shown in [@Taniyama; @1995] that the linking module $L(K_{5})=\langle [E^{e_{1},e_{3}}]\rangle \cong {\mathbb Z}$ and the Wu invariant is given by: $$\begin{aligned}
{\mathcal L}(f)
=\sum_{a\cap b=\emptyset}\varepsilon(a,b)\ell(f(a),f(b))[E^{e_{1},e_{3}}], \end{aligned}$$ where $\varepsilon(a,b)$ is defined by $\varepsilon(e_{i},e_{j})=1$, $\varepsilon(d_{i},d_{j})=-1$ and $\varepsilon(e_{i},d_{j})=-1$. ]{}

We work out the following example which it is given in [@Taniyama; @1995] without details.

\[K33\]

Let $K_{3,3}$ denote the complete bipartite graph, labeled and oriented as illustrated in Figure \[2K3K5K33\], and let $f$ be a spatial embedding of $K_{3,3}$. Then $Z(K_{3,3})$ is a free ${\mathbb Z}$-module generated by $$\begin{aligned}
&&E^{c_{1},c_{3}},E^{c_{2},c_{4}},E^{c_{3},c_{5}},E^{c_{4},c_{6}},E^{c_{5},c_{1}},E^{c_{6},c_{2}},E^{c_{1},c_{4}},E^{c_{2},c_{5}},E^{c_{3},c_{6}},\\
&&E^{b_{1},c_{2}},E^{b_{1},c_{5}},E^{b_{3},c_{4}},E^{b_{3},c_{1}},E^{b_{2},c_{3}},E^{b_{2},c_{6}},E^{b_{1},b_{2}},E^{b_{2},b_{3}},E^{b_{3},b_{1}}\end{aligned}$$ and $B(K_{3,3})$ is a submodule of $Z(K_{3,3})$ generated by $$\begin{aligned}
&&E^{b_{1},c_{2}}-E^{c_{6},c_{2}},\ E^{c_{1},c_{3}}-E^{c_{3},c_{6}},\ E^{c_{1},c_{4}}-E^{c_{4},c_{6}},\\
&&E^{c_{5},c_{1}}+E^{b_{1},c_{5}},\ E^{b_{3},c_{1}}+E^{b_{3},b_{1}},\ E^{b_{1},b_{2}}-E^{b_{2},c_{6}},\\
&&-E^{b_{2},c_{3}}-E^{c_{1},c_{3}},\ E^{c_{2},c_{4}}-E^{c_{1},c_{4}},\ E^{c_{2},c_{5}}-E^{c_{5},c_{1}},\\
&&E^{c_{6},c_{2}}-E^{b_{2},c_{6}},\ -E^{b_{2},b_{3}}-E^{b_{3},c_{1}},\ E^{b_{1},c_{2}}-E^{b_{1},b_{2}},\\
&&E^{b_{3},c_{4}}-E^{c_{2},c_{4}},\ E^{c_{3},c_{5}}-E^{c_{2},c_{5}},\ E^{c_{3},c_{6}}-E^{c_{6},c_{2}},\\
&&E^{c_{1},c_{3}}+E^{b_{3},c_{1}},\ E^{b_{2},c_{3}}+E^{b_{2},b_{3}},\ E^{b_{3},b_{1}}-E^{b_{1},c_{2}},\\
&&-E^{b_{1},c_{5}}-E^{c_{3},c_{5}},\ E^{c_{4},c_{6}}-E^{c_{3},c_{6}},\ E^{c_{1},c_{4}}-E^{c_{1},c_{3}},\\
&&E^{c_{2},c_{4}}-E^{b_{1},c_{2}},\ -E^{b_{1},b_{2}}-E^{b_{2},c_{3}},\ E^{b_{3},c_{4}}-E^{b_{3},b_{1}},\\
&&E^{b_{2},c_{6}}-E^{c_{4},c_{6}},\ E^{c_{5},c_{1}}-E^{c_{1},c_{4}},\ E^{c_{2},c_{5}}-E^{c_{2},c_{4}},\\
&&E^{c_{3},c_{5}}+E^{b_{2},c_{3}},\ E^{b_{1},c_{5}}+E^{b_{1},b_{2}},\ E^{b_{2},b_{3}}-E^{b_{3},c_{4}},\\
&&-E^{b_{3},c_{1}}-E^{c_{5},c_{1}},\ E^{c_{6},c_{2}}-E^{c_{2},c_{5}},\ E^{c_{3},c_{6}}-E^{c_{3},c_{5}},\\
&&E^{c_{4},c_{6}}-E^{d_{3},c_{4}},\ -E^{b_{3},b_{1}}-E^{b_{1},c_{5}},\ E^{b_{2},c_{6}}-E^{b_{2},b_{3}}. \end{aligned}$$ Then we have $$\begin{aligned}
&& [E^{c_{1},c_{3}}]=[E^{c_{2},c_{4}}]=[E^{c_{3},c_{5}}]=[E^{c_{4},c_{6}}]=[E^{c_{5},c_{1}}]=[E^{c_{6},c_{2}}]\\
&=& [E^{c_{1},c_{4}}]=[E^{c_{2},c_{5}}]=[E^{c_{3},c_{6}}]=[E^{b_{1},b_{2}}]=[E^{b_{2},b_{3}}]=[E^{b_{3},b_{1}}]\\
&=& [E^{b_{1},c_{2}}]=[E^{b_{3},c_{4}}]=[E^{b_{2},c_{6}}]\\
&=& -[E^{b_{1},c_{5}}] = -[E^{b_{3},c_{1}}] = -[E^{b_{2},c_{3}}], \end{aligned}$$ Then the linking module $L(K_{3,3})=\langle [E^{c_{1},c_{3}}]\rangle \cong {\mathbb Z}$ and the Wu invariant is given by: $$\begin{aligned}
{\mathcal L}(f)
=\sum_{a\cap b=\emptyset}\varepsilon(a,b)\ell(f(a),f(b))[E^{c_{1},c_{3}}], \end{aligned}$$ where $\varepsilon(a,b))$ is defined by $\varepsilon(c_{i},c_{j})=1$, $\varepsilon(b_{i},b_{j})=1$, and

$$\varepsilon(c_i,b_j) = 
    \begin{cases}
        1 &\text{if } c_i \text{ and } b_j \text { are parallel in Figure~\ref{2K3K5K33}}
\\
        