Suppose I have a 3-manifold obtained via face identifications of a polyhedron (e.g. the Poincaré sphere presented as a dodecahedron with opposite faces glued). Is there a program that exists for easily working with such manifolds? The only thing I really know how to do is to make a triangulation in Regina, but I'd like to avoid having to input a triangulation with an unholy number of simplices, if possible.
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1$\begingroup$ You could write a python script to generate the triangulation from the polyhedron identifications, then feed it to Snappy or Regina. I think that's how several of the triangulations in Regina were generated -- for example, the Seifert-Weber dodecahedral space. If your polyhedron is convex then the algorithm would be pretty straightforward: subdivide the faces then cone off in the polyhedron. $\endgroup$– Ryan BudneyCommented Aug 25, 2023 at 20:17
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1$\begingroup$ I think if you look around the Regina source code, you can find the python script to generate all the 3-manifold (triangulations) you get by gluing opposite faces of a dodecahedron. That code could be readily adapted to essentially any convex polyhedron gluing. It is ?likely? in the census code, but it might be in the utilities. $\endgroup$– Ryan BudneyCommented Aug 25, 2023 at 20:28
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$\begingroup$ @RyanBudney I think this is what I’ll end up doing. I just wanted to check if anyone had made a program already. $\endgroup$– mrburchCommented Aug 25, 2023 at 21:58
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1 Answer
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The two standard answers to this question are Snappy and Regina. If you have a famous manifold, then it will likely be in one of the many censuses that come with the programs. If you have a less-than-famous manifold, then find a surgery description of it and feed that to Snappy.
