{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:PSUVKE6RKJV4FAYSVWBPFCFLYS","short_pith_number":"pith:PSUVKE6R","schema_version":"1.0","canonical_sha256":"7ca95513d1526bc28312ad82f288abc4bc798c802ad429b87bdc8899b7c18b52","source":{"kind":"arxiv","id":"1101.0066","version":2},"attestation_state":"computed","paper":{"title":"Bigraded Betti numbers of some simple polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ivan Limonchenko","submitted_at":"2010-12-30T10:30:09Z","abstract_excerpt":"The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology of the corresponding moment-angle manifold \\mathcal Z_P, and therefore they find numerous applications in combinatorial commutative algebra and toric topology. Here we calculate some bigraded Betti numbers of the type \\beta^{-i,2(i+1)} for associahedra, and relate the calculation of the bigraded Betti numbers for truncation polytopes to the topology of their"},"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":"1101.0066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-12-30T10:30:09Z","cross_cats_sorted":[],"title_canon_sha256":"e0581c69b94e149b0a64b8efe3b02e82a7e9b1d38194890bf6dc1c1010d7a4cc","abstract_canon_sha256":"3c6cb99f04ebaa6ca173b9f22433ef398d08817765ee01b6f92a43c53c2780c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:49.892191Z","signature_b64":"6UV1XXBXp4K9OGgDgaFwQPGb9aw0VVF5RAXnOIdAanQsWN5ToFhuYOJU6rE+PdjP4G8WtaUlkFPzD3PquNjCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ca95513d1526bc28312ad82f288abc4bc798c802ad429b87bdc8899b7c18b52","last_reissued_at":"2026-05-18T00:30:49.891604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:49.891604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bigraded Betti numbers of some simple polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ivan Limonchenko","submitted_at":"2010-12-30T10:30:09Z","abstract_excerpt":"The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology of the corresponding moment-angle manifold \\mathcal Z_P, and therefore they find numerous applications in combinatorial commutative algebra and toric topology. Here we calculate some bigraded Betti numbers of the type \\beta^{-i,2(i+1)} for associahedra, and relate the calculation of the bigraded Betti numbers for truncation polytopes to the topology of their"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.0066","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":"1101.0066","created_at":"2026-05-18T00:30:49.891696+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.0066v2","created_at":"2026-05-18T00:30:49.891696+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.0066","created_at":"2026-05-18T00:30:49.891696+00:00"},{"alias_kind":"pith_short_12","alias_value":"PSUVKE6RKJV4","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"PSUVKE6RKJV4FAYS","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"PSUVKE6R","created_at":"2026-05-18T12:26:12.377268+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/PSUVKE6RKJV4FAYSVWBPFCFLYS","json":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS.json","graph_json":"https://pith.science/api/pith-number/PSUVKE6RKJV4FAYSVWBPFCFLYS/graph.json","events_json":"https://pith.science/api/pith-number/PSUVKE6RKJV4FAYSVWBPFCFLYS/events.json","paper":"https://pith.science/paper/PSUVKE6R"},"agent_actions":{"view_html":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS","download_json":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS.json","view_paper":"https://pith.science/paper/PSUVKE6R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.0066&json=true","fetch_graph":"https://pith.science/api/pith-number/PSUVKE6RKJV4FAYSVWBPFCFLYS/graph.json","fetch_events":"https://pith.science/api/pith-number/PSUVKE6RKJV4FAYSVWBPFCFLYS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS/action/storage_attestation","attest_author":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS/action/author_attestation","sign_citation":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS/action/citation_signature","submit_replication":"https://pith.science/pith/PSUVKE6RKJV4FAYSVWBPFCFLYS/action/replication_record"}},"created_at":"2026-05-18T00:30:49.891696+00:00","updated_at":"2026-05-18T00:30:49.891696+00:00"}