\geq 3.
    \end{cases}$$

For any embedding $f: M_{2N+1} \rightarrow {S}^3$ with a regular projection, define:

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

[ This definition of $\widehat{L}_{\varepsilon}(f)$ does not reduce to the original Simon invariant for $N=1$.]{}

\[mobius invariant\] For $N \geq 2$ and any embedding $f$ of $M_{2N+1}$ in ${S}^3$, $\widehat{L}_{\varepsilon}(f) $ is independent of labeling and orientation, and invariant under ambient isotopy of $f(M_{2N+1})$.

We first show that $\widehat{L}_{\varepsilon}(f) $ is independent of labeling and orientation. Since $N \geq 2$, it follows from Simon [@S86] that any automorphism of $M_{2N+1}$ with $N \geq 2$ takes the cycle of outer edges $\overline{x_1 x_2 ... x_{4N+2}}$ to itself, preserving the order of the edges $x_1, x_2, ..., x_{4N+2}$ and thus the edges $y_1, y_2, ..., y_{2N+1}$ as well. Thus any automorphism either preserves all the arrows in the orientation of $M_{2N+1}$, or reverses all the arrows. Reversing every arrow would have no effect on the signs of the crossings, so $\widehat{L}_{\varepsilon}(f) $ is independent of labeling and orientation.

As before, it is easy to see that $\widehat{L}_{\varepsilon}(f) $ is invariant under the first four Reidemeister moves. We show that $\widehat{L}_{\varepsilon}(f) $ is unchanged under the fifth Reidemeister move. Without loss of 