I have a Seifert fibered 3-manifold with base manifold $S^2$ and 3 exceptional fibers, say $M(1/p, 1/q,1/r)$. Say $p,q,r$ are relatively prime. Is there an easy way to understand which Seifert fibered spaces with base manifold $S^2$ and 3 exceptional fibers occur as regular cover of this space. Also, can one explicitly compute the degree of this cover.
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Not a complete answer, but a family of examples.
Suppose that $M$ is a Seifert fibered space. Suppose that $B$ is the base orbifold of $M$. Then any orbifold cover $B'$ of $B$ gives a Seifert cover $M'$ of $M$.
So, suppose that $T$ is the triangle for $B$. If $T$ "black-and-white" tiles a larger triangle $T'$, then we can double $T'$ to get $B'$ and so obtain an example.
Here are two concrete examples.
- Two copies of the $(2, 4, 4)$ triangle black-and-white tile a larger $(2, 4, 4)$ triangle.
- Twenty-four copies of the $(2, 3, 7)$ triangle black-and-white tile the $(7, 7, 7)$ triangle.
I got the second example from this answer to a question about reflection groups. In the reflection groups case, we don't need the black-and-white tiling. But it seems that all of those examples have it anyway.
