{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:C2VZKH7N37D4XLPHRIXKVDRJOF","short_pith_number":"pith:C2VZKH7N","canonical_record":{"source":{"id":"1305.6534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-05-28T15:31:17Z","cross_cats_sorted":[],"title_canon_sha256":"9cb38457bf19de6227d110cdc13786d452fd257cb8486463b28a8508b2cd1199","abstract_canon_sha256":"480655552c5806ffb487a2bce7b160158bb4ef67791f4b205ff99b7775b89e3f"},"schema_version":"1.0"},"canonical_sha256":"16ab951feddfc7cbade78a2eaa8e29714197109526853359dc802745f25aca36","source":{"kind":"arxiv","id":"1305.6534","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6534","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6534v2","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6534","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"C2VZKH7N37D4","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"C2VZKH7N37D4XLPH","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"C2VZKH7N","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:C2VZKH7N37D4XLPHRIXKVDRJOF","target":"record","payload":{"canonical_record":{"source":{"id":"1305.6534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-05-28T15:31:17Z","cross_cats_sorted":[],"title_canon_sha256":"9cb38457bf19de6227d110cdc13786d452fd257cb8486463b28a8508b2cd1199","abstract_canon_sha256":"480655552c5806ffb487a2bce7b160158bb4ef67791f4b205ff99b7775b89e3f"},"schema_version":"1.0"},"canonical_sha256":"16ab951feddfc7cbade78a2eaa8e29714197109526853359dc802745f25aca36","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:48.473996Z","signature_b64":"DsPNI8Mv9iyQO9bZ7YEZKdqTuuO0CJbr92JksKs30KT6GgyZIwqFTRg97pDY6ZPHQE3eDzOB2bReQSzJUBGnDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16ab951feddfc7cbade78a2eaa8e29714197109526853359dc802745f25aca36","last_reissued_at":"2026-05-18T02:55:48.473348Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:48.473348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.6534","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":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2c3pP8NADb0s+SejtTWofo03FJdmbzaYyL32djr9cp01jG2qLubo1OXAebH/Vxb5xtWbhhnvpMTnUF1Oc6oWDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:56:08.087384Z"},"content_sha256":"e79618c91d1cd7e32d61aba4f9636e229122e82f89f447274ec36da9b88f6009","schema_version":"1.0","event_id":"sha256:e79618c91d1cd7e32d61aba4f9636e229122e82f89f447274ec36da9b88f6009"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:C2VZKH7N37D4XLPHRIXKVDRJOF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sangbum Cho, Yuya Koda","submitted_at":"2013-05-28T15:31:17Z","abstract_excerpt":"Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6534","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":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HfT6oQCCu4b21mKMlxRLBSdQre4Mid0Kmm68Sr5NcGC+0lZy6rt66ykDkHqfdtVp95RYUOHjb/Fvz7mbu+DKDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:56:08.088079Z"},"content_sha256":"a612c4dcb9bdbdbfd5d3a9b23242fa40cb05c430ec69291b97c8236b694ee565","schema_version":"1.0","event_id":"sha256:a612c4dcb9bdbdbfd5d3a9b23242fa40cb05c430ec69291b97c8236b694ee565"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/bundle.json","state_url":"https://pith.science/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/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":"2026-06-12T02:56:08Z","links":{"resolver":"https://pith.science/pith/C2VZKH7N37D4XLPHRIXKVDRJOF","bundle":"https://pith.science/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/bundle.json","state":"https://pith.science/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C2VZKH7N37D4XLPHRIXKVDRJOF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:C2VZKH7N37D4XLPHRIXKVDRJOF","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":"480655552c5806ffb487a2bce7b160158bb4ef67791f4b205ff99b7775b89e3f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-05-28T15:31:17Z","title_canon_sha256":"9cb38457bf19de6227d110cdc13786d452fd257cb8486463b28a8508b2cd1199"},"schema_version":"1.0","source":{"id":"1305.6534","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6534","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6534v2","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6534","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"C2VZKH7N37D4","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"C2VZKH7N37D4XLPH","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"C2VZKH7N","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:a612c4dcb9bdbdbfd5d3a9b23242fa40cb05c430ec69291b97c8236b694ee565","target":"graph","created_at":"2026-05-18T02:55:48Z","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":"Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely ","authors_text":"Sangbum Cho, Yuya Koda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-05-28T15:31:17Z","title":"Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6534","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:e79618c91d1cd7e32d61aba4f9636e229122e82f89f447274ec36da9b88f6009","target":"record","created_at":"2026-05-18T02:55:48Z","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":"480655552c5806ffb487a2bce7b160158bb4ef67791f4b205ff99b7775b89e3f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-05-28T15:31:17Z","title_canon_sha256":"9cb38457bf19de6227d110cdc13786d452fd257cb8486463b28a8508b2cd1199"},"schema_version":"1.0","source":{"id":"1305.6534","kind":"arxiv","version":2}},"canonical_sha256":"16ab951feddfc7cbade78a2eaa8e29714197109526853359dc802745f25aca36","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16ab951feddfc7cbade78a2eaa8e29714197109526853359dc802745f25aca36","first_computed_at":"2026-05-18T02:55:48.473348Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:48.473348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DsPNI8Mv9iyQO9bZ7YEZKdqTuuO0CJbr92JksKs30KT6GgyZIwqFTRg97pDY6ZPHQE3eDzOB2bReQSzJUBGnDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:48.473996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.6534","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e79618c91d1cd7e32d61aba4f9636e229122e82f89f447274ec36da9b88f6009","sha256:a612c4dcb9bdbdbfd5d3a9b23242fa40cb05c430ec69291b97c8236b694ee565"],"state_sha256":"6003db3e86f48953c8672634935a1b9e0cf7239433ccf036943ded532b76c2af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8hEfap40dc/+gRuIf4FIV8gPdloXbmTyTBCDviIQQdpxSW77zx+aajfFlzaDT/heLUyBoNk8u1bGkvyeSLBWDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T02:56:08.091980Z","bundle_sha256":"3de31b8c64c242701360daa808100aecf283a159c1c22ab48ad5e5c4bf7fe2a2"}}