{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4DOENLAESU4RLQXQUFRNWTMLBB","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":"2205f9a27aba85e85a249836719389b258d8906e6ebe6242941b92607abe9b9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-03-07T02:11:25Z","title_canon_sha256":"8b76f32a9c402e6fb133746b11b37ce66fd970cba632d2e41ba1ca359ea4cf51"},"schema_version":"1.0","source":{"id":"1303.1591","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.1591","created_at":"2026-05-18T00:03:59Z"},{"alias_kind":"arxiv_version","alias_value":"1303.1591v2","created_at":"2026-05-18T00:03:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.1591","created_at":"2026-05-18T00:03:59Z"},{"alias_kind":"pith_short_12","alias_value":"4DOENLAESU4R","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"4DOENLAESU4RLQXQ","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"4DOENLAE","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:0db67eae3d6cbf1ff58a03bfd94bbd42b5503159b5e3f7fd8df9b5f431c261fe","target":"graph","created_at":"2026-05-18T00:03:59Z","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":"This paper extends Alexander duality to the setting of parametrized homology. Let X with be a compact subset of R^n x R (n \\geq 2) satisfying certain conditions, let Y be its complement, and let p be the projection onto the second factor. Both X and Y are parametrized spaces with respect to the projection. The parametrized homology is a variant of zigzag persistent homology that measures how the homology of the level sets of the space changes as we vary the parameter. We show that if (X, p|_X) has a well-defined parametrized homology, then the pair (Y, p|_Y) has a well-defined reduced parametr","authors_text":"Sara Kalisnik Verovsek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-03-07T02:11:25Z","title":"Alexander Duality for Parametrized Homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.1591","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":"sha256:0e62730c77f5d71d70dd50d21f1febc151ff02f42659fc354f064153e162d473","target":"record","created_at":"2026-05-18T00:03:59Z","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":"2205f9a27aba85e85a249836719389b258d8906e6ebe6242941b92607abe9b9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-03-07T02:11:25Z","title_canon_sha256":"8b76f32a9c402e6fb133746b11b37ce66fd970cba632d2e41ba1ca359ea4cf51"},"schema_version":"1.0","source":{"id":"1303.1591","kind":"arxiv","version":2}},"canonical_sha256":"e0dc46ac04953915c2f0a162db4d8b085ab89a6e80b83ce9ecd839fa09759110","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0dc46ac04953915c2f0a162db4d8b085ab89a6e80b83ce9ecd839fa09759110","first_computed_at":"2026-05-18T00:03:59.792497Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:59.792497Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6QRZarLG/jtCHYBPxpcpyGBRZ/UW2IYpJvQyl3DCj2b0VZv4gM9FI+KLCUvAQw6zIu7akl0hkZg+cDaSRmfaBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:59.793109Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.1591","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e62730c77f5d71d70dd50d21f1febc151ff02f42659fc354f064153e162d473","sha256:0db67eae3d6cbf1ff58a03bfd94bbd42b5503159b5e3f7fd8df9b5f431c261fe"],"state_sha256":"b104a571876f93a1270d1921a8d3198a64d99a99ad7d63f52d10b7f1804ad558"}