is an outer edge

    -   If $d(e,v) = 1$, then\
        $\Delta \widehat{L}_{\varepsilon}(f) = (2 + 0) - 2 = 0$

    -   If $d(e,v) = 2, 4, ..., 2N-2$, then\
        $\Delta \widehat{L}_{\varepsilon}(f) = (1 + 2) - 3 = 0$

    -   If $d(e,v) = 3, 5, ..., 2N-3$, then\
        $\Delta \widehat{L}_{\varepsilon}(f) = (2 + 1) - 3 = 0$

    -   If $d(e,v) = 2N-1$, then\
        $\Delta \widehat{L}_{\varepsilon}(f)= (1 + 1) - 2 = 0$

    -   If $d(e,v) = 2N$, then\
        $\Delta \widehat{L}_{\varepsilon}(f) = (-1 + 1) - 0 = 0$.

-   $e$ is an inner edge

    -   If $d(e,v) = 1$, then\
        $\Delta L(f) = (2 + 0) - 2 = 0$

    -   If $d(e,v) = 2$, then\
        $\Delta L(f) = (3 + 2) - 5 = 0$

    -   If $d(e,v) \geq 3$, then\
        $\Delta L(f) = (3 + 3) - 6 = 0$.

Thus $\widehat{L}_{\varepsilon}(f)$ is invariant under the fifth Reidemeister move, and so it is invariant under ambient isotopy

It follows that $\widehat{L}_{\varepsilon}(f)$ is a generalized Simon invariant for $M_{2N+1}$.

\[mobius odd\] For any $N \geq 2$ and any embedding $f$ of $M_{2N+1}$ in ${S}^3$, the generalized Simon invariant $\widehat{L}_{\varepsilon}(f)$ is an odd number.

Note that any crossing change 