{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XMYZB64IKSUI5VMFUPYL6R752D","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":"d19c93e8ac050b7106f0750cb3918e86ed3760fd191feaf48812a4a1b8939ade","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-08-07T10:20:22Z","title_canon_sha256":"b111d9d3f1161089acc3e2d58c073dcc17f680acd732383b6038402b422d5e88"},"schema_version":"1.0","source":{"id":"1208.1389","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.1389","created_at":"2026-05-18T03:25:35Z"},{"alias_kind":"arxiv_version","alias_value":"1208.1389v2","created_at":"2026-05-18T03:25:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1389","created_at":"2026-05-18T03:25:35Z"},{"alias_kind":"pith_short_12","alias_value":"XMYZB64IKSUI","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XMYZB64IKSUI5VMF","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XMYZB64I","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:b20537ecaafc974b1d3d46ea53450a4736b6830a0ac9eb60c89746e61cf767c7","target":"graph","created_at":"2026-05-18T03:25:35Z","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 introduce the class $\\Sigma_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \\leq k \\leq d + 1$) and compare and contrast it with the class ${\\cal S}_k(d)$ ($0 \\leq k \\leq d$) of $k$-stacked homology $d$-spheres. We have $\\Sigma_1(d) = {\\cal S}_1(d)$, and $\\Sigma_k(d) \\subseteq {\\cal S}_k(d)$ for $d \\geq 2k - 1$. However, for each $k \\geq 2$ there are $k$-stacked spheres which are not $k$-stellated. The existence of $k$-stellated spheres which are not $k$-stacked remains an open question.\n  We also consider the class ${\\cal W}_k(d)$ (and ${\\cal K}_k(d)$) of simplicial com","authors_text":"Basudeb Datta, Bhaskar Bagchi","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-08-07T10:20:22Z","title":"On $k$-stellated and $k$-stacked spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1389","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:f29619abe41a5fdb19f5304649825e853d20ed16909867d7bb91857daf15e98a","target":"record","created_at":"2026-05-18T03:25:35Z","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":"d19c93e8ac050b7106f0750cb3918e86ed3760fd191feaf48812a4a1b8939ade","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-08-07T10:20:22Z","title_canon_sha256":"b111d9d3f1161089acc3e2d58c073dcc17f680acd732383b6038402b422d5e88"},"schema_version":"1.0","source":{"id":"1208.1389","kind":"arxiv","version":2}},"canonical_sha256":"bb3190fb8854a88ed585a3f0bf47fdd0fe8fa42a36f9cff4bf8531946745800f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb3190fb8854a88ed585a3f0bf47fdd0fe8fa42a36f9cff4bf8531946745800f","first_computed_at":"2026-05-18T03:25:35.055243Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:35.055243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RHP3yqoRPdbOCIMGsPnvhU8Y3MnoA3k3QCDLYkObYi4xo+Kx4/FWOQN13ashZwRit7ehjnA5Ki+XFHIWMS9OCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:35.055749Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.1389","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f29619abe41a5fdb19f5304649825e853d20ed16909867d7bb91857daf15e98a","sha256:b20537ecaafc974b1d3d46ea53450a4736b6830a0ac9eb60c89746e61cf767c7"],"state_sha256":"2c117a23e21b25dfee86744ae499e79ae2baf4233185746ad681a726b16a639b"}