\begin{cases}       -1 & \text{if } x_i \text{ and } z_j \text { are anti-parallel in Figure~\ref{K6Simon}}
        
\\
1 &\text{if } x_i \text{ and } z_j \text { are parallel in Figure~\ref{K6Simon}}
        \end{cases}$$

In addition, we define $\varepsilon(z_{i},z_{j})=1$, $\varepsilon(x_{i},y_{j})=-1$, and $\varepsilon(y_{i},z_{j})=0$. Then it can be checked that $\varepsilon$ gives a homomorphism from $L(K_{6})$ to ${\mathbb Z}$. It follows that $\tilde{\mathcal L}_{\varepsilon}(f)=\sum_{a\cap b=\emptyset}\varepsilon(a,b)\ell(f(a),f(b))$ is a reduced Wu invariant for $K_6$.

Generalized Simon Invariants {#GSI}
============================

Simon introduced the following function of embeddings $f$ of the graphs $K_5$ and $K_{3,3}$, labeled and oriented as in Figure \[2K3K5K33\]. Let

$$\widehat{L}_{\varepsilon}(f) = \sum_{a\cap b=\emptyset} \varepsilon(a,b) \ell(f(a),f(b))$$

where $\varepsilon(a,b)$ is defined as $\varepsilon(e_{i},e_{j})=1$, and $\varepsilon(d_{i},d_{j})=\varepsilon(e_{i},d_{j})=-1$ for $K_5$; and $\varepsilon(a,b)$ is defined as $\varepsilon(c_{i},c_{j})=1$, $\varepsilon(b_{i},b_{j})=1$ $$\varepsilon(c_i,b_j) = 
    \begin{cases}
        1 &\text{if } c_i \text{ and } b_j \text { are parallel in Figure~\ref{2K3K5K33}}
\\
        -1 & \text{if } c_i \text{ and } b_j \text { are anti-parallel in Figure~\ref{2K3K5K33}}
        \end{cases}$$ for $K_{3,3}$.

Simon then proved that for any projection of an embedding $f$ of the oriented labeled graphs $K_5$ and $K_{3,3}$, the value of $$\sum_{a,b \in G} \varepsilon(a,b) \ell(f(a),f(b))$$

is invariant under the five Reidemeister moves for spatial graphs given in Figure \[Figuremy5radmoves\]. This invariant is known as 