dca7e63b607f24af7135e4555b40e9ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iSdUwTONZUT4xcV0xUgaKVVOw7F6J9U8uwOEBD8hSRHTYU5fK9F+/93gKey650nW2i65PhRVeEP+3tXfRh0RCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T00:04:38.126775Z","bundle_sha256":"623d8c820bc277caed900bf701b2b77e63ae844e60b72a6b74fdb94f646b060e"}}