# Triangulations of 3-manifolds in Regina and SnapPy

I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a single triangulation into SnapPy format, but is mum on any batch way of doing this. Any help appreciated...

• I imagine one can do a lot of things using Regina's Python interface - did you try asking on github.com/regina-normal/regina ? – Dima Pasechnik Mar 24 at 6:40
• Thanks, @DimaPasechnik! I did not know it HAD a python interface (it is not super well-documented). – Igor Rivin Mar 24 at 16:09
• @DimaPasechnik Actually, the python interface is kind of terrible and seems to violate all known (and some unknown) python design standards. sigh. – Igor Rivin Mar 24 at 20:42
• well, I gave up on trying to use any boost-dependent C++ projects. Aren’t there other tools available that can replace Regina here? – Dima Pasechnik Mar 24 at 22:05
• @DimaPasechnik Good question. The morally pure thing to do is to read the paper (which exists) and to implement the thing myself. but this is obviously slightly less time efficient. – Igor Rivin Mar 25 at 2:01

So I have done some of the hand-holding of getting regina and snappy to talk to each other. (Actually, I did this a few years ago, they are better integrated now thanks to the hard work of both development teams, especially with regards to moving isosigs back and forth.)

More specifically, I looked at all ideal (which for regina means at least one ideal vertex) orientable triangulations with 6 or fewer tetrahedra.

Then I threw out the triangulations with finite vertices and the triangulations of solid tori.

The complete data summary is available in Tables 1,2, and 3 at the end of

Garoufalidis, Stavros; Hodgson, Craig D.; Hoffman, Neil R.; Rubinstein, J. Hyam, The 3D-index and normal surfaces, Ill. J. Math. 60, No. 1, 289-352 (2016). ZBL1378.57030.

But the interesting data is also here:

http://math.okstate.edu/people/nhoffman/smalltriangulations.html

I tried to classify things as best as I could with the contemporary tools. However, toroidal mean that it contained an essential embedded tori or Klein bottle, but it could be SFS with over the S^2 with 4 exceptional fibers.

Unfortunately, Regina went through a major update since I implemented the code so it might take a little work to get it running again.

Yes, Regina has such a feature. Unfortunately the documentation has become a bit more difficult to read since Regina 5.0. The software we use to generate the documentation (Doxygen) doesn't deal well with highly-templated code, it seems.

You can also import from Orb (perhaps this isn't maintained anymore?) and Matveev's Recogniser.

Here is the link in the Regina 5.0 API docs for what you want. The command is in the Triangulation<3> class, and its called fromSnapPea().

• Thanks! I will try to check it out! – Igor Rivin Apr 2 at 21:22