{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:RE2D7EVCDWUXOEQM7JKYSI7DMW","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":"885b5c12ff9ddae2eef155b7cc344e622326f91fd22b7fb19d3ff72f4a35ee85","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-08-05T22:06:55Z","title_canon_sha256":"4b483e3f2888c0a876575dc1cfc1e8ea87495736df75b6c91f2b730e0dad27d1"},"schema_version":"1.0","source":{"id":"1008.1095","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1095","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1095v3","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1095","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"pith_short_12","alias_value":"RE2D7EVCDWUX","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RE2D7EVCDWUXOEQM","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RE2D7EVC","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:d9c4ab76c1f19206d1c3754cbee10fa61b3589d32f925e44c86383ea6a076863","target":"graph","created_at":"2026-05-18T02:41:50Z","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 determine for which $m$, the complete graph $K_m$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.","authors_text":"Blake Mellor, Erica Flapan, Ramin Naimi","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-08-05T22:06:55Z","title":"Complete graphs whose topological symmetry groups are polyhedral"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1095","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:ece27647241ab11fa58ffe6887ed3a53b7f9ed9a4855db060b59bd666e07cb99","target":"record","created_at":"2026-05-18T02:41:50Z","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":"885b5c12ff9ddae2eef155b7cc344e622326f91fd22b7fb19d3ff72f4a35ee85","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-08-05T22:06:55Z","title_canon_sha256":"4b483e3f2888c0a876575dc1cfc1e8ea87495736df75b6c91f2b730e0dad27d1"},"schema_version":"1.0","source":{"id":"1008.1095","kind":"arxiv","version":3}},"canonical_sha256":"89343f92a21da977120cfa558923e365a9bcd95237af138717b65e2738abb0dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89343f92a21da977120cfa558923e365a9bcd95237af138717b65e2738abb0dd","first_computed_at":"2026-05-18T02:41:50.796380Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:50.796380Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"niZzQabIozzCExOOYjnkB/3HUNNQvBC+714NjJyZGP3WOkVaTDsfK26UDsJVG2EZyeCfa+CCcJN28IuG4EJpBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:50.796947Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.1095","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ece27647241ab11fa58ffe6887ed3a53b7f9ed9a4855db060b59bd666e07cb99","sha256:d9c4ab76c1f19206d1c3754cbee10fa61b3589d32f925e44c86383ea6a076863"],"state_sha256":"a5d50216ea5dc9a59c02c77b215ca1b25e1092220404a60bd46286040e81b08d"}