{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:5VFXWLUGMQY7AI6CR5RO5VERA6","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":"4460d7c1b514995837aeb7451ffbeb168118aec6a73523aea8d654e8f52994ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-18T18:51:08Z","title_canon_sha256":"9b92b00bcc08e5bedb171ef06b0dbb63db4c450a0325675753bd25d5b8de12ae"},"schema_version":"1.0","source":{"id":"1011.4255","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4255","created_at":"2026-05-18T02:23:36Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4255v3","created_at":"2026-05-18T02:23:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4255","created_at":"2026-05-18T02:23:36Z"},{"alias_kind":"pith_short_12","alias_value":"5VFXWLUGMQY7","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5VFXWLUGMQY7AI6C","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5VFXWLUG","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:37d16a36292eeca228aa83e98c8b103427302c1eea71ea524e8b12174936243e","target":"graph","created_at":"2026-05-18T02:23:36Z","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":"We prove that a Cayley graph can be embedded in the euclidean plane without accumulation points of vertices if and only if it is the 1-skeleton of a Cayley complex that can be embedded in the plane after removing redundant simplices. We also give a characterisation of these Cayley graphs in term of group presentations, and deduce that they can be effectively enumerated.","authors_text":"Agelos Georgakopoulos","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-18T18:51:08Z","title":"Characterising planar Cayley graphs and Cayley complexes in terms of group presentations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4255","kind":"arxiv","version":3},"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:c3544d6298ac90a7370a0564912aad1dd819a78f4bf2a204f37f42229373afd3","target":"record","created_at":"2026-05-18T02:23:36Z","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":"4460d7c1b514995837aeb7451ffbeb168118aec6a73523aea8d654e8f52994ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-18T18:51:08Z","title_canon_sha256":"9b92b00bcc08e5bedb171ef06b0dbb63db4c450a0325675753bd25d5b8de12ae"},"schema_version":"1.0","source":{"id":"1011.4255","kind":"arxiv","version":3}},"canonical_sha256":"ed4b7b2e866431f023c28f62eed491078ea2107e9d778cd3857d1cc74f98c28c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed4b7b2e866431f023c28f62eed491078ea2107e9d778cd3857d1cc74f98c28c","first_computed_at":"2026-05-18T02:23:36.277202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:23:36.277202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gqVWZaKdY+uvF0IJzcI7zY3I855UkbJJxY+RMN5AHYuNHEG/goS6rKXQRgnRPLQHTOITjCRYPhkUoMvVBw8XCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:23:36.277860Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.4255","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3544d6298ac90a7370a0564912aad1dd819a78f4bf2a204f37f42229373afd3","sha256:37d16a36292eeca228aa83e98c8b103427302c1eea71ea524e8b12174936243e"],"state_sha256":"9ebdf637d800d17db6aa8ce0db0d81ff8de3938311a5871d245c16da9868bcce"}