Computation of index, underlying topological space, number of vertex points and number of S1 in the orbifolds in families 8–14 of Table 3. Some spaces are left blank, since those invariants are not necessary in the proof of Theorem B. The index |$|\pi_1(\mathcal{O}):N|$| is computed for the maximal normal abelian subgroup N, when it is unique.
| . | Fibrations . | |$|\pi_1(\mathcal{O}):N|$| . | |$|\mathcal{O}|$| . | Number of vertex points . | Number of circles in |$\Sigma_{\mathcal{O}}$| . |
|---|---|---|---|---|---|
| 8 | |$(2_02_0n_0)$| |$(2_12_1n_0)$| |$(2_02_1n_{n/2})$| | $\begin{cases}1\,\,\,\text{if }n=2 \\ 2\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^2\times S^1$| |$S^2\times S^1$| |$S^2\times S^1$| | 3 1 2 | |
| 9 | |$(\ast_02_02_0n_0)$| |$(\ast_12_12_1n_0)$||$(\ast_12_02_1n_{n/2})$| | $\begin{cases}2\,\,\,\text{if }n=2\\ 4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$S^3$| |$S^3$| | 6 2 4 | |
| 10 | |$(2_0\ast_0n_0)$| |$(2_1\ast_1n_0)$| | $\begin{cases}2\,\,\,\text{if }n=2\\4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$\mathbb{R} P^3$| | 2 2 | |
| 11 | |$(n_0n_0)$| | 1 | |$S^2\times S^1$| | $\begin{cases}0\,\text{if }n=1\\2\,\text{if }n\not=1\end{cases}$ | |
| 12 | |$(\ast_0n_0n_0)$| | 2 | S3 | $\begin{cases}0\,\text{ if }n=1\\4\,\text{ if }n\not=1\end{cases}$ | $\begin{cases}2\,\text{if }n=1\\0\,\text{if }n\not=1\end{cases}$ |
| 13 | |$(n_0\ast_0)$| |$(n_{(n/2)}\ast_1)$| | 2 | |$S^3$| |$\mathbb{R} P^3$| | 0 0 | 3 2 |
| 14 | |$(n_0\overline\times)$| | 2 | |$\mathbb{R} P^3\#\mathbb{R} P^3$| |
| . | Fibrations . | |$|\pi_1(\mathcal{O}):N|$| . | |$|\mathcal{O}|$| . | Number of vertex points . | Number of circles in |$\Sigma_{\mathcal{O}}$| . |
|---|---|---|---|---|---|
| 8 | |$(2_02_0n_0)$| |$(2_12_1n_0)$| |$(2_02_1n_{n/2})$| | $\begin{cases}1\,\,\,\text{if }n=2 \\ 2\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^2\times S^1$| |$S^2\times S^1$| |$S^2\times S^1$| | 3 1 2 | |
| 9 | |$(\ast_02_02_0n_0)$| |$(\ast_12_12_1n_0)$||$(\ast_12_02_1n_{n/2})$| | $\begin{cases}2\,\,\,\text{if }n=2\\ 4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$S^3$| |$S^3$| | 6 2 4 | |
| 10 | |$(2_0\ast_0n_0)$| |$(2_1\ast_1n_0)$| | $\begin{cases}2\,\,\,\text{if }n=2\\4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$\mathbb{R} P^3$| | 2 2 | |
| 11 | |$(n_0n_0)$| | 1 | |$S^2\times S^1$| | $\begin{cases}0\,\text{if }n=1\\2\,\text{if }n\not=1\end{cases}$ | |
| 12 | |$(\ast_0n_0n_0)$| | 2 | S3 | $\begin{cases}0\,\text{ if }n=1\\4\,\text{ if }n\not=1\end{cases}$ | $\begin{cases}2\,\text{if }n=1\\0\,\text{if }n\not=1\end{cases}$ |
| 13 | |$(n_0\ast_0)$| |$(n_{(n/2)}\ast_1)$| | 2 | |$S^3$| |$\mathbb{R} P^3$| | 0 0 | 3 2 |
| 14 | |$(n_0\overline\times)$| | 2 | |$\mathbb{R} P^3\#\mathbb{R} P^3$| |
Computation of index, underlying topological space, number of vertex points and number of S1 in the orbifolds in families 8–14 of Table 3. Some spaces are left blank, since those invariants are not necessary in the proof of Theorem B. The index |$|\pi_1(\mathcal{O}):N|$| is computed for the maximal normal abelian subgroup N, when it is unique.
| . | Fibrations . | |$|\pi_1(\mathcal{O}):N|$| . | |$|\mathcal{O}|$| . | Number of vertex points . | Number of circles in |$\Sigma_{\mathcal{O}}$| . |
|---|---|---|---|---|---|
| 8 | |$(2_02_0n_0)$| |$(2_12_1n_0)$| |$(2_02_1n_{n/2})$| | $\begin{cases}1\,\,\,\text{if }n=2 \\ 2\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^2\times S^1$| |$S^2\times S^1$| |$S^2\times S^1$| | 3 1 2 | |
| 9 | |$(\ast_02_02_0n_0)$| |$(\ast_12_12_1n_0)$||$(\ast_12_02_1n_{n/2})$| | $\begin{cases}2\,\,\,\text{if }n=2\\ 4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$S^3$| |$S^3$| | 6 2 4 | |
| 10 | |$(2_0\ast_0n_0)$| |$(2_1\ast_1n_0)$| | $\begin{cases}2\,\,\,\text{if }n=2\\4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$\mathbb{R} P^3$| | 2 2 | |
| 11 | |$(n_0n_0)$| | 1 | |$S^2\times S^1$| | $\begin{cases}0\,\text{if }n=1\\2\,\text{if }n\not=1\end{cases}$ | |
| 12 | |$(\ast_0n_0n_0)$| | 2 | S3 | $\begin{cases}0\,\text{ if }n=1\\4\,\text{ if }n\not=1\end{cases}$ | $\begin{cases}2\,\text{if }n=1\\0\,\text{if }n\not=1\end{cases}$ |
| 13 | |$(n_0\ast_0)$| |$(n_{(n/2)}\ast_1)$| | 2 | |$S^3$| |$\mathbb{R} P^3$| | 0 0 | 3 2 |
| 14 | |$(n_0\overline\times)$| | 2 | |$\mathbb{R} P^3\#\mathbb{R} P^3$| |
| . | Fibrations . | |$|\pi_1(\mathcal{O}):N|$| . | |$|\mathcal{O}|$| . | Number of vertex points . | Number of circles in |$\Sigma_{\mathcal{O}}$| . |
|---|---|---|---|---|---|
| 8 | |$(2_02_0n_0)$| |$(2_12_1n_0)$| |$(2_02_1n_{n/2})$| | $\begin{cases}1\,\,\,\text{if }n=2 \\ 2\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^2\times S^1$| |$S^2\times S^1$| |$S^2\times S^1$| | 3 1 2 | |
| 9 | |$(\ast_02_02_0n_0)$| |$(\ast_12_12_1n_0)$||$(\ast_12_02_1n_{n/2})$| | $\begin{cases}2\,\,\,\text{if }n=2\\ 4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$S^3$| |$S^3$| | 6 2 4 | |
| 10 | |$(2_0\ast_0n_0)$| |$(2_1\ast_1n_0)$| | $\begin{cases}2\,\,\,\text{if }n=2\\4\,\,\,\text{if }n\not\in\{2,4\}\end{cases}$ | |$S^3$| |$\mathbb{R} P^3$| | 2 2 | |
| 11 | |$(n_0n_0)$| | 1 | |$S^2\times S^1$| | $\begin{cases}0\,\text{if }n=1\\2\,\text{if }n\not=1\end{cases}$ | |
| 12 | |$(\ast_0n_0n_0)$| | 2 | S3 | $\begin{cases}0\,\text{ if }n=1\\4\,\text{ if }n\not=1\end{cases}$ | $\begin{cases}2\,\text{if }n=1\\0\,\text{if }n\not=1\end{cases}$ |
| 13 | |$(n_0\ast_0)$| |$(n_{(n/2)}\ast_1)$| | 2 | |$S^3$| |$\mathbb{R} P^3$| | 0 0 | 3 2 |
| 14 | |$(n_0\overline\times)$| | 2 | |$\mathbb{R} P^3\#\mathbb{R} P^3$| |
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