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.