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There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist we receive the Seifert-Weber space, and with a $\frac{5}{10}$ twist we receive the 3-dimensional real projective space. I've been using the CurvedSpaces software for visualizing these manifolds and coming from a Convex Geometry perspective I have some questions that I thought I would pass by here before I spent more time researching it if the idea is complete garbage.

It was explained to me that as a Corollary of Moise's Theorem there must exist a tetrahedralization of any orientable closed 3-manifold. I am interested in what an explicit tetrahedralization of some closed 3-manifold would look like, especially for the three manifolds mentioned above that are generated by gluing opposite faces of the dodecahedron. The reason why I am interested in this, is that I want to have a concrete understanding of a small class of examples for which I can investigate the explicit tetrahedra which tessellate the manifold. With a deeper understanding and some explicit examples I was hoping to work on classifying hyperbolic knot complements by the convex polytopes that tessellate them. So, my question is the following:

Question:

What do explicit tetrahedralizations of closed 3-manifolds look like? In particular, for the Poincaré dodecahedral space, the Seifert-Weber space or any hyperbolic knot complements. How can I determine the explicit vertex coordinates of the tetrahedra that tetrahedralize the manifold?

I've come up with a method for determining "Intermediary Polytopes" given two distinct polytopes, and a string of conjectures relating to how these intermediary polytopes may tetrahedralize certain manifolds, but I need some sort of explicit vertex positions in order for some of the machinery I have constructed to work.

If anyone has done research on this in the past I will accept a reference to a paper as an answer, otherwise I would like an actual answer to my question if it not too vague. If it is too vague, please ask questions in the comments and I can sharpen my question in response.

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    $\begingroup$ Can't you triangulate these dodecahedral manifolds by taking the central subdivision of each face, coning to the center, and then barycentrically subdividing? $\endgroup$
    – Jim Conant
    Commented Jan 20, 2012 at 3:00
  • $\begingroup$ Thank you for that suggestion, I had not thought of that. $\endgroup$
    – Samuel Reid
    Commented Jan 20, 2012 at 3:08
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    $\begingroup$ "vertex coordinates of the tetrahedra" -- this doesn't make much sense. In full generality (whatever that means) triangulations are topological objects, not geometric ones. You have to restrict the setting to get to triangulations that have geometric meaning. The standard way to do this, following Thurston, is to consider ideal triangulations of knot complements. There is a discussion of this at the beginning of chapter three of Thurston's book. $\endgroup$
    – Sam Nead
    Commented Jan 20, 2012 at 4:44
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    $\begingroup$ @Samuel: Do you want an ideal triangulation? (Meaning vertices at infinity and edges geodesic.) That's a bit harder. Otherwise, yes you can just look at your manifold as a quotient space of a polyhedron and subdivide. $\endgroup$
    – Jim Conant
    Commented Jan 20, 2012 at 14:50
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    $\begingroup$ The software "Regina" by Ben Burton has over 16,000 explicit triangulations of 3-manifolds. $\endgroup$
    – Ryan Budney
    Commented Mar 9, 2012 at 21:23

2 Answers 2

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The software "Regina" by Ben Burton has the census of all triangulations containing 11 or less tetrahedra. There are more than 16,000 distinct manifolds in this list of triangulations. So if you're interested in getting a sense for what 3-manifold triangulations look like, this is a start.

It sounds like one of the main features you'd be interested in is the "Skeletons" feature and the "Composition" feature. This latter feature will help you recognise common sub-triangulations, such as certain standard triangulations of a solid torus (layered solid tori is the term).

https://regina-normal.github.io/

You can also export the triangulations to Jeff Weeks SnapPea and compute and visualize a Dirichlet domain (if the manifold is hyperbolic) there.

The most actively maintained version of SnapPea is Culler and Dunfield's SnapPy, which is available here:

https://www.math.uic.edu/t3m/SnapPy/

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Ryan's answer above will be more comprehensive and general than this one. But if you are just looking for some examples, especially highly symmetric examples, of explicit shapes for closed hyperbolic 3-manifolds (like the Seifert-Weber dodecahedral space), then you might also consider using Damien Heard's orb (which will give approximate shapes), or looking at the enumeration of compact hyperbolic tetrahedral orbifolds in the appendix of

Maclachlan, Colin; Reid, Alan W., The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics. 219. New York, NY: Springer. xiii, 463 p. (2003). ZBL1025.57001.

For explicit representations and shapes of tetrahedral orbifolds see Lakeland's unpublished note (http://www.ux1.eiu.edu/~gslakeland/tetrahedral_realizations.pdf).

For example, the Seifert-Weber dodecahedral orbifold is a 120 fold cover of the orbifold $T_3$ (page 415 of Maclachlan-Reid) or 60 cover of the orientable quotient $H^3/\Gamma(5,3,2,2,5,2)$ in Lakeland's notation.

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