the same. Also, since $\alpha$ either preserves all the edge orientations or reverses all the edge orientations, the sign of every crossing in the projection of the oriented embedded graph $h\circ f(G)$ is the same as it is in the projection of the oriented embedded graph $f(G)$. Furthermore, by hypothesis each crossing has the same epsilon coefficient, whether the crossing is considered in $f\circ \alpha(G)$ or in $f(G)$. It follows that $\widehat{L}_{\varepsilon}(h^\circ f)=\widehat{L}_{\varepsilon}(\alpha\circ f)=\widehat{L}_{\varepsilon}(f)$.

Let $\rho$ denote a reflection of $S^3$ which pointwise fixes the sphere of projection $P$. Using orientations induced by $G$, the sign of every crossing in the projection of $\rho\circ f(G)$ is the reverse of that of the corresponding crossing in $f(G)$. It follows that $\widehat{L}_{\varepsilon}(\rho\circ f)=-\widehat{L}_{\varepsilon}(f)$. On the other hand, since $h$ is orientation reversing it is isotopic to $\rho$. Hence by by definition of a generalized Simon invariant, $\widehat{L}_{\varepsilon}(\rho\circ f)=\widehat{L}_{\varepsilon}(h\circ f)$. Consequently, $\widehat{L}_{\varepsilon}(h\circ f)=-\widehat{L}_{\varepsilon}(f)$. Thus $\widehat{L}_{\varepsilon}(f)=0$, which contradicts our hypothesis that $\widehat{L}_{\varepsilon}(f)$ is odd. Hence $G$ is intrinsically chiral.

Flapan [@Flapan; @1989] showed that $M_{2N+1}$ is intrinsically chiral. However, now that result follows as an immediate corollary of Lemmas \[mobius odd\], \[mobius automorphism\], and Proposition \[intrinsic chirality\].

\[mobius chiral\] $M_{2N+1}$ is intrinsically chiral for 