{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3EKNGBFXCNNCK5OBF5DPVMJYRJ","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":"b3e25369c07c7ac6f597c803548a90d036af102a0f9d47a614312d013e839660","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-02-13T00:49:31Z","title_canon_sha256":"35527ba18a200ede2cd12237064700928a791d0a3071602ffe0db9947c9ff6ac"},"schema_version":"1.0","source":{"id":"1102.2547","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2547","created_at":"2026-05-18T03:06:13Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2547v2","created_at":"2026-05-18T03:06:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2547","created_at":"2026-05-18T03:06:13Z"},{"alias_kind":"pith_short_12","alias_value":"3EKNGBFXCNNC","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3EKNGBFXCNNCK5OB","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3EKNGBFX","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:6e13310180ef10cf467ad02ef18d334562af1dd569c73b18350c30521e095f7a","target":"graph","created_at":"2026-05-18T03:06:13Z","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":"In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed ","authors_text":"Filippo Viviani, Jesse Leo Kass, Sebastian Casalaina-Martin","cross_cats":["math.AG","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-02-13T00:49:31Z","title":"The geometry and combinatorics of cographic toric face rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2547","kind":"arxiv","version":2},"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":"sha256:19a588addabc521f1d288596acd03a9fb5d7d5c76328b9b2fb12521092f83fb0","target":"record","created_at":"2026-05-18T03:06:13Z","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":"b3e25369c07c7ac6f597c803548a90d036af102a0f9d47a614312d013e839660","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-02-13T00:49:31Z","title_canon_sha256":"35527ba18a200ede2cd12237064700928a791d0a3071602ffe0db9947c9ff6ac"},"schema_version":"1.0","source":{"id":"1102.2547","kind":"arxiv","version":2}},"canonical_sha256":"d914d304b7135a2575c12f46fab1388a7e0e68e4eeb7b2694bdaa7033ed03c30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d914d304b7135a2575c12f46fab1388a7e0e68e4eeb7b2694bdaa7033ed03c30","first_computed_at":"2026-05-18T03:06:13.926688Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:13.926688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t/268L0hNrScopBN4eZxrDd6vm37qILrMZIYWYu35jV+Jw/tsYuaGLWu/aLujzsJfS3GpgR3+V/vGaA9a4mnDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:13.927216Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.2547","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19a588addabc521f1d288596acd03a9fb5d7d5c76328b9b2fb12521092f83fb0","sha256:6e13310180ef10cf467ad02ef18d334562af1dd569c73b18350c30521e095f7a"],"state_sha256":"4bf3fb9f987bbb995130a0d170c899ec4c4e57f1543cb1fee9d6e7b84c309a4b"}