{"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"}