{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2000:VEYFWZPYQX3AYZZUTLF5NO6N3L","short_pith_number":"pith:VEYFWZPY","canonical_record":{"source":{"id":"math/0003132","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2000-03-22T17:07:24Z","cross_cats_sorted":[],"title_canon_sha256":"648823a61d14e12f7f0f51c383558b27acb243f3ab35737a9e3fe4c32461749e","abstract_canon_sha256":"fda632b9bca7225e6c377ce625bc85b32be66e2cd13faf37f23432961db3ca5a"},"schema_version":"1.0"},"canonical_sha256":"a9305b65f885f60c67349acbd6bbcddaf2352ed726b6202110d4b270d8f115c1","source":{"kind":"arxiv","id":"math/0003132","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0003132","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"arxiv_version","alias_value":"math/0003132v2","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0003132","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"pith_short_12","alias_value":"VEYFWZPYQX3A","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"VEYFWZPYQX3AYZZU","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"VEYFWZPY","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2000:VEYFWZPYQX3AYZZUTLF5NO6N3L","target":"record","payload":{"canonical_record":{"source":{"id":"math/0003132","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2000-03-22T17:07:24Z","cross_cats_sorted":[],"title_canon_sha256":"648823a61d14e12f7f0f51c383558b27acb243f3ab35737a9e3fe4c32461749e","abstract_canon_sha256":"fda632b9bca7225e6c377ce625bc85b32be66e2cd13faf37f23432961db3ca5a"},"schema_version":"1.0"},"canonical_sha256":"a9305b65f885f60c67349acbd6bbcddaf2352ed726b6202110d4b270d8f115c1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:01.026251Z","signature_b64":"Feu40RrAJN+IWPdRBupxwp4thkSC4SI9QAlBno8QNhrT89Tf/l2Q9lrnxmzBmyDd2cotZSr5Hel/xGyMh58OAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9305b65f885f60c67349acbd6bbcddaf2352ed726b6202110d4b270d8f115c1","last_reissued_at":"2026-05-18T02:38:01.025679Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:01.025679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0003132","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":"2026-05-18T02:38:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PCtU4XgzJKOEKsr9wkJQKDzdv2df2PKewZYsqu0qNMMQj6IcWswBTJLPDJ0Nw6TdosKkKyW5cJBqbBLgvuhoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:41:49.152012Z"},"content_sha256":"62aebe62e047803cbebaecb3158cf7355e1471741827dec85fd3cd3e5cab0311","schema_version":"1.0","event_id":"sha256:62aebe62e047803cbebaecb3158cf7355e1471741827dec85fd3cd3e5cab0311"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2000:VEYFWZPYQX3AYZZUTLF5NO6N3L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Taut ideal triangulations of 3-manifolds","license":"","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Marc Lackenby","submitted_at":"2000-03-22T17:07:24Z","abstract_excerpt":"A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behaviour is very similar to that of a taut foliation. For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai's result that the singular genus of a knot in the 3-sphere is equal to its genus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0003132","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":"2026-05-18T02:38:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0fPTKV+RZHpU+oWLpDuOXhTgLHxxPUtFgCwPHyWvxfiU6GGk0TnSjjkOtoth0/GNDJH5KdoJjvIf2Xd/4kpXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:41:49.152368Z"},"content_sha256":"161b26e1a081a197e878e725f07539b2cf2be97301a8bb312466ec194afb44de","schema_version":"1.0","event_id":"sha256:161b26e1a081a197e878e725f07539b2cf2be97301a8bb312466ec194afb44de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/bundle.json","state_url":"https://pith.science/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/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":"2026-06-01T02:41:49Z","links":{"resolver":"https://pith.science/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L","bundle":"https://pith.science/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/bundle.json","state":"https://pith.science/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VEYFWZPYQX3AYZZUTLF5NO6N3L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:VEYFWZPYQX3AYZZUTLF5NO6N3L","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":"fda632b9bca7225e6c377ce625bc85b32be66e2cd13faf37f23432961db3ca5a","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2000-03-22T17:07:24Z","title_canon_sha256":"648823a61d14e12f7f0f51c383558b27acb243f3ab35737a9e3fe4c32461749e"},"schema_version":"1.0","source":{"id":"math/0003132","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0003132","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"arxiv_version","alias_value":"math/0003132v2","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0003132","created_at":"2026-05-18T02:38:01Z"},{"alias_kind":"pith_short_12","alias_value":"VEYFWZPYQX3A","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"VEYFWZPYQX3AYZZU","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"VEYFWZPY","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:161b26e1a081a197e878e725f07539b2cf2be97301a8bb312466ec194afb44de","target":"graph","created_at":"2026-05-18T02:38:01Z","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":"A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behaviour is very similar to that of a taut foliation. For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai's result that the singular genus of a knot in the 3-sphere is equal to its genus.","authors_text":"Marc Lackenby","cross_cats":[],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2000-03-22T17:07:24Z","title":"Taut ideal triangulations of 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0003132","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":"sha256:62aebe62e047803cbebaecb3158cf7355e1471741827dec85fd3cd3e5cab0311","target":"record","created_at":"2026-05-18T02:38:01Z","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":"fda632b9bca7225e6c377ce625bc85b32be66e2cd13faf37f23432961db3ca5a","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2000-03-22T17:07:24Z","title_canon_sha256":"648823a61d14e12f7f0f51c383558b27acb243f3ab35737a9e3fe4c32461749e"},"schema_version":"1.0","source":{"id":"math/0003132","kind":"arxiv","version":2}},"canonical_sha256":"a9305b65f885f60c67349acbd6bbcddaf2352ed726b6202110d4b270d8f115c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9305b65f885f60c67349acbd6bbcddaf2352ed726b6202110d4b270d8f115c1","first_computed_at":"2026-05-18T02:38:01.025679Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:01.025679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Feu40RrAJN+IWPdRBupxwp4thkSC4SI9QAlBno8QNhrT89Tf/l2Q9lrnxmzBmyDd2cotZSr5Hel/xGyMh58OAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:01.026251Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0003132","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62aebe62e047803cbebaecb3158cf7355e1471741827dec85fd3cd3e5cab0311","sha256:161b26e1a081a197e878e725f07539b2cf2be97301a8bb312466ec194afb44de"],"state_sha256":"1c50085365506453a57d872ea666c2909756b190e371fe01439f8b8dda373ce4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sFx3Da4SmLS6cibQ94M1cmBBzYhLfMvLTH/ZaCRPDx0+6oq++NFIZhQdtPXi57OdTkctEgidLTIOO8yehx5OAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:41:49.154625Z","bundle_sha256":"28772a6c396fb21f88150589c50e7ca3c7b57e72ab3356057a1f2b5cd48e2a93"}}