{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6GSU45VLDHNSGEVHRO44OZRH2H","short_pith_number":"pith:6GSU45VL","canonical_record":{"source":{"id":"1308.5638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-26T16:57:46Z","cross_cats_sorted":[],"title_canon_sha256":"bb72e3eab3adbaa41e2222d60cc7bd063f710caaf0781c14bef1b0ff59f39cbd","abstract_canon_sha256":"286696336837a762128b2aca746ee511a0c27449724a6943cbb42ce4f006a937"},"schema_version":"1.0"},"canonical_sha256":"f1a54e76ab19db2312a78bb9c76627d1e59c43518d300f6b9db4fe414cf487e6","source":{"kind":"arxiv","id":"1308.5638","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5638","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5638v1","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5638","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"pith_short_12","alias_value":"6GSU45VLDHNS","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6GSU45VLDHNSGEVH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6GSU45VL","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6GSU45VLDHNSGEVHRO44OZRH2H","target":"record","payload":{"canonical_record":{"source":{"id":"1308.5638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-26T16:57:46Z","cross_cats_sorted":[],"title_canon_sha256":"bb72e3eab3adbaa41e2222d60cc7bd063f710caaf0781c14bef1b0ff59f39cbd","abstract_canon_sha256":"286696336837a762128b2aca746ee511a0c27449724a6943cbb42ce4f006a937"},"schema_version":"1.0"},"canonical_sha256":"f1a54e76ab19db2312a78bb9c76627d1e59c43518d300f6b9db4fe414cf487e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:03.930916Z","signature_b64":"90WfCs/vGoULF1ECiUQgWXHB9gpK49Us3aHvedC7QTsz5qsOaQMYdpqaS+uaMWtISCoLVP9zQw/Mf/tfQzYtCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1a54e76ab19db2312a78bb9c76627d1e59c43518d300f6b9db4fe414cf487e6","last_reissued_at":"2026-05-18T03:15:03.930134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:03.930134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.5638","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:15:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1hursiB2U33BtBfSbExTBaL8Pnb+k2OxRnBxQFg5wrNgRodYxRERSV2jjQ37VRKZtVaBHQc0wLv7pNF/ghdkBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:45:09.222956Z"},"content_sha256":"dca08dc413427015324937f76f8e43e6428f22ca7e770dc4d959e07f05ca2ec1","schema_version":"1.0","event_id":"sha256:dca08dc413427015324937f76f8e43e6428f22ca7e770dc4d959e07f05ca2ec1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6GSU45VLDHNSGEVHRO44OZRH2H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Splitting numbers of links","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jae Choon Cha, Mark Powell, Stefan Friedl","submitted_at":"2013-08-26T16:57:46Z","abstract_excerpt":"The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with 9 or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by J. Batson and C. Seed using Khovanov homology."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5638","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:15:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rbucCsHoezaLgECDg1P3LbDIe3YlDz4AeSmxlTVQHrequCBY8WnOHGxKlQ+6HAO0HsSCJYpjKtUTIT06O2x/BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:45:09.223358Z"},"content_sha256":"153b08d1310b4162392e7c2e0831226ca4dcfd88bf2c2c1f370d94ca5398349c","schema_version":"1.0","event_id":"sha256:153b08d1310b4162392e7c2e0831226ca4dcfd88bf2c2c1f370d94ca5398349c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6GSU45VLDHNSGEVHRO44OZRH2H/bundle.json","state_url":"https://pith.science/pith/6GSU45VLDHNSGEVHRO44OZRH2H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6GSU45VLDHNSGEVHRO44OZRH2H/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":"2026-05-27T14:45:09Z","links":{"resolver":"https://pith.science/pith/6GSU45VLDHNSGEVHRO44OZRH2H","bundle":"https://pith.science/pith/6GSU45VLDHNSGEVHRO44OZRH2H/bundle.json","state":"https://pith.science/pith/6GSU45VLDHNSGEVHRO44OZRH2H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6GSU45VLDHNSGEVHRO44OZRH2H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6GSU45VLDHNSGEVHRO44OZRH2H","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":"286696336837a762128b2aca746ee511a0c27449724a6943cbb42ce4f006a937","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-26T16:57:46Z","title_canon_sha256":"bb72e3eab3adbaa41e2222d60cc7bd063f710caaf0781c14bef1b0ff59f39cbd"},"schema_version":"1.0","source":{"id":"1308.5638","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5638","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5638v1","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5638","created_at":"2026-05-18T03:15:03Z"},{"alias_kind":"pith_short_12","alias_value":"6GSU45VLDHNS","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6GSU45VLDHNSGEVH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6GSU45VL","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:153b08d1310b4162392e7c2e0831226ca4dcfd88bf2c2c1f370d94ca5398349c","target":"graph","created_at":"2026-05-18T03:15:03Z","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":"The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with 9 or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by J. Batson and C. Seed using Khovanov homology.","authors_text":"Jae Choon Cha, Mark Powell, Stefan Friedl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-26T16:57:46Z","title":"Splitting numbers of links"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5638","kind":"arxiv","version":1},"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:dca08dc413427015324937f76f8e43e6428f22ca7e770dc4d959e07f05ca2ec1","target":"record","created_at":"2026-05-18T03:15:03Z","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":"286696336837a762128b2aca746ee511a0c27449724a6943cbb42ce4f006a937","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-26T16:57:46Z","title_canon_sha256":"bb72e3eab3adbaa41e2222d60cc7bd063f710caaf0781c14bef1b0ff59f39cbd"},"schema_version":"1.0","source":{"id":"1308.5638","kind":"arxiv","version":1}},"canonical_sha256":"f1a54e76ab19db2312a78bb9c76627d1e59c43518d300f6b9db4fe414cf487e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1a54e76ab19db2312a78bb9c76627d1e59c43518d300f6b9db4fe414cf487e6","first_computed_at":"2026-05-18T03:15:03.930134Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:03.930134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"90WfCs/vGoULF1ECiUQgWXHB9gpK49Us3aHvedC7QTsz5qsOaQMYdpqaS+uaMWtISCoLVP9zQw/Mf/tfQzYtCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:03.930916Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.5638","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dca08dc413427015324937f76f8e43e6428f22ca7e770dc4d959e07f05ca2ec1","sha256:153b08d1310b4162392e7c2e0831226ca4dcfd88bf2c2c1f370d94ca5398349c"],"state_sha256":"ef8c9f490f5e7fccaea7bcfce42307176b496fe3a5951bac716156d18ee7192d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tUkjz/kkssm7dfQFWPDQTPD4R32My0/DKCXkdbSzByCvFXWO2MAH+g5UYZuQGyrTqfBy/54R8qVK9PDvfZTXCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:45:09.226099Z","bundle_sha256":"32c6e23d1e6058f1c8c49d0ee902509a0bd46e219fe71c8f695040a24584e5e7"}}