{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:R34XTQGARBH5ZXSGPIUHGWTN2S","short_pith_number":"pith:R34XTQGA","canonical_record":{"source":{"id":"1407.2148","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-07-08T16:06:41Z","cross_cats_sorted":[],"title_canon_sha256":"c4fff1f9a2ea87bf8a4c10527a1b29c467f80214c95753da04cd50e888656595","abstract_canon_sha256":"173fcd1ce3ad0ac0741de60be65221884a9f6f6652fc8cd9fd220326db4cdf6b"},"schema_version":"1.0"},"canonical_sha256":"8ef979c0c0884fdcde467a28735a6dd48c45c5579d9890fc9ceb4239cfbb1f48","source":{"kind":"arxiv","id":"1407.2148","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.2148","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"arxiv_version","alias_value":"1407.2148v1","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.2148","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"pith_short_12","alias_value":"R34XTQGARBH5","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R34XTQGARBH5ZXSG","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R34XTQGA","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:R34XTQGARBH5ZXSGPIUHGWTN2S","target":"record","payload":{"canonical_record":{"source":{"id":"1407.2148","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-07-08T16:06:41Z","cross_cats_sorted":[],"title_canon_sha256":"c4fff1f9a2ea87bf8a4c10527a1b29c467f80214c95753da04cd50e888656595","abstract_canon_sha256":"173fcd1ce3ad0ac0741de60be65221884a9f6f6652fc8cd9fd220326db4cdf6b"},"schema_version":"1.0"},"canonical_sha256":"8ef979c0c0884fdcde467a28735a6dd48c45c5579d9890fc9ceb4239cfbb1f48","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:03.185622Z","signature_b64":"u/a4hQPbkMcFCZtaxiO9S/qHUcEk+3P5HKRGySCEQIbT2H6ak8txNXJIOgreHBlAY6kvCHk6W8meD4Q636qkAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ef979c0c0884fdcde467a28735a6dd48c45c5579d9890fc9ceb4239cfbb1f48","last_reissued_at":"2026-05-18T02:48:03.185202Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:03.185202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.2148","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":"2026-05-18T02:48:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xWg7y6bhOk/pcxj3WHxNdGSmn0s1V0yc9zSw4Wb9DwLanAoxkRnjtcui91TqNzeLW1qu9kFGiHXLWrQShfc5Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:25:10.038840Z"},"content_sha256":"b97a6ad4c6aea318f1d5ae0f1b10723000cb9c9ea45e1489933a91c21df12b54","schema_version":"1.0","event_id":"sha256:b97a6ad4c6aea318f1d5ae0f1b10723000cb9c9ea45e1489933a91c21df12b54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:R34XTQGARBH5ZXSGPIUHGWTN2S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Open book decompositions versus prime factorizations of closed, oriented 3-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Paolo Ghiggini, Paolo Lisca","submitted_at":"2014-07-08T16:06:41Z","abstract_excerpt":"Let $M$ be a closed, oriented, connected 3--manifold and $(B,\\pi)$ an open book decomposition on $M$ with page $\\Sigma$ and monodromy $\\varphi$. It is easy to see that the first Betti number of $\\Sigma$ is bounded below by the number of $S^2\\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $\\varphi$ is trivial and $M$ is a connected sum of $S^2\\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2148","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":"2026-05-18T02:48:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d/tYHJmYwLP0SXwoCN7Pblr7/+9MdU4pYsjfjPQesgtAg8lkPLE17HYfhp7w9NiXxcETP0h3Bki0qDpYBi2+Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:25:10.039186Z"},"content_sha256":"2d2af0d90d59706a726ecb55e0fef2d77a488dfd5837be75867664fc225877bd","schema_version":"1.0","event_id":"sha256:2d2af0d90d59706a726ecb55e0fef2d77a488dfd5837be75867664fc225877bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/bundle.json","state_url":"https://pith.science/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/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":"2026-05-28T10:25:10Z","links":{"resolver":"https://pith.science/pith/R34XTQGARBH5ZXSGPIUHGWTN2S","bundle":"https://pith.science/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/bundle.json","state":"https://pith.science/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R34XTQGARBH5ZXSGPIUHGWTN2S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:R34XTQGARBH5ZXSGPIUHGWTN2S","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":"173fcd1ce3ad0ac0741de60be65221884a9f6f6652fc8cd9fd220326db4cdf6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-07-08T16:06:41Z","title_canon_sha256":"c4fff1f9a2ea87bf8a4c10527a1b29c467f80214c95753da04cd50e888656595"},"schema_version":"1.0","source":{"id":"1407.2148","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.2148","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"arxiv_version","alias_value":"1407.2148v1","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.2148","created_at":"2026-05-18T02:48:03Z"},{"alias_kind":"pith_short_12","alias_value":"R34XTQGARBH5","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R34XTQGARBH5ZXSG","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R34XTQGA","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:2d2af0d90d59706a726ecb55e0fef2d77a488dfd5837be75867664fc225877bd","target":"graph","created_at":"2026-05-18T02:48:03Z","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":"Let $M$ be a closed, oriented, connected 3--manifold and $(B,\\pi)$ an open book decomposition on $M$ with page $\\Sigma$ and monodromy $\\varphi$. It is easy to see that the first Betti number of $\\Sigma$ is bounded below by the number of $S^2\\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $\\varphi$ is trivial and $M$ is a connected sum of $S^2\\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with","authors_text":"Paolo Ghiggini, Paolo Lisca","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-07-08T16:06:41Z","title":"Open book decompositions versus prime factorizations of closed, oriented 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2148","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:b97a6ad4c6aea318f1d5ae0f1b10723000cb9c9ea45e1489933a91c21df12b54","target":"record","created_at":"2026-05-18T02:48:03Z","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":"173fcd1ce3ad0ac0741de60be65221884a9f6f6652fc8cd9fd220326db4cdf6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-07-08T16:06:41Z","title_canon_sha256":"c4fff1f9a2ea87bf8a4c10527a1b29c467f80214c95753da04cd50e888656595"},"schema_version":"1.0","source":{"id":"1407.2148","kind":"arxiv","version":1}},"canonical_sha256":"8ef979c0c0884fdcde467a28735a6dd48c45c5579d9890fc9ceb4239cfbb1f48","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ef979c0c0884fdcde467a28735a6dd48c45c5579d9890fc9ceb4239cfbb1f48","first_computed_at":"2026-05-18T02:48:03.185202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:03.185202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u/a4hQPbkMcFCZtaxiO9S/qHUcEk+3P5HKRGySCEQIbT2H6ak8txNXJIOgreHBlAY6kvCHk6W8meD4Q636qkAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:03.185622Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.2148","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b97a6ad4c6aea318f1d5ae0f1b10723000cb9c9ea45e1489933a91c21df12b54","sha256:2d2af0d90d59706a726ecb55e0fef2d77a488dfd5837be75867664fc225877bd"],"state_sha256":"67338f1782c832708753ee1bf9d08d3aedb029b72abe8db73bc10da2700e622f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Can7IHpBUWKF3EAuOL6Mmu+7WDSP6IcxAGhSYeuRYr1nAEQERQ2SRYTzlikca0hBoGR9+5KbZ69RnuS/PKTWCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T10:25:10.041353Z","bundle_sha256":"c900c76320bdcede4017df4b24db75fa63d63c0ee37047b2f4e9ce35015bc3ec"}}