the signs of the crossings between $f(e_{i})$ and $f(e_{j})$. Taniyama [@Taniyama; @1995] showed that the equivalence class $$\begin{aligned}
{\mathcal L}(f)=
\left[
\sum_{e_{i}\cap e_{j}=\emptyset}\ell(f(e_{i}),f(e_{j}))E^{e_{i},e_{j}}
\right]
\in L(G)\end{aligned}$$ coincides with $(f\times f)^{*}(\Sigma)$ through the isomorphism from $H^{2}(C_{2}(G),\sigma)$ to $L(G)$. Thus we may regard ${\mathcal L}(f)$ as the Wu invariant of $f$. Furthermore, $H^{2}(C_{2}(G),\sigma)$ is torsion free, namely $L(G)$ is a free ${\mathbb Z}$-module, and for an orientation-reversing self-homeomorphism $\Phi$ of $S^{3}$, it follows that ${\mathcal L}(\Phi\circ f)=-{\mathcal L}(f)$.

![The Wu invariants for embeddings of these graphs are given in Examples \[2K3\], \[K5\], and \[K33\].[]{data-label="2K3K5K33"}](2K3K5K33bis.eps){width="\textwidth"}

\[2K3\] [ Let $2K_{3}$ denote the graph consisting of two copies of $K_{3}$, labeled and oriented as illustrated in Figure \[2K3K5K33\], and let $f$ be a spatial embedding of $2K_{3}$. It was shown in [@Taniyama; @1995] that the linking module $L(2K_{3})=\langle [E^{e_{1},d_{1}}]\rangle \cong {\mathbb Z}$, and the Wu invariant of $f$ is given by: $$\begin{aligned}
{\mathcal L}(f)
=\sum_{1\le i,j\le 3}\ell(f(e_{i}),f(d_{j}))[E^{e_{1},d_{1}}]
=2{\rm lk}(f)[E^{e_{1},d_{1}}], \end{aligned}$$ where ${\rm lk}(f)$ denotes the [*linking number*]{} of the pair of triangles in $S^{3}$. ]{}

\[K5\] [ Let $K_{5}$ denote the complete graph on five vertices, labeled and oriented as illustrated in Figure \[2K3K5K33\], and let $f$ be a spatial embedding of $K_{5}$. It was 