{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:CDIIBPMRVT7OWGVW7LKV6ZX53B","short_pith_number":"pith:CDIIBPMR","schema_version":"1.0","canonical_sha256":"10d080bd91acfeeb1ab6fad55f66fdd87ce2df256c53608cb1669ec88c242cad","source":{"kind":"arxiv","id":"1712.03869","version":3},"attestation_state":"computed","paper":{"title":"On the Space of 2-Linkages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guantao Chen, Hein van der Holst, Robin Thomas, Serguei Norine","submitted_at":"2017-12-11T16:20:59Z","abstract_excerpt":"Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\\in V$ to $r_2\\in V$, we define $\\partial(P) = r_2-r_1$. If $R, S\\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\\partial P\\otimes \\partial Q$ with $P$ and $Q$ disjoint oriented paths of $G$ connecting vertices in $R$ and $S$, respectively. By $L(R, S)$, we denote the submodule of $\\mathbb{Z}\\langle R\\rangle\\otimes\\mathbb{Z}\\langle S\\rangle$ consisting all $\\sum_{r\\in R, s\\in S} c(r,s)r\\otimes s$ such that $c(r,r) = 0$ for all $r\\in R\\cap S$, $\\sum_{r\\in R} c(r, s) = 0$ for all $s\\in S$, and $\\s"},"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":"1712.03869","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-11T16:20:59Z","cross_cats_sorted":[],"title_canon_sha256":"16966c13e8bf8be966a971acb1857ee47c403271373c3244b0430abc21e7e6f8","abstract_canon_sha256":"399a3a9f2046c036543010cbd503d4fe979cc46cee4583523ef267259d3256a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:17.552409Z","signature_b64":"qIQ0D+tCELQU+PvTfP4GLP5IwpGZBQ9UldTbZQFxhQlE1I0dnnBKLfuX0dsXPnohK/kWAt6R59uPxFOpO0+4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10d080bd91acfeeb1ab6fad55f66fdd87ce2df256c53608cb1669ec88c242cad","last_reissued_at":"2026-05-17T23:48:17.551798Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:17.551798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Space of 2-Linkages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guantao Chen, Hein van der Holst, Robin Thomas, Serguei Norine","submitted_at":"2017-12-11T16:20:59Z","abstract_excerpt":"Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\\in V$ to $r_2\\in V$, we define $\\partial(P) = r_2-r_1$. If $R, S\\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\\partial P\\otimes \\partial Q$ with $P$ and $Q$ disjoint oriented paths of $G$ connecting vertices in $R$ and $S$, respectively. By $L(R, S)$, we denote the submodule of $\\mathbb{Z}\\langle R\\rangle\\otimes\\mathbb{Z}\\langle S\\rangle$ consisting all $\\sum_{r\\in R, s\\in S} c(r,s)r\\otimes s$ such that $c(r,r) = 0$ for all $r\\in R\\cap S$, $\\sum_{r\\in R} c(r, s) = 0$ for all $s\\in S$, and $\\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03869","kind":"arxiv","version":3},"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":"1712.03869","created_at":"2026-05-17T23:48:17.551914+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.03869v3","created_at":"2026-05-17T23:48:17.551914+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03869","created_at":"2026-05-17T23:48:17.551914+00:00"},{"alias_kind":"pith_short_12","alias_value":"CDIIBPMRVT7O","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CDIIBPMRVT7OWGVW","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CDIIBPMR","created_at":"2026-05-18T12:31:10.602751+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/CDIIBPMRVT7OWGVW7LKV6ZX53B","json":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B.json","graph_json":"https://pith.science/api/pith-number/CDIIBPMRVT7OWGVW7LKV6ZX53B/graph.json","events_json":"https://pith.science/api/pith-number/CDIIBPMRVT7OWGVW7LKV6ZX53B/events.json","paper":"https://pith.science/paper/CDIIBPMR"},"agent_actions":{"view_html":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B","download_json":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B.json","view_paper":"https://pith.science/paper/CDIIBPMR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.03869&json=true","fetch_graph":"https://pith.science/api/pith-number/CDIIBPMRVT7OWGVW7LKV6ZX53B/graph.json","fetch_events":"https://pith.science/api/pith-number/CDIIBPMRVT7OWGVW7LKV6ZX53B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B/action/storage_attestation","attest_author":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B/action/author_attestation","sign_citation":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B/action/citation_signature","submit_replication":"https://pith.science/pith/CDIIBPMRVT7OWGVW7LKV6ZX53B/action/replication_record"}},"created_at":"2026-05-17T23:48:17.551914+00:00","updated_at":"2026-05-17T23:48:17.551914+00:00"}