On the Space of 2-Linkages

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 $\mathrm{\partial }(P)=r_2-r_1$. If $R, S\subseteq V$,wedenoteby$P(G; R, S)$thespanofthesetofall$\partial P\otimes \partial Q$with$P$and$Q$disjointorientedpathsof$G$ connecting vertices in $R$ and $S$, respectively. By $L(R,S)$, we denote the submodule of $\mathbb{Z}⟨R⟩\otimes \mathbb{Z}⟨S⟩$ consisting all ${\sum }_{r\in R,s\in S}c(r,s)r\otimes s$ such that $c(r,r)=0$ for all $r\in R\cap S$,$\sum_{r\in R} c(r, s) = 0$forall$s\in S$,and$\sum_{s\in S} c(r, s) = 0$forall$r\in R$. In this paper, we provide, when $G$ is sufficiently connected, characterizations when $P(G;R,S)$ is a proper subset of $L(R,S)$.