{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:P5IUTZROED42SYFC256VDTN2N2","short_pith_number":"pith:P5IUTZRO","canonical_record":{"source":{"id":"1110.1102","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-05T20:58:14Z","cross_cats_sorted":[],"title_canon_sha256":"db69b1b7f4f334865eb84ac0ed9dca5d2740be85cf43ca94c7dd82ec4c03a4bf","abstract_canon_sha256":"15ddeeecb97bd86253be782e1b65f56c64108d61ff5667db6ae2c92b4eab3177"},"schema_version":"1.0"},"canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","source":{"kind":"arxiv","id":"1110.1102","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1102","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1102v1","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1102","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"P5IUTZROED42","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"P5IUTZROED42SYFC","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"P5IUTZRO","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:P5IUTZROED42SYFC256VDTN2N2","target":"record","payload":{"canonical_record":{"source":{"id":"1110.1102","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-05T20:58:14Z","cross_cats_sorted":[],"title_canon_sha256":"db69b1b7f4f334865eb84ac0ed9dca5d2740be85cf43ca94c7dd82ec4c03a4bf","abstract_canon_sha256":"15ddeeecb97bd86253be782e1b65f56c64108d61ff5667db6ae2c92b4eab3177"},"schema_version":"1.0"},"canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:39.281546Z","signature_b64":"jenXp3kYp0hTJ84BatNh6w9+0MQwxWZ0ZAed0RPfwunHh3zaeIWCtiqFSP3/BAX94mB26sW/B2hka7NVplPLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","last_reissued_at":"2026-05-18T04:11:39.280995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:39.280995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.1102","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hMdlNT+ZI5iD4klXvZLvz3ULB3ZslTqn7hwNKFiG6hkaX8KCw5GEJi0MP/K5ByG5ojNvOMURfaG+qmQvjpieBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T07:18:53.009919Z"},"content_sha256":"94489c3cc67cdc251105e3941ec03406f35bf69167e7871a68cd0b9ad0919f2e","schema_version":"1.0","event_id":"sha256:94489c3cc67cdc251105e3941ec03406f35bf69167e7871a68cd0b9ad0919f2e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:P5IUTZROED42SYFC256VDTN2N2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\ell^2$-homology and planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Timothy A. Schroeder","submitted_at":"2011-10-05T20:58:14Z","abstract_excerpt":"In his 1930 paper, Kuratowksi categorized planar graphs, proving that a finite graph $\\Gamma$ is planar if and only if it does not contain a subgraph that is homeomorphic to $K_5$, the complete graph on 5 vertices, or $K_{3,3}$, the complete bipartite graph on six vertices. In their 2001 paper, Davis and Okun point out that the $K_{3,3}$ graph can be understood as the nerve of a right-angled Coxeter system and prove that this graph is not planar using results from $\\ell^2$-homology. In this paper, we employ a similar method proving $K_5$ is not planar."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1102","kind":"arxiv","version":1},"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":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pj9QjpjICPoYFdnnZtbafyZ+QS3Bg65UB7oK078OK0UGq0mzlGbu1PtIY04962I3Nt5u7oCeor0gIauYI299Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T07:18:53.010257Z"},"content_sha256":"92789e0a296d3c17da3818071f4511151e0d74d2d317d2360b88fdb85a3877bd","schema_version":"1.0","event_id":"sha256:92789e0a296d3c17da3818071f4511151e0d74d2d317d2360b88fdb85a3877bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/bundle.json","state_url":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P5IUTZROED42SYFC256VDTN2N2/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":"2026-06-27T07:18:53Z","links":{"resolver":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2","bundle":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/bundle.json","state":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P5IUTZROED42SYFC256VDTN2N2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:P5IUTZROED42SYFC256VDTN2N2","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":"15ddeeecb97bd86253be782e1b65f56c64108d61ff5667db6ae2c92b4eab3177","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-05T20:58:14Z","title_canon_sha256":"db69b1b7f4f334865eb84ac0ed9dca5d2740be85cf43ca94c7dd82ec4c03a4bf"},"schema_version":"1.0","source":{"id":"1110.1102","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1102","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1102v1","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1102","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"P5IUTZROED42","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"P5IUTZROED42SYFC","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"P5IUTZRO","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:92789e0a296d3c17da3818071f4511151e0d74d2d317d2360b88fdb85a3877bd","target":"graph","created_at":"2026-05-18T04:11:39Z","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 his 1930 paper, Kuratowksi categorized planar graphs, proving that a finite graph $\\Gamma$ is planar if and only if it does not contain a subgraph that is homeomorphic to $K_5$, the complete graph on 5 vertices, or $K_{3,3}$, the complete bipartite graph on six vertices. In their 2001 paper, Davis and Okun point out that the $K_{3,3}$ graph can be understood as the nerve of a right-angled Coxeter system and prove that this graph is not planar using results from $\\ell^2$-homology. In this paper, we employ a similar method proving $K_5$ is not planar.","authors_text":"Timothy A. Schroeder","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-05T20:58:14Z","title":"$\\ell^2$-homology and planar graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1102","kind":"arxiv","version":1},"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":"sha256:94489c3cc67cdc251105e3941ec03406f35bf69167e7871a68cd0b9ad0919f2e","target":"record","created_at":"2026-05-18T04:11:39Z","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":"15ddeeecb97bd86253be782e1b65f56c64108d61ff5667db6ae2c92b4eab3177","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-05T20:58:14Z","title_canon_sha256":"db69b1b7f4f334865eb84ac0ed9dca5d2740be85cf43ca94c7dd82ec4c03a4bf"},"schema_version":"1.0","source":{"id":"1110.1102","kind":"arxiv","version":1}},"canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","first_computed_at":"2026-05-18T04:11:39.280995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:39.280995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jenXp3kYp0hTJ84BatNh6w9+0MQwxWZ0ZAed0RPfwunHh3zaeIWCtiqFSP3/BAX94mB26sW/B2hka7NVplPLBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:39.281546Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1102","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94489c3cc67cdc251105e3941ec03406f35bf69167e7871a68cd0b9ad0919f2e","sha256:92789e0a296d3c17da3818071f4511151e0d74d2d317d2360b88fdb85a3877bd"],"state_sha256":"9982196ad13472baba61ece3c31f2cf2af0bf90d9715184ea28b2250adc04e5b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RhntiQSFtozD16p4kmenE/77c2Wf6BTwxep7XoaSrjc+9TavPsXeSM8mX30dSVCJVpcx6T/RyttWFrmetQTEDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T07:18:53.012099Z","bundle_sha256":"fdfa46e5cf2f49daabcf01786d6117fcf1cb0a30cce98dfab39321f2532c8e93"}}