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Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\\cdot {2n+k\\choose 2n}$.\n  Denote $\\mathcal{F}_n^{[m]}$ to be the collection of $n$-subsets of $[m]$ equipped with the symmetric difference metric. We prove tha"},"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":"2404.10566","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-04-16T13:40:22Z","cross_cats_sorted":[],"title_canon_sha256":"cde55c0e6c734eb16f827c27bfaa9b4c50f802b0dbb0302611d7c765d9613bd1","abstract_canon_sha256":"9a140649c037f1b69fbc4e7151a77bfe822e9fe92fda352592e2705ebe0a846a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:08:42.088236Z","signature_b64":"FGARArx3ld5aHe6r78/aH7+hGaooFx6/4laoGwZiHdoztm7Y1IvaoHZM7wIX+uOcXO7mEZ7Uz8j0NJh0iaVTBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2bf34a3c563e2c74e4a2793bc35cbd49f9c78dd0e8dbefda82543d28a0cb7ec","last_reissued_at":"2026-07-05T08:08:42.087763Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:08:42.087763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exploring Homological Properties of Independent Complexes of Kneser Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guanghui Wang, Ziqin Feng","submitted_at":"2024-04-16T13:40:22Z","abstract_excerpt":"We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\\geq 3$ and $k\\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3, k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\\cdot {2n+k\\choose 2n}$.\n  Denote $\\mathcal{F}_n^{[m]}$ to be the collection of $n$-subsets of $[m]$ equipped with the symmetric difference metric. 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