I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question


[can you fool SnapPea?](https://mathoverflow.net/questions/4918/can-you-fool-snappea)


but in this case I'm dealing with closed, orientable 3-manifolds instead of knot and link complements. 


The 3-manifolds I've come across have 11, 12 and 13 tetrahedra in their triangulations. One of them SnapPea finds a solution to the gluing equations but it has "negatively oriented tetrahedra". Does this mean what I think it means -- that once you've put the geometric structure on the tetrahedra, you have a tetrahedron folded-over? If you view the gluing equations from the upper half-space model, they say that the sum of a bunch of angles should be $2\pi$. Is this the case where one of those angles is negative? 


The other two triangulations SnapPea finds geometric structures with degenerate tetrahedra. In the upper half-space model this is where one of the angles is zero, I believe. Is there a way to fix this, so that I could get a Dirichlet domain, drill and fill, etc? This is my primary question. 


Here are the triangulations, both in SnapPea format and Regina format. 


[Regina file, all triangulations](http://dl.dropbox.com/u/46424505/triangulations/found_hyp_tri.rga)


[11-tet, SnapPea](http://dl.dropbox.com/u/46424505/triangulations/tri11)


[12-tet, SnapPea](http://dl.dropbox.com/u/46424505/triangulations/tri12)


[13-tet, SnapPea](http://dl.dropbox.com/u/46424505/triangulations/tri13)


With the 12-tetrahedron example, SnapPea can generate a Dirichlet Domain. This one has what looks almost like bevelled-edges. Is there a way to find a better basepoint to grow the Dirichlet domain from? 


[12-tet Dirichlet domain http://dl.dropbox.com/u/46424505/triangulations/pic.png](http://dl.dropbox.com/u/46424505/triangulations/pic.png)


edit: After working a bit with Nathan's answer, I've identified these three manifolds and got a little closer to understanding how to work SnapPea to maximal advantage. 


tri11, as Nathan mentioned, is hyperbolic. It has a fairly pretty Dirichlet domain.


[Tri11 Dirichlet Domain http://dl.dropbox.com/u/46424505/triangulations/tri11.jpg](http://dl.dropbox.com/u/46424505/triangulations/tri11.jpg)


Another common name for this manifold would be the 0-surgery on the 2-component link $7a\_6$ (in the Thistlethwaite table). Similarly tri12 can be identified as Nathan says. 


After playing around with Regina a bit I found an incompressible torus in tri13 that splits tri13 into the union of an orientable $I$-bundle over the Klein bottle and a figure-8 complement. So this answers the core of my question. 



Some mid-sized ¿hyperbolic? manifolds and SnapPea
The first two of these manifolds are hyperbolic. The problem, I suspect, is that they are being specified as Dehn fillings on 1-cusped manifolds with essential tori. So one needs to pick a different knot in the closed manifold to actually see the hyperbolic structure. One way to force SnapPy to do this is to create a 1-vertex non-ideal triangulation and then use one of the edges there as the knot. For example: 



```
sage: M = Manifold('tri11.txt')
sage: N = Manifold(M.filled_triangulation()._to_string())
sage: N.solution_type()
'all tetrahedra positively oriented'
sage: closed = snappy.OrientableClosedCensus()
sage: closed.identify(N)
m038(1,2)

```

The tri12 manifold is "m032(5,2)". I strongly suspect the tri13 manifold is *not* hyperbolic. Instead, I suspect it has a nontrivial JSJ splitting where one piece is the figure-8 complement, since sometimes the volume appears to be 2.0988... 


