# 3-orbifolds with a Seifert geometry that are not actually Seifert fibered

It is well-known that Seifert fibered $3$--manifolds are geometric: they admit one of the Thurston geometries $S^2 \times R$, $R^3$, $H^2 \times R$, $S^3$, $Nil$, and $PSL(2,R)$. Furthermore, the converse is also true: every $3$--manifold admitting one of these 6 geometries has a Seifert fibration over a $2$--orbifold.

In the realm of $3$--orbifolds, it is still true that every Seifert fibered $3$--orbifold is geometric, with the same 6 geometries. However, the converse is false: there exist geometric $3$--orbifolds with one of the 6 "Seifert" geometries, which do not Seifert fiber. My question is:

What are the orientable $3$--orbifolds that have a Seifert geometry, but are not Seifert fibered? Do we know a complete list?

One example of this weird phenomenon is the figure-8 knot, labeled 3. This orbifold is Euclidean. But it is not Seifert fibered, because any order-3 singular locus must be vertical in a Seifert fibration. Thus, if there was a Seifert fibering of the orbifold, drilling out the singular locus would produce a Seifert fibration on the knot complement, which is absurd because the knot complement is hyperbolic.

More generally, Dunbar has classified the spherical $3$--orbifolds that are not Seifert fibered. There are 21 such examples in total: http://www.ams.org/mathscinet-getitem?mr=1118824

What is known about the other 5 Seifert geometries?

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Remark. There are orbifolds with geometry $E^3$ and $S^3$ which are not orbifold-Seifert fibered [52, 54].