{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:YPKJRQ5OPFGWWO7YQUNL42O3YX","short_pith_number":"pith:YPKJRQ5O","canonical_record":{"source":{"id":"1106.5797","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-28T20:37:56Z","cross_cats_sorted":[],"title_canon_sha256":"32e2cf6f4f3ea0633c4dc02c7ee76cd00a15504644fd5f065582f0926a39b591","abstract_canon_sha256":"a5515dcc8f404cab382370bc0658e49f0ad99246acc20a487958bda821e712fe"},"schema_version":"1.0"},"canonical_sha256":"c3d498c3ae794d6b3bf8851abe69dbc5c5d53a6011057e0b792ab91c4e15086d","source":{"kind":"arxiv","id":"1106.5797","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5797","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5797v2","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5797","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"pith_short_12","alias_value":"YPKJRQ5OPFGW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YPKJRQ5OPFGWWO7Y","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YPKJRQ5O","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:YPKJRQ5OPFGWWO7YQUNL42O3YX","target":"record","payload":{"canonical_record":{"source":{"id":"1106.5797","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-28T20:37:56Z","cross_cats_sorted":[],"title_canon_sha256":"32e2cf6f4f3ea0633c4dc02c7ee76cd00a15504644fd5f065582f0926a39b591","abstract_canon_sha256":"a5515dcc8f404cab382370bc0658e49f0ad99246acc20a487958bda821e712fe"},"schema_version":"1.0"},"canonical_sha256":"c3d498c3ae794d6b3bf8851abe69dbc5c5d53a6011057e0b792ab91c4e15086d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:40.707682Z","signature_b64":"WrmfFrgq2NkS6J1SkOAK00mef0k8IteZGB21Y3Dg8GRAtL24zmDWH6sSy2NQNzyobH5jd50J/KBmSfRpgDdwBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3d498c3ae794d6b3bf8851abe69dbc5c5d53a6011057e0b792ab91c4e15086d","last_reissued_at":"2026-05-18T02:30:40.707223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:40.707223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.5797","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+lUR5J4tjgg2oAqYmgYhA4bh5/XwRfXNKho6w8HI/xGb4TyiR/IJfTLHQEBySfDTREcH7MVr3h1GmZLX8lIMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:40:14.313155Z"},"content_sha256":"e9a660021fb846f379286b240ea467458483c998fba6a61e97717b03c41cefa5","schema_version":"1.0","event_id":"sha256:e9a660021fb846f379286b240ea467458483c998fba6a61e97717b03c41cefa5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:YPKJRQ5OPFGWWO7YQUNL42O3YX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complete bipartite graphs whose topological symmetry groups are polyhedral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Blake Mellor","submitted_at":"2011-06-28T20:37:56Z","abstract_excerpt":"We determine for which $n$, the complete bipartite graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5797","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F3aRhOZ7ZUJu44cdncvLbwkUdwcpK+WNO8h69ouwt+OHcidMwB1aSRbkWdyJpOsNpvhH/WoVBinKEpzDq3plCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:40:14.313516Z"},"content_sha256":"3b14c468f5a6f488b74ba8643e0d9e176c11b77f18914f869fcf34651d046db4","schema_version":"1.0","event_id":"sha256:3b14c468f5a6f488b74ba8643e0d9e176c11b77f18914f869fcf34651d046db4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/bundle.json","state_url":"https://pith.science/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T05:40:14Z","links":{"resolver":"https://pith.science/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX","bundle":"https://pith.science/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/bundle.json","state":"https://pith.science/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YPKJRQ5OPFGWWO7YQUNL42O3YX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:YPKJRQ5OPFGWWO7YQUNL42O3YX","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":"a5515dcc8f404cab382370bc0658e49f0ad99246acc20a487958bda821e712fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-28T20:37:56Z","title_canon_sha256":"32e2cf6f4f3ea0633c4dc02c7ee76cd00a15504644fd5f065582f0926a39b591"},"schema_version":"1.0","source":{"id":"1106.5797","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5797","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5797v2","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5797","created_at":"2026-05-18T02:30:40Z"},{"alias_kind":"pith_short_12","alias_value":"YPKJRQ5OPFGW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YPKJRQ5OPFGWWO7Y","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YPKJRQ5O","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:3b14c468f5a6f488b74ba8643e0d9e176c11b77f18914f869fcf34651d046db4","target":"graph","created_at":"2026-05-18T02:30:40Z","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 $n$, the complete bipartite graph $K_{n,n}$ 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","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-28T20:37:56Z","title":"Complete bipartite graphs whose topological symmetry groups are polyhedral"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5797","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:e9a660021fb846f379286b240ea467458483c998fba6a61e97717b03c41cefa5","target":"record","created_at":"2026-05-18T02:30:40Z","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":"a5515dcc8f404cab382370bc0658e49f0ad99246acc20a487958bda821e712fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-28T20:37:56Z","title_canon_sha256":"32e2cf6f4f3ea0633c4dc02c7ee76cd00a15504644fd5f065582f0926a39b591"},"schema_version":"1.0","source":{"id":"1106.5797","kind":"arxiv","version":2}},"canonical_sha256":"c3d498c3ae794d6b3bf8851abe69dbc5c5d53a6011057e0b792ab91c4e15086d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3d498c3ae794d6b3bf8851abe69dbc5c5d53a6011057e0b792ab91c4e15086d","first_computed_at":"2026-05-18T02:30:40.707223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:40.707223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WrmfFrgq2NkS6J1SkOAK00mef0k8IteZGB21Y3Dg8GRAtL24zmDWH6sSy2NQNzyobH5jd50J/KBmSfRpgDdwBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:40.707682Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.5797","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9a660021fb846f379286b240ea467458483c998fba6a61e97717b03c41cefa5","sha256:3b14c468f5a6f488b74ba8643e0d9e176c11b77f18914f869fcf34651d046db4"],"state_sha256":"221fff0fbd96e0d20d52449bf84170530cf54909ac823dcd6aa5a393b665b15c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1rKlLV+VMuamWSfoTOhnc+Px7VG5aTmwd0qCO3dQx3300gD6NjeG6SN0BcIZ644oT7sLT9mzWsVUnguCOuIaAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T05:40:14.315458Z","bundle_sha256":"cc571f23fa28cc1b0951bf25338a18120715ea3a4627995199826d1e44b029ba"}}