{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:DKPE6AY34VOGUYO2F7VMCDEK5H","short_pith_number":"pith:DKPE6AY3","schema_version":"1.0","canonical_sha256":"1a9e4f031be55c6a61da2feac10c8ae9ec591ae5cf4a028269e26ca657c9cc46","source":{"kind":"arxiv","id":"1606.08656","version":3},"attestation_state":"computed","paper":{"title":"Markov numbers and Lagrangian cell complexes in the complex projective plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.GT"],"primary_cat":"math.SG","authors_text":"Ivan Smith, Jonathan David Evans","submitted_at":"2016-06-28T11:29:51Z","abstract_excerpt":"We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\\ldots,N$, cannot be made disjoint unless $N\\leq 3$ and the $p_i$ form part "},"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":"1606.08656","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2016-06-28T11:29:51Z","cross_cats_sorted":["math.AG","math.GT"],"title_canon_sha256":"0aed13a2c0a8396efad14afb540ba5622cafcc90d9f46dde744a5de2a3a43e19","abstract_canon_sha256":"018652b2d89c4bf7fb00d8398f75272c66c7ccd5550b58e46fb3c3c36f73f63b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:59.412518Z","signature_b64":"sOHBQCmjMNSEsO1phuuM+nzfLebZnT6xKvHL2O+Sil7s9x3KP0pkso8AAyZD2NbzeAq8B37LKHQ4ufORzknKCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a9e4f031be55c6a61da2feac10c8ae9ec591ae5cf4a028269e26ca657c9cc46","last_reissued_at":"2026-05-18T00:20:59.412058Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:59.412058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Markov numbers and Lagrangian cell complexes in the complex projective plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.GT"],"primary_cat":"math.SG","authors_text":"Ivan Smith, Jonathan David Evans","submitted_at":"2016-06-28T11:29:51Z","abstract_excerpt":"We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\\ldots,N$, cannot be made disjoint unless $N\\leq 3$ and the $p_i$ form part "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08656","kind":"arxiv","version":3},"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":"1606.08656","created_at":"2026-05-18T00:20:59.412135+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.08656v3","created_at":"2026-05-18T00:20:59.412135+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.08656","created_at":"2026-05-18T00:20:59.412135+00:00"},{"alias_kind":"pith_short_12","alias_value":"DKPE6AY34VOG","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DKPE6AY34VOGUYO2","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DKPE6AY3","created_at":"2026-05-18T12:30:12.583610+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/DKPE6AY34VOGUYO2F7VMCDEK5H","json":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H.json","graph_json":"https://pith.science/api/pith-number/DKPE6AY34VOGUYO2F7VMCDEK5H/graph.json","events_json":"https://pith.science/api/pith-number/DKPE6AY34VOGUYO2F7VMCDEK5H/events.json","paper":"https://pith.science/paper/DKPE6AY3"},"agent_actions":{"view_html":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H","download_json":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H.json","view_paper":"https://pith.science/paper/DKPE6AY3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.08656&json=true","fetch_graph":"https://pith.science/api/pith-number/DKPE6AY34VOGUYO2F7VMCDEK5H/graph.json","fetch_events":"https://pith.science/api/pith-number/DKPE6AY34VOGUYO2F7VMCDEK5H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H/action/storage_attestation","attest_author":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H/action/author_attestation","sign_citation":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H/action/citation_signature","submit_replication":"https://pith.science/pith/DKPE6AY34VOGUYO2F7VMCDEK5H/action/replication_record"}},"created_at":"2026-05-18T00:20:59.412135+00:00","updated_at":"2026-05-18T00:20:59.412135+00:00"}