Low Dimensional Topology

November 10, 2012

SnapPy 1.7: Ptolemy and reps to PSL(n, C).

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Triangulations — Nathan Dunfield @ 2:45 pm

SnapPy 1.7 is out. The main new feature is the ptolemy module for studying representations into PSL(n, C). This code was contributed by Mattias Görner, and is based on the the following two very interesting papers:

  1. Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert: Gluing equations for PGL(n,C)-representations of 3-manifolds.
  2. Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert: The complex volume of SL(n,C)-representations of 3-manifolds.

You can get the latest version of SnapPy at the usual place.


August 22, 2012

Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

April 25, 2012

Regina on Windows and MacOS

Filed under: 3-manifolds,Computation and experiment,Triangulations — Benjamin Burton @ 4:03 am

For those non-Linux users who’ve wanted to tinker with Regina:

Regina 4.92 came out a couple of weeks ago, and has some big portability improvements.  Mac users now have a simple drag-and-drop install (no need for fink), and for the first time there is an installer for MS Windows.  As always, there are also ready-made packages for several GNU/Linux distributions.

The new version adds features such as fundamental normal surfaces and boundary slopes for spun-normal surfaces, and the user interface is cleaner.  For more information or to have a play, hop over to regina.sourceforge.net.

March 5, 2012

Rethinking the basics

Filed under: Misc.,Quantum topology,Triangulations — dmoskovich @ 3:36 am

Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds really?

This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of \mathbb{R}^n by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it really is.

In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually exist, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.

It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s On piecewise linear cell decompositions). (more…)

February 20, 2012

New SnapPy 1.5, now with better manifolds

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Triangulations — Nathan Dunfield @ 12:47 pm

Marc Culler and I have released version 1.5 of SnapPy, which you can get from the usual place. The main new feature is greatly improved manifold censuses, including the ability check if a given manifold is in one; here are many examples of now to use the new censuses.

I never blogged about version 1.4, since while there were major changes they were mostly under the hood, principally moving to the current version of IPython and making snappy (mostly) compatible with Python 3.2.

February 5, 2012

The 3-sphere has interesting triangulations

Filed under: 3-manifolds,Combinatorics,Triangulations — dmoskovich @ 1:06 am
Tags: ,

There is a recent interesting question on MO regarding a paper by Benedetti and Ziegler which I found most interesting.

Upon reading the question, I right away downloaded and printed Benedetti and Ziegler’s paper, after which I sat down to glance through it. My first impression is that it constitutes a really high-class piece of mathematics; the exposition is clear enough that a non-specialist can sit down and enjoy it, and the results are deep and interesting. There’s something really inspirational about a paper like that. (more…)

September 17, 2011

Regina is back!

Filed under: 3-manifolds,Computation and experiment,Triangulations — Benjamin Burton @ 5:32 pm

For those interested in computation with triangulations and normal surfaces:

It’s been a while coming, but Regina 4.90 finally went live last week!  This is the release that was previewed at the HyamFest.  There have been many changes since the last release 2+ years ago: see what’s new in this version, or check out the user handbook to learn more about what Regina can do.

If you use GNU/Linux, Regina is easy to install (just download a package).  If you use MacOS, it’s still complex (you need fink) but the end result is much cleaner than ever before (it feels and behaves like a native Mac app).

See some screenshots after the jump, or head across to regina.sourceforge.net to learn more.


February 7, 2011

New SnapPy 1.3, now with better horospheres!

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Triangulations — Nathan Dunfield @ 5:25 pm

Marc Culler and I have released version 1.3 of SnapPy, which you can get from the usual place. The main change is a much improved cusp viewer, including labels on the edges and vertices of the associated canonical ideal cellulation. Previous versions of SnapPy gave you a 3D view of the horoball packing which, while pretty, was not very practical; the perspective meant that there were not actually any 2D symmetries of the resulting picture! The new version uses an orthographic perspective that puts the viewers eye at infinity and thus gives the picture a euclidean lattice of symmetries. Also, due to internal changes, the cusp viewer is much faster than it used to be, and should be much less likely to crash on complicated examples.   Some pretty pictures from the new version are follow the jump.


February 23, 2010

SnapPy 1.1 released

Marc Culler and I have released version 1.1 of SnapPy.   Recently added features include:

  • Better Chern-Simons. The original SnapPea kernel could only compute the CS invariant some of the time.   Matthias Goerner has contributed an implementation of Zickert’s algorithm which works in many more cases. This is the first major code addition by someone other than Marc or I, so many thank to Matthias for making this available!
  • Now with even more manifolds. Due to the kindness of Morwen Thistlethwaite, SnapPy now comes with almost 200,000 more manifolds: a census of links through 14 crossings (MorwenLinks) and all orientable cusped manifolds with 8 tetrahedra (denoted e.g. “t01234” and part of OrientableCuspedCensus).
  • General Improvements: Improved UI features relating to saving and loading links, better Windows 7 support, various bug fixes.

There’s an example of all the new features after the jump. Enjoy!

August 20, 2009

Canonical triangulations of surface bundles

Filed under: 3-manifolds,Hyperbolic geometry,Triangulations,WYSIWYG topology — Nathan Dunfield @ 11:56 am

Hyperbolic 3-manifolds with a single cusp have canonical ideal triangulations constructed by Epstein and Penner via a certain convex hull construction involving the light-cone in the hyperboloid model of hyperbolic space. (Occasionally, these “triangulations” are really cellulations more complicated cells.) When such a 3-manifold fibers over the circle, there is another type of natural triangulation, called a layered triangulation. Roughly, one starts with a certain ideal triangulation of the fiber surface, looks at the image of this triangulation under the bundle monodromy, interpolates between these by a series of Pachner moves which can then be realized geometrically by layering on tetrahedra.

When the fiber is a once-punctured torus, these two type of triangulations coincide. This was shown by Marc Lackenby using a remarkably soft and elegant argument. It’s natural to wonder whether this phenomena occurs more broadly. For instance Sakuma suggested considering the following:

Conjecture: Canonical triangulations of punctured surface bundles are always layered.

Saul Schleimer and I have discovered that this is false in general. In particular, the manifold v1348 from the SnapPea census is fibered by a once-punctured surface of genus 5 yet has a canonical triangulation which is not layered. Precisely, the canonical triangulation does not admit one of Lackenby’s taut structures so that the resulting branched surface carries something with positive weights. Note that v1349 is in fact the complement of a certain knot in the 3-sphere [CFP].

Technical details: The canonical triangulation of v1348 is in fact just the triangulation encoded in the SnapPea census (and it is a triangulation, not a celluation). It’s easy to check that it fibers using the BNS invariant (cf. [DT]), and compute the genus of the fiber from the Alexander polynomial. One then checks that there are no taut structures of this type using Marc Culler and I’s t3m Python package.

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