On the Space of 2-Linkages

Abstract

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 $\mathbb{Z}⟨R⟩\otimes \mathbb{Z}⟨S⟩$ consisting all ${\sum }_{r\in R,s\in S}c(r,s)r\otimes s$ such that $c(r,r)=0$ for all $r\in R\cap S$, ${\sum }_{r\in R}c(r,s)=0$ for all $s\in S$, and ${\sum }_{s\in S}c(r,s)=0$ for all $r\in R$. In this paper, we provide, when $G$ is sufficiently connected, characterizations when $P(G;R,S)$ is a proper subset of $L(R,S)$.