$v_{1},v_{2},\ldots,v_{m}$ and oriented edges labeled $e_{1},e_{2},\ldots,e_{n}$. For each pair of disjoint edges $e_{i}$ and $e_{j}$, we define a variable $E^{e_{i},e_{j}}=E^{e_{j},e_{i}}$; and for each edge $e_{i}$ and vertex $v_{s}$ which is disjoint from $e_i$, we define a variable $V^{e_{i},v_{s}}$. Let $Z(G)$ be the free ${\mathbb Z}$-module generated by the collection of $E^{e_{i},e_{j}}$’s. For each $V^{e_{i},v_{s}}$, let $\delta(V^{e_{i},v_{s}})$ be the element of $Z(G)$ given by the sum of all $E^{e_{i},e_{k}}$ such that $e_k$ is disjoint from $e_i$ and has initial vertex $v_s$, minus the sum of all $E^{e_{i},e_{k}}$ such that $e_k$ is disjoint from $e_i$ and has terminal vertex $v_s$. Thus

$$\begin{aligned}
\delta(V^{e_{i},v_{s}})
= 
\sum_{\substack{
{I(k)=s} 
\\ e_{i}\cap e_{k}=\emptyset
}
}E^{e_{i},e_{k}}
-
\sum_{\substack{
{T(l)=s} 
\\ e_{i}\cap e_{l}=\emptyset
} 
}E^{e_{i},e_{l}}, \end{aligned}$$

where $I(k)=s$ indicates that the initial vertex of $e_{k}$ is $v_{s}$, and $T(l)=s$ indicates that the terminal vertex of $e_{l}$ is $v_{s}$. Let $B(G)$ be the submodule of $Z(G)$ generated by the collection of $\delta(V^{e_{i},v_{s}})$’s. We let $L(G)$ denote the quotient module $Z(G) / B(G)$, and call it a [*linking module*]{} of $G$. Then $L(G)\cong H^{2}(C_{2}(G),\sigma)$.

Now let $f$ be an embedding of the labeled oriented graph $G$ in $S^3$. Fix a projection of $f(G)$ and let $\ell(f(e_{i}),f(e_{j}))=\ell(f(e_{j}),f(e_{i}))$ denote the sum of 