-1 & \text{if } c_i \text{ and } b_j \text { are anti-parallel in Figure~\ref{2K3K5K33}}
        \end{cases}$$

It was shown in [@Taniyama; @1995] that $L(G)=0$ if and only if $G$ is a planar graph which does not contain a pair of two disjoint cycles.

[ It was shown in [@N00] that if the graph $G$ is [*$3$-connected*]{}, then $$\begin{aligned}
{\rm rank}L(G)=\frac{1}{2}
\left\{
\beta_{1}(G)^{2}+\beta_{1}(G)+4|E(G)|-\sum_{v\in V(G)}\left({\rm deg}(v)\right)^{2}
\right\}, \end{aligned}$$ where $\beta_{1}(G)$ denotes the first Betti number of $G$ and ${\rm deg}(v)$ denotes the valency of a vertex $v$. For example, ${\rm rank}(L(K_{6}))=10$ and ${\rm rank}(L(K_{7}))=36$. ]{}

Let $f$ be a spatial embedding of an oriented graph $G$ with linking module $L(G)$ and Wu invariant ${\mathcal L}(f)\in L(G)$. Let $\varepsilon:L(G)\to {\mathbb Z}$ be a homomorphism. Then we call the integer $\varepsilon({\mathcal L}(f))$ the [**reduced Wu invariant of $f$ with respect to $\varepsilon$**]{} and denote it by $\tilde{\mathcal L}_{\varepsilon}(f)$.

For a pair of disjoint edges $e_i$ and $e_j$, we denote $\varepsilon([E^{e_{i},e_{j}}])$ by $\varepsilon(e_i,e_j)$. Thus

$$\begin{aligned}
\tilde{\mathcal L}_{\varepsilon}(f)=\varepsilon\left(\left[
\sum_{e_{i}\cap e_{j}=\emptyset}\ell(f(e_{i}),f(e_{j}))E^{e_{i},e_{j}}
\right]\right)=\sum_{e_{i}\cap e_{j}=\emptyset}\ell(f(e_{i}),f(e_{j}))\varepsilon(e_i,e_j).\end{aligned}$$

\[2K3\_2\] [ Consider $2K_{3}$, labeled and oriented as in Figure \[2K3K5K33\], and let $f$ be an embedding of $2K_{3}$ in $S^3$. Let $\varepsilon$ be the isomorphism from $L(2K_{3})$ to ${\mathbb Z}$ defined by $\varepsilon(e_{1},d_{1})=1$. 