to $\mathbb{Z}$. Thus $\widehat{L}_{\varepsilon}(f)$ also gives us a reduced Wu invariant for $K_7$.]{}

We now apply the generalized Simon invariant of $K_7$ to prove that $K_7$ is intrinsically chiral. This result was previously proven by Flapan and Weaver [@Flapan; @1992], but using the generalized Simon invariant allows us to give a simpler proof which can be generalized to apply to many other graphs. We begin with a lemma.

\[k7 odd\] For any embedding $f$ of $K_7$ in ${S}^3$, the generalized Simon invariant $\widehat{L}_{\varepsilon}(f)$ is an odd number.

Since any crossing change will change the signed crossing number between two edges by $\pm 2$, we only need to find an embedding $f$ where $\widehat{L}_{\varepsilon}(f)$ is odd. Consider an embedding of $K_7$ which has Figure \[FigureK7\_labeled\] as its projection with the intersections between edges replaced by crossings. Note that there are 35 crossings in this embedding of $K_7$: 14 crossings of the 2-star with itself, and 21 crossings between the 1-star and the 2-star. The epsilon coefficient for every one of these crossings is 1. Since there is an odd number of crossings, regardless of their signs, $\widehat{L}_{\varepsilon}(f)$ must be odd. Because any crossing change will change $\widehat{L}_{\varepsilon}(f)$ by an even number, 