## Happenings – 2011 Dec 10

The blog had passed 140,000 hits since inception at the end of the day Tuesday Dec 6.

I have a post written for this coming Monday; as usual, when I say “written” I mean that it awaits final edits — typically, links and more transition and summary sentences.

It is, as I forecast, a regression post about fitting polynomials – but it’s only one of the two examples I had in mind; the second example is even juicier than I thought, and definitely warrants a post of its own.

My crucial Xmas shopping is done: what had to be shipped to the East Coast has, in fact, been shipped. Everything else is local… oh, there’s one thing I need to order on line.

As for the week that was, I’ve been browsing.

One of the things I came across was a 14-element triangulation of the torus. I have shown that we need at least 14 triangles (instead of the easy triangulation with 18), but I had not exhibited a solution. Here it is, based on Nash & Sen “Topology and Geometry for Physicists” ISBN 012-514081-9.

Posted in diary. 5 Comments »

### 5 Responses to “Happenings – 2011 Dec 10”

1. Andrej Says:

Thanks for both this one and the original triangulation post!

You might be interested to know that this paper has a more “elegant” (symmetric) torus triangulation with 7 vertices:
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Teo.pdf

I was searching for a clarification on this topic, because it seems that sometimes it is valid to consider triangulations with multiple edges sharing two vertices. For example, having one vertex and two triangles for a two-dimensional torus. In particular, the Dijkgraaf-Witten topological invariants of a manifold M are defined using group elements that live on edges of a triangulation of M. Even the original paper considers such “invalid” triangulations, in particular for a three-dimensional torus (of course, using tetrahedra instead of triangles), but still obtains a valid expression for the topological invariant.

My question is: Which topological properties of a manifold can be obtained correctly even when studying slightly “invalid” triangulations of the manifold, e.g. having multiple edges between two vertices? It seems mathematicians dislike using such triangulations, but physicists are comfortable believing they work in general 🙂

• Mildred Bonk Says:

It’s possible to get an even more elegant triangulation using only right triangles, which is important if you care about the aspect ratio or the maximum angles of your triangles. I’ll leave it as an exercise for the reader; it’s pretty easy if you start from Teo’s diagram.

2. rip Says:

Thanks, Andrej.

The symmetric drawing in that link does look nicer than mine… and shows me that I should probably have numbered the two remaining vertices, so that it is clear I have 7 vertices. The paper itself looks like a senior thesis, or possibly Master’s thesis, working out the relevant material from Lee’s “Introduction to Topological Manifolds”.

I know absolutely nothing about Djikgraaf-Witten invariants… and I don’t think I’m ready to start.

• Andrej Says:

Thanks for the reply! I did not mean to judge your drawing, it’s just useful to have a representation that exhibits as much symmetry as possible.