{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:P5IUTZROED42SYFC256VDTN2N2","short_pith_number":"pith:P5IUTZRO","schema_version":"1.0","canonical_sha256":"7f5149e62e20f9a960a2d77d51cdba6e97ef64c1056c034712375f310a296ea1","source":{"kind":"arxiv","id":"1110.1102","version":1},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.1102","created_at":"2026-05-18T04:11:39.281102+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.1102v1","created_at":"2026-05-18T04:11:39.281102+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1102","created_at":"2026-05-18T04:11:39.281102+00:00"},{"alias_kind":"pith_short_12","alias_value":"P5IUTZROED42","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"P5IUTZROED42SYFC","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"P5IUTZRO","created_at":"2026-05-18T12:26:39.201973+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2","json":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2.json","graph_json":"https://pith.science/api/pith-number/P5IUTZROED42SYFC256VDTN2N2/graph.json","events_json":"https://pith.science/api/pith-number/P5IUTZROED42SYFC256VDTN2N2/events.json","paper":"https://pith.science/paper/P5IUTZRO"},"agent_actions":{"view_html":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2","download_json":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2.json","view_paper":"https://pith.science/paper/P5IUTZRO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.1102&json=true","fetch_graph":"https://pith.science/api/pith-number/P5IUTZROED42SYFC256VDTN2N2/graph.json","fetch_events":"https://pith.science/api/pith-number/P5IUTZROED42SYFC256VDTN2N2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/action/storage_attestation","attest_author":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/action/author_attestation","sign_citation":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/action/citation_signature","submit_replication":"https://pith.science/pith/P5IUTZROED42SYFC256VDTN2N2/action/replication_record"}},"created_at":"2026-05-18T04:11:39.281102+00:00","updated_at":"2026-05-18T04:11:39.281102+00:00"}