{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZODPKDWHXZ3TLXX7LFESWJXA7Z","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":"34b14e29b4e040aadeb6f0c568ebd1642f97fc071ed8aa4868ddd8b4fcf2780f","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-22T19:26:23Z","title_canon_sha256":"62076ae3721e5b09d3c6893f364f34435146f6b68749f6b3e3fd0be193ac3055"},"schema_version":"1.0","source":{"id":"1112.5423","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5423","created_at":"2026-05-18T03:17:25Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5423v3","created_at":"2026-05-18T03:17:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5423","created_at":"2026-05-18T03:17:25Z"},{"alias_kind":"pith_short_12","alias_value":"ZODPKDWHXZ3T","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"ZODPKDWHXZ3TLXX7","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"ZODPKDWH","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:767b22517c844d4e5aa36e1794cab461f3e8c32d61b7f6e18bf7a3b9fa86e22e","target":"graph","created_at":"2026-05-18T03:17:25Z","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":"Let $G$ be a triangulation of the sphere with vertex set $V$, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $A_W$ to be the abelian group generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $A_B$ can be defined similarly, using black triangles.\n  The paper shows that $A_W$ and $A_B$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the struct","authors_text":"Simon R. Blackburn, Thomas A. McCourt","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-22T19:26:23Z","title":"Triangulations of the sphere, bitrades and abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5423","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:db2938874c942355084ba55b67f973bcbced3e04c5d3e9f152d7d934a747e23c","target":"record","created_at":"2026-05-18T03:17:25Z","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":"34b14e29b4e040aadeb6f0c568ebd1642f97fc071ed8aa4868ddd8b4fcf2780f","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-22T19:26:23Z","title_canon_sha256":"62076ae3721e5b09d3c6893f364f34435146f6b68749f6b3e3fd0be193ac3055"},"schema_version":"1.0","source":{"id":"1112.5423","kind":"arxiv","version":3}},"canonical_sha256":"cb86f50ec7be7735deff59492b26e0fe7d379319913084dd9941e86ec846d6fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb86f50ec7be7735deff59492b26e0fe7d379319913084dd9941e86ec846d6fa","first_computed_at":"2026-05-18T03:17:25.080084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:25.080084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ndWM80q0yxOkKQtAuPn7VpisTx59Ik0uVkLLZQPDLOYBTlqxqFdxAbY3TbR4iGBfYbPwp8SqNnIFls1htrLbAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:25.080536Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.5423","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db2938874c942355084ba55b67f973bcbced3e04c5d3e9f152d7d934a747e23c","sha256:767b22517c844d4e5aa36e1794cab461f3e8c32d61b7f6e18bf7a3b9fa86e22e"],"state_sha256":"d71eeff00739ba2e57ddedc032a6a1a9893b35ea4a222a16fc9706cc7189b26e"}