These triangulations are also the basic building block in the layered triangulations studied by Jaco-Rubinstein-Tillman, and are used in the inflation/filling theory that Jaco-Rubinstein use to construct triangulations of filled knot complements. There’s always a trade-off in clarity between the two perspectives – triangulations or special spines – and while the spines may be easier to visualize in this case, there are benefits to understanding the triangulation later on in these constructions.

]]>Hi Dylan, I came across these triangulations quite naturally. In my paper with Burton and Hillman we triangulated the complement of a knotted S^2 in S^4, a Cappell-Shaneson knot. The triangulation has two 4-dimensional simplices. The induced triangulation of the “cusp” S^1xS^2 has the above as a subcomplex. I needed to visualize this triangulation to see how the whole S^1xS^2 was built — actually, in order to prove the cusp was really an S^1xS^2.

]]>Just the 1-skeleton is a little disorienting. But that’s it. The 0-cell is white. Then there’s the three 1-cells. The orangy one is slope zero. The yellow one is slope 1/2, and the light brown one is slope 1/3. So by “slope 0” I mean the slope of the longitude of the torus — parallel to a planar curve. Slope 1/0 would be the meridian of the torus.

Up to an isomorphism of the torus (S^1xS^1) this is just your standard triangulation with two triangles — i.e. cut the “square” CW-decomposition in half. There’s two triangles, three edges, one vertex, but it does not sit on S^1xD^2 in the standard way in that the meridian is not part of this triangulation.

The above image is the 1-skeleton but with the two triangles on the boundary torus coloured, one blue, the other red. They’re transparent so if you’re looking through both the colour filters.

]]>Very cool. You should post this as a regular blog entry because I don’t know how many people are looking at these old comment sections.

]]>I’m going to put all these observations into an animation that slowly builds-up and strips away the cells for this triangulation of the solid torus. Here are a couple preliminary shots (I still have to adjust the lighting and tweak a few things here and there). In particular, my parametrization of the Moebius band has a singularity near the 0-cell (running all the way across the band). I’ll have to fix that before the finished product, but here they are:

I used geodesics in the Poincare disc to construct the two arcs in the {z}xD^2 slices… maybe I should have used straight lines in a Euclidean disc instead? Perhaps making the 1-skeleton thicker would help…

Okay, so a summary: the first five shots have the 1-skeleton and the Moebius band (the internal triangle).

The second five shots have the 1-skeleton, the Moebius band and *one* of the triangles that sits on the boundary of S^1 x D^2. These pictures look red and blue.

The last five shots have the 1-skeleton, the Moebius band and the *other* of the boundary triangles. These pictures look red and green.

]]>Can you see the other 2-cells in a picture like this, or does it get too messy?

]]>That’s a good point. Not only is this much easier to “see”, but it’s also easier to generalize to higher genus handlebodies: Start with any surface with a single boundary component and a one-vertex triangulation, then layer on tetrahedra until the link of the vertex becomes a disk.

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