{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:7SQL7ECX7EXYMC4AUWISYWFM5T","short_pith_number":"pith:7SQL7ECX","schema_version":"1.0","canonical_sha256":"fca0bf9057f92f860b80a5912c58acece2a6733b8d938a80cb9d3bfbe511b678","source":{"kind":"arxiv","id":"1610.05343","version":1},"attestation_state":"computed","paper":{"title":"Secondary Upsilon invariants of knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Charles Livingston, Se-Goo Kim","submitted_at":"2016-10-17T20:30:46Z","abstract_excerpt":"The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented."},"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":"1610.05343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-17T20:30:46Z","cross_cats_sorted":[],"title_canon_sha256":"88ac5af91f92e5d78fbca9a8b33207c0267ddfa483be01e76b4bb99ebeba8d39","abstract_canon_sha256":"f7e8802957772630c5d2bff77ee84cd79b77da4d1c2855a10eb93cd56d0b2864"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:03.390928Z","signature_b64":"xKmGDAvpkNFvk8ET4hsCSBt87JuZZH3X5hmKNEAdKSiTdnqSOA19hMtQdDtXs9GIE8Yz3isBGXMrCElsSo45BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fca0bf9057f92f860b80a5912c58acece2a6733b8d938a80cb9d3bfbe511b678","last_reissued_at":"2026-05-17T23:54:03.390429Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:03.390429Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Secondary Upsilon invariants of knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Charles Livingston, Se-Goo Kim","submitted_at":"2016-10-17T20:30:46Z","abstract_excerpt":"The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05343","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":"1610.05343","created_at":"2026-05-17T23:54:03.390522+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.05343v1","created_at":"2026-05-17T23:54:03.390522+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.05343","created_at":"2026-05-17T23:54:03.390522+00:00"},{"alias_kind":"pith_short_12","alias_value":"7SQL7ECX7EXY","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"7SQL7ECX7EXYMC4A","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"7SQL7ECX","created_at":"2026-05-18T12:30:04.600751+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/7SQL7ECX7EXYMC4AUWISYWFM5T","json":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T.json","graph_json":"https://pith.science/api/pith-number/7SQL7ECX7EXYMC4AUWISYWFM5T/graph.json","events_json":"https://pith.science/api/pith-number/7SQL7ECX7EXYMC4AUWISYWFM5T/events.json","paper":"https://pith.science/paper/7SQL7ECX"},"agent_actions":{"view_html":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T","download_json":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T.json","view_paper":"https://pith.science/paper/7SQL7ECX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.05343&json=true","fetch_graph":"https://pith.science/api/pith-number/7SQL7ECX7EXYMC4AUWISYWFM5T/graph.json","fetch_events":"https://pith.science/api/pith-number/7SQL7ECX7EXYMC4AUWISYWFM5T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T/action/storage_attestation","attest_author":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T/action/author_attestation","sign_citation":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T/action/citation_signature","submit_replication":"https://pith.science/pith/7SQL7ECX7EXYMC4AUWISYWFM5T/action/replication_record"}},"created_at":"2026-05-17T23:54:03.390522+00:00","updated_at":"2026-05-17T23:54:03.390522+00:00"}