{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:BGDISHSPQSBYUHUS6LYXII53SN","short_pith_number":"pith:BGDISHSP","schema_version":"1.0","canonical_sha256":"0986891e4f84838a1e92f2f17423bb9374178103d54fadd0983c00b8bdd6130a","source":{"kind":"arxiv","id":"math/0612213","version":2},"attestation_state":"computed","paper":{"title":"Cluster-Cyclic Quivers with three Vertices and the Markov Equation","license":"","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Andre Beineke, Lutz Hille, Thomas Br\\\"ustle","submitted_at":"2006-12-08T11:53:03Z","abstract_excerpt":"Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic.\n  In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov."},"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":"math/0612213","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2006-12-08T11:53:03Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"1e6da950fcf11410aceba0407b1a8dfa837001bc17b7db5f62d72d61785c2ee5","abstract_canon_sha256":"032397160188a2dd743b60465e1299a4a5170b604b4c96dbb6d96cd54d5f5926"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:53.093196Z","signature_b64":"N6FnleWleFZJ+1fty2FeXn6R+4blocR9RFQQno6tpfiWyNqCv6ju+HwW079u/GY1m/2jvDGu/3DTG92QCUR2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0986891e4f84838a1e92f2f17423bb9374178103d54fadd0983c00b8bdd6130a","last_reissued_at":"2026-05-18T04:08:53.092759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:53.092759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cluster-Cyclic Quivers with three Vertices and the Markov Equation","license":"","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Andre Beineke, Lutz Hille, Thomas Br\\\"ustle","submitted_at":"2006-12-08T11:53:03Z","abstract_excerpt":"Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic.\n  In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612213","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":"math/0612213","created_at":"2026-05-18T04:08:53.092828+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0612213v2","created_at":"2026-05-18T04:08:53.092828+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612213","created_at":"2026-05-18T04:08:53.092828+00:00"},{"alias_kind":"pith_short_12","alias_value":"BGDISHSPQSBY","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"BGDISHSPQSBYUHUS","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"BGDISHSP","created_at":"2026-05-18T12:25:53.939244+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/BGDISHSPQSBYUHUS6LYXII53SN","json":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN.json","graph_json":"https://pith.science/api/pith-number/BGDISHSPQSBYUHUS6LYXII53SN/graph.json","events_json":"https://pith.science/api/pith-number/BGDISHSPQSBYUHUS6LYXII53SN/events.json","paper":"https://pith.science/paper/BGDISHSP"},"agent_actions":{"view_html":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN","download_json":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN.json","view_paper":"https://pith.science/paper/BGDISHSP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0612213&json=true","fetch_graph":"https://pith.science/api/pith-number/BGDISHSPQSBYUHUS6LYXII53SN/graph.json","fetch_events":"https://pith.science/api/pith-number/BGDISHSPQSBYUHUS6LYXII53SN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN/action/storage_attestation","attest_author":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN/action/author_attestation","sign_citation":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN/action/citation_signature","submit_replication":"https://pith.science/pith/BGDISHSPQSBYUHUS6LYXII53SN/action/replication_record"}},"created_at":"2026-05-18T04:08:53.092828+00:00","updated_at":"2026-05-18T04:08:53.092828+00:00"}