{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:GYJ4HMIN7XWBR74BI6OYD57K5V","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":"d62f11189ef6b9c0e80306ca3eda1ab9fde69691c59727dd6ce185c160715eb5","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-31T18:24:13Z","title_canon_sha256":"7882b8a6b7a22970dcebbbeb4e518007371f52b9298eeb9da16dd38d9414bd97"},"schema_version":"1.0","source":{"id":"1708.00032","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.00032","created_at":"2026-05-18T00:39:06Z"},{"alias_kind":"arxiv_version","alias_value":"1708.00032v1","created_at":"2026-05-18T00:39:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.00032","created_at":"2026-05-18T00:39:06Z"},{"alias_kind":"pith_short_12","alias_value":"GYJ4HMIN7XWB","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GYJ4HMIN7XWBR74B","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GYJ4HMIN","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:a341a616556ded187a85b30be60ff794fa3266dbd2812e718cced47021efd7bc","target":"graph","created_at":"2026-05-18T00:39:06Z","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 consider the topology of real no $k$-equal spaces via the theory of cellular spanning trees. Our main theorem proves that the rank of the $(k-2)$-dimensional homology of the no $k$-equal subspace of $\\mathbb{R}$ is equal to the number of facets in a $k$-dimensional spanning tree of the $k$-skeleton of the $n$-dimensional hypercube.","authors_text":"Caroline Klivans, Nicholas Kosar, Yuliy Baryshnikov","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-31T18:24:13Z","title":"On the topology of no $k$-equal spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00032","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:fe002b1a3384ff5628477842d1c93a8c728d422ed7fab725c8d24cb65565abbd","target":"record","created_at":"2026-05-18T00:39:06Z","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":"d62f11189ef6b9c0e80306ca3eda1ab9fde69691c59727dd6ce185c160715eb5","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-31T18:24:13Z","title_canon_sha256":"7882b8a6b7a22970dcebbbeb4e518007371f52b9298eeb9da16dd38d9414bd97"},"schema_version":"1.0","source":{"id":"1708.00032","kind":"arxiv","version":1}},"canonical_sha256":"3613c3b10dfdec18ff81479d81f7eaed66b050f3f32e43b845385db6ab9329cc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3613c3b10dfdec18ff81479d81f7eaed66b050f3f32e43b845385db6ab9329cc","first_computed_at":"2026-05-18T00:39:06.209846Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:06.209846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jtswJ41HCMWHnMeOyAsaAJ46EueAFDxBCVxsKY6laYRZujOWNRICG4ybhj2unrh/pNp7clavmS1pDc4VSFz5CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:06.210494Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.00032","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe002b1a3384ff5628477842d1c93a8c728d422ed7fab725c8d24cb65565abbd","sha256:a341a616556ded187a85b30be60ff794fa3266dbd2812e718cced47021efd7bc"],"state_sha256":"31b411fdeb9570e4c2b29a89b5ba32a5dcd1b94b1f5992ecf44b650c1400cabf"}