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List of Equations

Euler Characteristic Formula: $\chi = V - E + F$ , where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces.

Genus Formula: $2-2g=\chi$ , where $g$ is the genus.

Equation 1. For each $x_i \in \rho$ , $\lambda(x_i)=\rho(x_i^{-1})$ .

Equation 2. For $n_e$ , where $n_e$ is an even integer, $L(n_e,n_e)=(\frac{n_e}{2})^2-n_e+1$ .

Equation 3. For $n_o$ , where $n_o$ is an odd integer, $L(n_o,n_o)=\frac{n_o^2+3}{4}-n_o+1$ .

Equation 4. Let $m$ be the order of the sum of the elements in $F_j$ . The total number of edges, $t$ , of the face produced by $\lambda_j$ is $(t = m|F_j|)$ .

Equation 5. The number of occurrences, $c$ , of each face is $c=\frac{2p|F_j|}{t} = \frac{2p}{m}$ , where $m$ is the order of the sum of the elements in $F_j$ .

Equation 6. The number of darts, $d$ , in $C_M(Z_{2p}, \rho)$ is $d=ct= pV$ .