give a lower bound on the minimal crossing number of embedded graphs.

Wu Invariants and Reduced Wu Invariants
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In 1960, Wu [@Wu; @1960] introduced an invariant as follows. Let $C_{2}(X)$ be the [*configuration space*]{} of ordered pairs of points from a topological space $X$, namely $$\begin{aligned}
C_{2}(X) = \left\{(x,y)\in X\times X~|~x\neq y\right\}. \end{aligned}$$ Let $\sigma$ be the involution of $C_{2}(X)$ given by $\sigma(x,y)=(y,x)$. The integral cohomology group of ${\rm
Ker}\left(1+\sigma_{\sharp}\right)$ denoted by $H^{*}\left(C_{2}(X),\sigma\right)$ is said to be the [*skew-symmetric integral cohomology group*]{} of the pair $\left(C_{2}(X),\sigma\right)$, where $\sigma_{\sharp}$ denotes the chain map induced by $\sigma$. Wu [@Wu; @1960] proved that $H^{2}(C_{2}({\mathbb R}^{3}),\sigma)\cong {\mathbb Z}$, and hence is generated by some element $\Sigma$. Let $f:G\to {\mathbb R}^{3}$ be a spatial embedding of a graph $G$ with labeled vertices and orientations on the edges. Then $f$ naturally induces an equivariant embedding $f\times f:C_{2}(G)\to C_{2}({\mathbb R}^{3})$ with respect to the action $\sigma$, and therefore induces a homomorphism $$\begin{aligned}
(f\times f)^{*}:H^{2}(C_{2}({\mathbb R}^{3}),\sigma)
\longrightarrow H^{2}(C_{2}(G),\sigma). \end{aligned}$$ The element $(f\times f)^{*}(\Sigma)$ is an ambient isotopy invariant known as the [*Wu invariant*]{}.

In order to explicitly calculate the Wu invariant, Taniyama [@Taniyama; @1995] developed the following combinatorial approach. Let $G$ be a graph with vertices labeled 