the circle). We denote the “outer edges” consecutively as $x_1, x_2, ..., x_{2(2N+1)}$, and the “inner edges” consecutively as $y_1, y_2, ..., y_{2N+1}$. Since $N \geq 2$, it follows from Simon [@S86] that there is no automorphism of $M_{2N+1}$ which takes an outer edge to an inner edge. Thus, the distinction between inner and outer edges does not depend on any particular labeling.

![An oriented $M_{2N+1}$.[]{data-label="FigureM_2N+1"}](M14_oriented3.eps){width="35.00000%"}

For any pair of edges $a$ and $b$, let the minimal outer edge distance $d(a,b)$ be defined as the minimum number of edges of any path between $a$ and $b$ using only outer edges (not counting $a$ and $b$). For $M_{2N+1}$, note that $d(x_i, x_j) \leq 2N$ for any $i, j$. We define the epsilon coefficient $\varepsilon(a,b)$ of a pair of disjoint edges $a$ and $b$ by:

$$\varepsilon(x_i,x_j) = 
    \begin{cases}
        2 & \text{if } d(x_i,x_j) \text{ is odd and } d(x_i,x_j) \neq 2N-1\\
        -1 & \text{if } d(x_i,x_j) = 2N\\
        1 & \text{otherwise}
    \end{cases}$$

$$\varepsilon(x_i,y_j) = 
    \begin{cases}
        2 & \text{if } d(x_i,y_j) = 1\\
        3 & \text{if } d(x_i,y_j) \geq 2
    \end{cases}$$

$$\varepsilon(y_i,y_j) = 
    \begin{cases}
        2 & \text{if } d(y_i,y_j) = 1\\
        5 & \text{if } d(y_i,y_j) = 2\\
        6 & \text{if } d(y_i,y_j) 