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Programmatically Drawing Cusp Diagrams of Hyperbolic 3-manifolds

20.01.2025

In this post I assume all hyperbolic manifolds are singly cusped and all triangulations are ideal.

Introduction

Over the past month, I've been working on a Python script that draws the cusp diagram of a given geometric triangulation of a hyperbolic 3-manifold. The project started when thinking about something else: geometric Pachner moves. It would be nice, for a handful of applications, if one could quickly find a bunch of geometric triangulations of a cusped hyperbolic manifold \(M\). Of course not all triangulations are geometric, or even essential, so the first idea would be to get a bunch of triangulations via 2-3 and 3-2 moves, and then check if these are geometric. The obvious issue with this is that it's slow, especially as the number of tetrahedra, and thus the number of triangulations, grows.

The bottleneck comes when calculating the shape parameters (by e.g. solving some non-linear complex equations). A better idea is to observe how the shape parameters of the tetrahedra change as you perform a 2-3 move or a 3-2 move. Then you can pass the old shape parameters to the new triangulation, updating it as necessary. This saves us (and SnapPy) from having to find solutions to the gluing equations.

It's not always possible, however, to do a geometric Pachner move, even if there are many available non-geometric Pachner moves. This paper gives some good background on this. If a geometric triangulation \(T\) has available Pachner moves but none of them are geometric, then we say that \(T\) is an isolated geometric triangulation (I also like to call these 'precarious' triangulations, but I don't think this will catch on). These are kind of interesting to study — somehow these triangulations are nicely geometric, but any local (2-3 or 3-2) move will collapse the geometry.

These isolated geometric triangulations can be found by brute search (see Remark 3.3 in the paper linked in the previous paragraph), but perhaps it's strange that these should exist at all; the tetrahedra must be arranged so delicately (hence 'precarious') in \(\mathbb{H}^3\) that these local moves either flatten or negatively orient one of the tetrahedra. So it's interesting to ask about what these look like. And while it's hard to visualize these in three dimensions, the tetrahedra intersect the cusps to triangulate a boundary torus. And, up to some scaling, they do this uniquely by Mostow Rigidity, so it really does seem like an appropriate thing to look at. Let's see what this looks like.

The Pictures

Here are three different triangulations of the figure-8 knot complement sister manifold (see the paper linked above to see why it's interesting to look at these):

cPcbbbdxm

cPcbbbdxm

dLQacccjgnb

dLQacccjgnb

eLAkaccddngbqw

eLAkaccddngbqw

The pictures are captioned with their isomorphism signature. One triangle in the image represents a vertex in the triangulation (the triangles are the truncated part of a truncated tetrahedron), and each tetrahedron gets one color. I've drawn these pictures so that a fundamental domain appears and its repetition is obvious. The images lose some information, though, since it doesn't show the edges or faces of the triangulation (this is fixable if one colors the vertices and edges of the picture, but this is less pretty).

The third triangulation (eLAkaccddngbqw) has the property that none of its five possible 2-3 moves are geometric. If you stare at the picture for long enough, seeing where the colors meet, you can kind of see this. A 2-3 move on the cusp looks like three pairs of two colors adjacent to each other (the fourth triangle of each color is split into three new colors). You would see a 2-2 move (a two-dimensional Pachner move) on the three pairs, and a 1-3 move on the other two. Note that in reducing to a 2D picture, the local moves become no longer local. Looking at the picture for that triangulation, you can find all five possible 2-3 moves and see that four of them give negatively oriented triangles and one of them gives a flat triangle.

So what do the isolated geometric triangulations look like? Well, they're not as pretty. Here is one of the figure-8 knot complement:

fLLQcacdedejbqqww

fLLQcacdedejbqqww

You could probably look at this for long enough in the same way and see that it's isolated, but it's just not as pleasant. And while there's more to say about these cusp diagrams, I want to finish this section by pivoting to aesthetics. These can really get quite pretty, and eventually I'll set up a small gallery of them on this website (sort of like my gallery of asymmetric graphs). For now, here are some nice ones:
jLLzzQQccdffihhiiqffofafoaa

jLLzzQQccdffihhiiqffofafoaa

gLLAQccedfefqfhfhsj

gLLAQccedfefqfhfhsj

eLPkbdcddhgggb

eLPkbdcddhgggb

The Program

The code repository can be found here. As it stands it's a bit messy, and there's still a bit to do (I need to refactor how it searches through the triangulation so that it doesn't break on a certain edge case). It doesn't do anything fancy: just gets the triangulation data from Regina and the geometric data from SnapPy, and puts it together in an image in Sage.

I found (after a long and frustrating bug) that SnapPy's M.isometry_signature(), which gives the isomorphism signature of the canonical retriangulation of the manifold (an extension of the Epstein-Penner cell decomposition), tends to have nicer triangulations than what's given in the census. Perhaps this makes sense: trying to minimize tetrahedra tends to clutter things?

In the direction of utility, I want to appropriately color the edges and vertices, and also try to have the drawings line up under Pachner moves. In the direction of aesthetics, I might make a script that tries to find triangulations where all triangles are not too small nor too obtuse. This might make much nicer pictures.