{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:N5HZNBR3FNXEBJ2OWU4FUD3VGC","short_pith_number":"pith:N5HZNBR3","canonical_record":{"source":{"id":"1512.06342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-12-20T09:55:28Z","cross_cats_sorted":[],"title_canon_sha256":"0cc3871d316a2369677b4682b734253ad890b033eed33c03260e00ecbde03979","abstract_canon_sha256":"12d5216f03acf6a3ce542e9c39fecf96fcaa431661e212dd1964c49cb2eb62e8"},"schema_version":"1.0"},"canonical_sha256":"6f4f96863b2b6e40a74eb5385a0f753081c83dc6e7e41c134d0c259a69c919c4","source":{"kind":"arxiv","id":"1512.06342","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.06342","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"1512.06342v1","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06342","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"N5HZNBR3FNXE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N5HZNBR3FNXEBJ2O","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N5HZNBR3","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:N5HZNBR3FNXEBJ2OWU4FUD3VGC","target":"record","payload":{"canonical_record":{"source":{"id":"1512.06342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-12-20T09:55:28Z","cross_cats_sorted":[],"title_canon_sha256":"0cc3871d316a2369677b4682b734253ad890b033eed33c03260e00ecbde03979","abstract_canon_sha256":"12d5216f03acf6a3ce542e9c39fecf96fcaa431661e212dd1964c49cb2eb62e8"},"schema_version":"1.0"},"canonical_sha256":"6f4f96863b2b6e40a74eb5385a0f753081c83dc6e7e41c134d0c259a69c919c4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:00.785634Z","signature_b64":"bTzrhDmksgS1iDAS37wNdJSi3lVQQyXsn70B4BkKHOPCmSqdg+nQ57FhfYG/Ts3up/CIY5TfkXDQuhYT5UjHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f4f96863b2b6e40a74eb5385a0f753081c83dc6e7e41c134d0c259a69c919c4","last_reissued_at":"2026-05-18T01:24:00.784913Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:00.784913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.06342","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":"2026-05-18T01:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AY92AfjOx2A7HyE8+rr3I2ziWRb4Rg1BkRS+JQwG0SHL5FB312iA+RA3xjs9A59rDPxqSLwbgWvUjuQq18vKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:58:25.131552Z"},"content_sha256":"e56a52d70f4d0af0d4f8546ed2364bbc2591e528c339ad2933d7bd140613b5e1","schema_version":"1.0","event_id":"sha256:e56a52d70f4d0af0d4f8546ed2364bbc2591e528c339ad2933d7bd140613b5e1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:N5HZNBR3FNXEBJ2OWU4FUD3VGC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Haken spheres for genus two Heegaard splittings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sangbum Cho, Yuya Koda","submitted_at":"2015-12-20T09:55:28Z","abstract_excerpt":"A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \\times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \\times S^1$ or the connected sum whose summands are lens spaces or $S^2 \\times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06342","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":"2026-05-18T01:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"50NikDLbSvvVr1jo9VQvwFfF063f4fbokBDIquSf5n/Pqkh3+yCvYvkSQbdmm5cXk/i0KFGD6ClhoVQRvqhQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:58:25.132196Z"},"content_sha256":"24b53084df1fb722962e4abe541a21918ae7f90a5a55def72c170f4973c2e913","schema_version":"1.0","event_id":"sha256:24b53084df1fb722962e4abe541a21918ae7f90a5a55def72c170f4973c2e913"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/bundle.json","state_url":"https://pith.science/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/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":"2026-06-11T04:58:25Z","links":{"resolver":"https://pith.science/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC","bundle":"https://pith.science/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/bundle.json","state":"https://pith.science/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N5HZNBR3FNXEBJ2OWU4FUD3VGC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:N5HZNBR3FNXEBJ2OWU4FUD3VGC","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":"12d5216f03acf6a3ce542e9c39fecf96fcaa431661e212dd1964c49cb2eb62e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-12-20T09:55:28Z","title_canon_sha256":"0cc3871d316a2369677b4682b734253ad890b033eed33c03260e00ecbde03979"},"schema_version":"1.0","source":{"id":"1512.06342","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.06342","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"1512.06342v1","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06342","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"N5HZNBR3FNXE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N5HZNBR3FNXEBJ2O","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N5HZNBR3","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:24b53084df1fb722962e4abe541a21918ae7f90a5a55def72c170f4973c2e913","target":"graph","created_at":"2026-05-18T01:24:00Z","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 manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \\times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \\times S^1$ or the connected sum whose summands are lens spaces or $S^2 \\times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precis","authors_text":"Sangbum Cho, Yuya Koda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-12-20T09:55:28Z","title":"Haken spheres for genus two Heegaard splittings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06342","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:e56a52d70f4d0af0d4f8546ed2364bbc2591e528c339ad2933d7bd140613b5e1","target":"record","created_at":"2026-05-18T01:24:00Z","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":"12d5216f03acf6a3ce542e9c39fecf96fcaa431661e212dd1964c49cb2eb62e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-12-20T09:55:28Z","title_canon_sha256":"0cc3871d316a2369677b4682b734253ad890b033eed33c03260e00ecbde03979"},"schema_version":"1.0","source":{"id":"1512.06342","kind":"arxiv","version":1}},"canonical_sha256":"6f4f96863b2b6e40a74eb5385a0f753081c83dc6e7e41c134d0c259a69c919c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f4f96863b2b6e40a74eb5385a0f753081c83dc6e7e41c134d0c259a69c919c4","first_computed_at":"2026-05-18T01:24:00.784913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:00.784913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bTzrhDmksgS1iDAS37wNdJSi3lVQQyXsn70B4BkKHOPCmSqdg+nQ57FhfYG/Ts3up/CIY5TfkXDQuhYT5UjHBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:00.785634Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.06342","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e56a52d70f4d0af0d4f8546ed2364bbc2591e528c339ad2933d7bd140613b5e1","sha256:24b53084df1fb722962e4abe541a21918ae7f90a5a55def72c170f4973c2e913"],"state_sha256":"d0f31ca277e8842c6b268500259993e2b148adf31cb84ba1446987306f71dabb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cRbsvs0zS7I2BnKyR/cYHRyrxoZrDx8byJEddb5akosYAYMG4PBByDuzeFBn+Ae2C0YkhyHoR9dGAmPpRb7CCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T04:58:25.136187Z","bundle_sha256":"221e9e69084696fe4260b26f7893e61da3d6ea0077562e5261f051099705c7bd"}}