{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:VIZSXW3X3RMP2W2B3ZXBGBZHR4","short_pith_number":"pith:VIZSXW3X","schema_version":"1.0","canonical_sha256":"aa332bdb77dc58fd5b41de6e1307278f002d625512439eaec825d366ce0819d2","source":{"kind":"arxiv","id":"1604.00163","version":2},"attestation_state":"computed","paper":{"title":"Asphericity of a length four relative group presentation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Abd Ghafur Bin Ahmad, Martin Edjvet, Muna A Al-Mulla","submitted_at":"2016-04-01T08:20:35Z","abstract_excerpt":"We consider the relative group presentation $\\mathcal{P} = \\langle G, \\mathbf{x} | \\mathbf{r} \\rangle$ where $\\mathbf{x} = \\{ x \\}$ and $\\mathbf{r} = \\{ xg_1 xg_2 xg_3 x^{-1} g_4 \\}$. We show modulo a small number of exceptional cases exactly when $\\mathcal{P}$ is aspherical. If $H = \\langle g_1^{-1} g_2, g_1^{-1} g_3 g_1 , g_4 \\rangle \\leq G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2 \\times C_4$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.00163","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-01T08:20:35Z","cross_cats_sorted":[],"title_canon_sha256":"43b92da37536241eea692f630fe727764e57367becad932bb55a8d3cdeabc9af","abstract_canon_sha256":"7bf9190cd0a521577da58f56b7c0c5242db8c0b57c5186a0d39862aa5eedc2e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:42.344184Z","signature_b64":"+D57UGddChldi3QPLGs22HpEAYDsjwtGkecGUMJ3J3XSryjEhcdw45vkecvU/zBR0gN9G7J+sqGBbOCJRVu5BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa332bdb77dc58fd5b41de6e1307278f002d625512439eaec825d366ce0819d2","last_reissued_at":"2026-05-18T00:38:42.343558Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:42.343558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asphericity of a length four relative group presentation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Abd Ghafur Bin Ahmad, Martin Edjvet, Muna A Al-Mulla","submitted_at":"2016-04-01T08:20:35Z","abstract_excerpt":"We consider the relative group presentation $\\mathcal{P} = \\langle G, \\mathbf{x} | \\mathbf{r} \\rangle$ where $\\mathbf{x} = \\{ x \\}$ and $\\mathbf{r} = \\{ xg_1 xg_2 xg_3 x^{-1} g_4 \\}$. We show modulo a small number of exceptional cases exactly when $\\mathcal{P}$ is aspherical. If $H = \\langle g_1^{-1} g_2, g_1^{-1} g_3 g_1 , g_4 \\rangle \\leq G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2 \\times C_4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00163","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.00163","created_at":"2026-05-18T00:38:42.343650+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.00163v2","created_at":"2026-05-18T00:38:42.343650+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00163","created_at":"2026-05-18T00:38:42.343650+00:00"},{"alias_kind":"pith_short_12","alias_value":"VIZSXW3X3RMP","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VIZSXW3X3RMP2W2B","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VIZSXW3X","created_at":"2026-05-18T12:30:48.956258+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4","json":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4.json","graph_json":"https://pith.science/api/pith-number/VIZSXW3X3RMP2W2B3ZXBGBZHR4/graph.json","events_json":"https://pith.science/api/pith-number/VIZSXW3X3RMP2W2B3ZXBGBZHR4/events.json","paper":"https://pith.science/paper/VIZSXW3X"},"agent_actions":{"view_html":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4","download_json":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4.json","view_paper":"https://pith.science/paper/VIZSXW3X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.00163&json=true","fetch_graph":"https://pith.science/api/pith-number/VIZSXW3X3RMP2W2B3ZXBGBZHR4/graph.json","fetch_events":"https://pith.science/api/pith-number/VIZSXW3X3RMP2W2B3ZXBGBZHR4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4/action/storage_attestation","attest_author":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4/action/author_attestation","sign_citation":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4/action/citation_signature","submit_replication":"https://pith.science/pith/VIZSXW3X3RMP2W2B3ZXBGBZHR4/action/replication_record"}},"created_at":"2026-05-18T00:38:42.343650+00:00","updated_at":"2026-05-18T00:38:42.343650+00:00"}