{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2003:VFJ23LDLOORVBCJUDSZLYHGVIG","short_pith_number":"pith:VFJ23LDL","canonical_record":{"source":{"id":"math/0311499","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2003-11-27T07:55:31Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"0067937051072fc63b3533d8ae7e26badf2a4d6f89a331f41691091840691378","abstract_canon_sha256":"c8259e868f9d4baa93e283061624d968ccd3f4cd65310c707caa9004d0036ba2"},"schema_version":"1.0"},"canonical_sha256":"a953adac6b73a35089341cb2bc1cd54182bb783d617ec6e44c5bbb143c9963fb","source":{"kind":"arxiv","id":"math/0311499","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0311499","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"arxiv_version","alias_value":"math/0311499v2","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0311499","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_12","alias_value":"VFJ23LDLOORV","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_16","alias_value":"VFJ23LDLOORVBCJU","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_8","alias_value":"VFJ23LDL","created_at":"2026-07-04T15:52:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2003:VFJ23LDLOORVBCJUDSZLYHGVIG","target":"record","payload":{"canonical_record":{"source":{"id":"math/0311499","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2003-11-27T07:55:31Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"0067937051072fc63b3533d8ae7e26badf2a4d6f89a331f41691091840691378","abstract_canon_sha256":"c8259e868f9d4baa93e283061624d968ccd3f4cd65310c707caa9004d0036ba2"},"schema_version":"1.0"},"canonical_sha256":"a953adac6b73a35089341cb2bc1cd54182bb783d617ec6e44c5bbb143c9963fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:52:40.143925Z","signature_b64":"N3/Qk22bOYCVpKOvkfjIQ+SYpsQi4eIs1w78omfLml5tGoP4m9P4lU4u/AZvHpVMB+WzKOnQRSyWf06pPYVlBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a953adac6b73a35089341cb2bc1cd54182bb783d617ec6e44c5bbb143c9963fb","last_reissued_at":"2026-07-04T15:52:40.143559Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:52:40.143559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0311499","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":"2026-07-04T15:52:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j+2QIBOpVmFnEpGWgzVYj0OWWEFRKJUhG03LcmhK1vuLnRjo0H3OeW0lA+IjdIaMnLQixB4+BSBml1KwdWebDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T01:45:19.794528Z"},"content_sha256":"42196227de519672d2d3277ed4f266142016d153df9ed32187e8aa088a88576c","schema_version":"1.0","event_id":"sha256:42196227de519672d2d3277ed4f266142016d153df9ed32187e8aa088a88576c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2003:VFJ23LDLOORVBCJUDSZLYHGVIG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the classification of rational tangles","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GT","authors_text":"Louis H. Kauffman, Sofia Lambropoulou","submitted_at":"2003-11-27T07:55:31Z","abstract_excerpt":"This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rationa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0311499","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/math/0311499/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":"2026-07-04T15:52:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rwKhzmV0cQBh4fpcmulS/0fcsVtwtBu0+77YMy5aK0wnS1coxG/15unoV9TwJJZx8JcI6S4Rb51ueHMhEg+OAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T01:45:19.795456Z"},"content_sha256":"980da9636006215e1ef44cf50ce3d827d60d26e33c8a2ecea70aefd7d5577621","schema_version":"1.0","event_id":"sha256:980da9636006215e1ef44cf50ce3d827d60d26e33c8a2ecea70aefd7d5577621"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/bundle.json","state_url":"https://pith.science/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/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":"2026-08-06T01:45:19Z","links":{"resolver":"https://pith.science/pith/VFJ23LDLOORVBCJUDSZLYHGVIG","bundle":"https://pith.science/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/bundle.json","state":"https://pith.science/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VFJ23LDLOORVBCJUDSZLYHGVIG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:VFJ23LDLOORVBCJUDSZLYHGVIG","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":"c8259e868f9d4baa93e283061624d968ccd3f4cd65310c707caa9004d0036ba2","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.GT","submitted_at":"2003-11-27T07:55:31Z","title_canon_sha256":"0067937051072fc63b3533d8ae7e26badf2a4d6f89a331f41691091840691378"},"schema_version":"1.0","source":{"id":"math/0311499","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0311499","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"arxiv_version","alias_value":"math/0311499v2","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0311499","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_12","alias_value":"VFJ23LDLOORV","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_16","alias_value":"VFJ23LDLOORVBCJU","created_at":"2026-07-04T15:52:40Z"},{"alias_kind":"pith_short_8","alias_value":"VFJ23LDL","created_at":"2026-07-04T15:52:40Z"}],"graph_snapshots":[{"event_id":"sha256:980da9636006215e1ef44cf50ce3d827d60d26e33c8a2ecea70aefd7d5577621","target":"graph","created_at":"2026-07-04T15:52:40Z","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/math/0311499/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rationa","authors_text":"Louis H. Kauffman, Sofia Lambropoulou","cross_cats":["math.NT"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2003-11-27T07:55:31Z","title":"On the classification of rational tangles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0311499","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":"sha256:42196227de519672d2d3277ed4f266142016d153df9ed32187e8aa088a88576c","target":"record","created_at":"2026-07-04T15:52:40Z","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":"c8259e868f9d4baa93e283061624d968ccd3f4cd65310c707caa9004d0036ba2","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.GT","submitted_at":"2003-11-27T07:55:31Z","title_canon_sha256":"0067937051072fc63b3533d8ae7e26badf2a4d6f89a331f41691091840691378"},"schema_version":"1.0","source":{"id":"math/0311499","kind":"arxiv","version":2}},"canonical_sha256":"a953adac6b73a35089341cb2bc1cd54182bb783d617ec6e44c5bbb143c9963fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a953adac6b73a35089341cb2bc1cd54182bb783d617ec6e44c5bbb143c9963fb","first_computed_at":"2026-07-04T15:52:40.143559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:52:40.143559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N3/Qk22bOYCVpKOvkfjIQ+SYpsQi4eIs1w78omfLml5tGoP4m9P4lU4u/AZvHpVMB+WzKOnQRSyWf06pPYVlBw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:52:40.143925Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0311499","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42196227de519672d2d3277ed4f266142016d153df9ed32187e8aa088a88576c","sha256:980da9636006215e1ef44cf50ce3d827d60d26e33c8a2ecea70aefd7d5577621"],"state_sha256":"096a08a1f581195fe3735b2bac9ab9384b983e877f954169dafbd17d78ba5149"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6eqBW/EzT6QzjN6e0tyCQHXIaUjzWpNLhN9vKZ7jPg2hjTueNMXJ1W2UCz66b9KX82Q9eIedbz7EIwnYF99WBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T01:45:19.812687Z","bundle_sha256":"3e6ab08eff9741a9a4da880d07585f43797c4930d95acfd47726d15af799e958"}}