{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:CU5KOLTHR6QWTB3GBITL4SPTCL","short_pith_number":"pith:CU5KOLTH","canonical_record":{"source":{"id":"1902.05405","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-02-14T15:02:00Z","cross_cats_sorted":[],"title_canon_sha256":"63f463080ff56c9001a2532b8a44945ed4161e45764adc2d5b6c4efd6f476c09","abstract_canon_sha256":"542b29f77002fb3e1b37ba9fe773ee45a91c2774a96f415df7b51193d7689b3f"},"schema_version":"1.0"},"canonical_sha256":"153aa72e678fa16987660a26be49f312f9729159e056d1d28692c22d15e40028","source":{"kind":"arxiv","id":"1902.05405","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.05405","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"arxiv_version","alias_value":"1902.05405v2","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05405","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_12","alias_value":"CU5KOLTHR6QW","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_16","alias_value":"CU5KOLTHR6QWTB3G","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_8","alias_value":"CU5KOLTH","created_at":"2026-07-05T02:58:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:CU5KOLTHR6QWTB3GBITL4SPTCL","target":"record","payload":{"canonical_record":{"source":{"id":"1902.05405","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-02-14T15:02:00Z","cross_cats_sorted":[],"title_canon_sha256":"63f463080ff56c9001a2532b8a44945ed4161e45764adc2d5b6c4efd6f476c09","abstract_canon_sha256":"542b29f77002fb3e1b37ba9fe773ee45a91c2774a96f415df7b51193d7689b3f"},"schema_version":"1.0"},"canonical_sha256":"153aa72e678fa16987660a26be49f312f9729159e056d1d28692c22d15e40028","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:58:32.948330Z","signature_b64":"f1zIpwur3aIlgc5VR2zFJvN2bE96K8epyqpyY2OYB3smx7DssjmktBf46FEy1121WDSRPciwB4IeouWKGXLSDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"153aa72e678fa16987660a26be49f312f9729159e056d1d28692c22d15e40028","last_reissued_at":"2026-07-05T02:58:32.947949Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:58:32.947949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1902.05405","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":"2026-07-05T02:58:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"691cjNh6hWuK3o8mIM0TVXExykAMBo9IZB8kA2IMuExkDZoSFmta37beBa4V/Dk+MS1/UKTBv+MrQhKo96D+Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T23:00:12.329403Z"},"content_sha256":"ee2e3cbc0d4c1e0bb7526e95df23b05cedb80f76b2b8d86b6ec6d2dbcdd1e753","schema_version":"1.0","event_id":"sha256:ee2e3cbc0d4c1e0bb7526e95df23b05cedb80f76b2b8d86b6ec6d2dbcdd1e753"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:CU5KOLTHR6QWTB3GBITL4SPTCL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Null-homologous unknottings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Charles Livingston","submitted_at":"2019-02-14T15:02:00Z","abstract_excerpt":"Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05405","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1902.05405/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:58:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aJiwUEaLGmolRAV5//Si+lQ0i0OdimLvb194sawF5jU+Uv1iv/IkhwDwn95GCZQbFsObFq9GDg32nLJLauOrCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T23:00:12.330280Z"},"content_sha256":"4f8a6d31785fe16022ed1dc7abf7eb94264d15141f76e5150df5f163a69d7b84","schema_version":"1.0","event_id":"sha256:4f8a6d31785fe16022ed1dc7abf7eb94264d15141f76e5150df5f163a69d7b84"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/bundle.json","state_url":"https://pith.science/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/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":"2026-08-16T23:00:12Z","links":{"resolver":"https://pith.science/pith/CU5KOLTHR6QWTB3GBITL4SPTCL","bundle":"https://pith.science/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/bundle.json","state":"https://pith.science/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CU5KOLTHR6QWTB3GBITL4SPTCL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:CU5KOLTHR6QWTB3GBITL4SPTCL","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":"542b29f77002fb3e1b37ba9fe773ee45a91c2774a96f415df7b51193d7689b3f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-02-14T15:02:00Z","title_canon_sha256":"63f463080ff56c9001a2532b8a44945ed4161e45764adc2d5b6c4efd6f476c09"},"schema_version":"1.0","source":{"id":"1902.05405","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.05405","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"arxiv_version","alias_value":"1902.05405v2","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05405","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_12","alias_value":"CU5KOLTHR6QW","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_16","alias_value":"CU5KOLTHR6QWTB3G","created_at":"2026-07-05T02:58:32Z"},{"alias_kind":"pith_short_8","alias_value":"CU5KOLTH","created_at":"2026-07-05T02:58:32Z"}],"graph_snapshots":[{"event_id":"sha256:4f8a6d31785fe16022ed1dc7abf7eb94264d15141f76e5150df5f163a69d7b84","target":"graph","created_at":"2026-07-05T02:58:32Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1902.05405/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.","authors_text":"Charles Livingston","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-02-14T15:02:00Z","title":"Null-homologous unknottings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05405","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:ee2e3cbc0d4c1e0bb7526e95df23b05cedb80f76b2b8d86b6ec6d2dbcdd1e753","target":"record","created_at":"2026-07-05T02:58:32Z","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":"542b29f77002fb3e1b37ba9fe773ee45a91c2774a96f415df7b51193d7689b3f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-02-14T15:02:00Z","title_canon_sha256":"63f463080ff56c9001a2532b8a44945ed4161e45764adc2d5b6c4efd6f476c09"},"schema_version":"1.0","source":{"id":"1902.05405","kind":"arxiv","version":2}},"canonical_sha256":"153aa72e678fa16987660a26be49f312f9729159e056d1d28692c22d15e40028","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"153aa72e678fa16987660a26be49f312f9729159e056d1d28692c22d15e40028","first_computed_at":"2026-07-05T02:58:32.947949Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:58:32.947949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f1zIpwur3aIlgc5VR2zFJvN2bE96K8epyqpyY2OYB3smx7DssjmktBf46FEy1121WDSRPciwB4IeouWKGXLSDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:58:32.948330Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.05405","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee2e3cbc0d4c1e0bb7526e95df23b05cedb80f76b2b8d86b6ec6d2dbcdd1e753","sha256:4f8a6d31785fe16022ed1dc7abf7eb94264d15141f76e5150df5f163a69d7b84"],"state_sha256":"93927fc24a8e94393e758841a90ff644760a9d11102a0f9876af1cb40083d632"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TQVxZ9Z4RKpF5vpPJe7P32Ho8v1qvNje1/p4+vwJHgoV7QCCKV8aUYXQR3coI+EoxWsOdnXQwzlxEHP5h0gcAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T23:00:12.353293Z","bundle_sha256":"1005da6397d0f97088e397c0622b373c9bcb1c8f149ed9b6baa06be6f0bdbbdf"}}