{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4SC46VATD57FFLAN5HVGFCLEOV","short_pith_number":"pith:4SC46VAT","canonical_record":{"source":{"id":"1508.05307","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-08-21T15:17:46Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"24db2a3d6e95fcc8ade7cbadffd2661bfb7982f296ed8a916f4aa5603f0eca2d","abstract_canon_sha256":"5b172aa17f4d8816cd98e9770ee538af36f2c55bb45372eb9873709957ea136e"},"schema_version":"1.0"},"canonical_sha256":"e485cf54131f7e52ac0de9ea6289647549e84a7b6eeae941e296efb03fc3d97e","source":{"kind":"arxiv","id":"1508.05307","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05307","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05307v1","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05307","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"pith_short_12","alias_value":"4SC46VATD57F","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4SC46VATD57FFLAN","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4SC46VAT","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4SC46VATD57FFLAN5HVGFCLEOV","target":"record","payload":{"canonical_record":{"source":{"id":"1508.05307","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-08-21T15:17:46Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"24db2a3d6e95fcc8ade7cbadffd2661bfb7982f296ed8a916f4aa5603f0eca2d","abstract_canon_sha256":"5b172aa17f4d8816cd98e9770ee538af36f2c55bb45372eb9873709957ea136e"},"schema_version":"1.0"},"canonical_sha256":"e485cf54131f7e52ac0de9ea6289647549e84a7b6eeae941e296efb03fc3d97e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:56.165171Z","signature_b64":"Ep8A96ggpDSpasvrHlwiqA02laYStVEe18lp2BuGB4NuL+NXduq8O6t+RnZoMAPgdR6HY5VU/YqVlUvT00GkCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e485cf54131f7e52ac0de9ea6289647549e84a7b6eeae941e296efb03fc3d97e","last_reissued_at":"2026-05-18T01:34:56.164435Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:56.164435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.05307","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":"2026-05-18T01:34:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mnkp0wDx9TIJJSy/21VjUIPYMzHECa3BFhAOy8+Hbe5bvBa1w8BHZjuJ6/639lIxyWrKGn9Jo7H10o5KJ4zMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T11:14:36.620760Z"},"content_sha256":"b80a9745b3828ceb4a8cf464b51ac60c5183088f80457a1b119042cfef4ca62a","schema_version":"1.0","event_id":"sha256:b80a9745b3828ceb4a8cf464b51ac60c5183088f80457a1b119042cfef4ca62a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4SC46VATD57FFLAN5HVGFCLEOV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Morse structures on open books","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.GT","authors_text":"David T. Gay, Joan E. Licata","submitted_at":"2015-08-21T15:17:46Z","abstract_excerpt":"We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $\\mathbb{R}^3$ to arbitrary closed contact $3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05307","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":"2026-05-18T01:34:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iF4aF5LQBgPqN2MPNm8h9It04fSEroISdSiNkj+ZyqEIor10UHFawdgeLiu3b7Nuszi5pLQk4LAA0quddR0HAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T11:14:36.621108Z"},"content_sha256":"584c2c7d4c317fe3ed4ea4210926b964cfe9b3f6661d4cde97896d91da79ec6f","schema_version":"1.0","event_id":"sha256:584c2c7d4c317fe3ed4ea4210926b964cfe9b3f6661d4cde97896d91da79ec6f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4SC46VATD57FFLAN5HVGFCLEOV/bundle.json","state_url":"https://pith.science/pith/4SC46VATD57FFLAN5HVGFCLEOV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4SC46VATD57FFLAN5HVGFCLEOV/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":"2026-06-03T11:14:36Z","links":{"resolver":"https://pith.science/pith/4SC46VATD57FFLAN5HVGFCLEOV","bundle":"https://pith.science/pith/4SC46VATD57FFLAN5HVGFCLEOV/bundle.json","state":"https://pith.science/pith/4SC46VATD57FFLAN5HVGFCLEOV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4SC46VATD57FFLAN5HVGFCLEOV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4SC46VATD57FFLAN5HVGFCLEOV","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":"5b172aa17f4d8816cd98e9770ee538af36f2c55bb45372eb9873709957ea136e","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-08-21T15:17:46Z","title_canon_sha256":"24db2a3d6e95fcc8ade7cbadffd2661bfb7982f296ed8a916f4aa5603f0eca2d"},"schema_version":"1.0","source":{"id":"1508.05307","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05307","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05307v1","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05307","created_at":"2026-05-18T01:34:56Z"},{"alias_kind":"pith_short_12","alias_value":"4SC46VATD57F","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4SC46VATD57FFLAN","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4SC46VAT","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:584c2c7d4c317fe3ed4ea4210926b964cfe9b3f6661d4cde97896d91da79ec6f","target":"graph","created_at":"2026-05-18T01:34:56Z","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":"We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $\\mathbb{R}^3$ to arbitrary closed contact $3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous L","authors_text":"David T. Gay, Joan E. Licata","cross_cats":["math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-08-21T15:17:46Z","title":"Morse structures on open books"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05307","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:b80a9745b3828ceb4a8cf464b51ac60c5183088f80457a1b119042cfef4ca62a","target":"record","created_at":"2026-05-18T01:34:56Z","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":"5b172aa17f4d8816cd98e9770ee538af36f2c55bb45372eb9873709957ea136e","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-08-21T15:17:46Z","title_canon_sha256":"24db2a3d6e95fcc8ade7cbadffd2661bfb7982f296ed8a916f4aa5603f0eca2d"},"schema_version":"1.0","source":{"id":"1508.05307","kind":"arxiv","version":1}},"canonical_sha256":"e485cf54131f7e52ac0de9ea6289647549e84a7b6eeae941e296efb03fc3d97e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e485cf54131f7e52ac0de9ea6289647549e84a7b6eeae941e296efb03fc3d97e","first_computed_at":"2026-05-18T01:34:56.164435Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:56.164435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ep8A96ggpDSpasvrHlwiqA02laYStVEe18lp2BuGB4NuL+NXduq8O6t+RnZoMAPgdR6HY5VU/YqVlUvT00GkCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:56.165171Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.05307","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b80a9745b3828ceb4a8cf464b51ac60c5183088f80457a1b119042cfef4ca62a","sha256:584c2c7d4c317fe3ed4ea4210926b964cfe9b3f6661d4cde97896d91da79ec6f"],"state_sha256":"212082eff43e857c609c9826e1674af1f82a67140b58f7d534c74561e3b2f337"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CNq/IlFkKe3UwYLSMR+HkIq28fXVmVX06qlW9kJTQ1WZ05gw+nTabtmthnAL8fUXeUSOQtVoYhn17oBKVnnhCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T11:14:36.623079Z","bundle_sha256":"91b0c8e7c5669e0dc0dfdb321e155fa0a844d61746660cfa2194887e43c26c38"}}