and $f(K_7)$ are identical as subsets of $S^3$, their projections on $P$ are the same. Furthermore, if $\alpha^n$ preserves the orientation of the 0-star, then $\alpha^n$ preserves the orientation of the 1-star and 2-star and hence of every edge. Otherwise, $\alpha^n$ reverses the orientation of every edge. In either case, a given crossing in the projection has the same sign whether it is considered with orientations induced by $\alpha^n(K_7)$ or with orientations induced by $K_7$. Furthermore, since $\alpha^n$ leaves the 0-star, 1-star, and 2-star of $K_7$ setwise invariant, each crossing has the same epsilon coefficient, whether the crossing is considered in $f\circ \alpha^n(K_7)$ or in $f(K_7)$. It follows that $\widehat{L}_{\varepsilon}(h^n\circ f)=\widehat{L}_{\varepsilon}(\alpha^n\circ f)=\widehat{L}_{\varepsilon}(f)$.

Let $\rho$ denote a reflection of $S^3$ which pointwise fixes the sphere of projection $P$. Using orientations induced by $K_7$, we see that the sign of every crossing in the projection of $\rho\circ f(K_7)$ on $P$ is the reverse of that of the corresponding crossing in the projection of $f(K_7)$. Using the oriented 0-star from $K_7$, it follows that $\widehat{L}_{\varepsilon}(\rho\circ f)=-\widehat{L}_{\varepsilon}(f)$. On the other hand, since $n$ is odd $h^n$ is orientation reversing and is thus isotopic to $\rho$. Hence by Lemma \[k7 invariant\], $\widehat{L}_{\varepsilon}(\rho\circ f)=\widehat{L}_{\varepsilon}(h^n\circ f)$. Consequently, 