Low Dimensional Topology

March 2, 2014

SnapPy 2.1: Now with extra precision!

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Knot theory — Nathan Dunfield @ 11:39 pm

Marc Culler and I released SnapPy 2.1 today. The main new feature is the ManifoldHP variant of Manifold which does all floating-point calculations in quad-double precision, which has four times as many significant digits as the ordinary double precision numbers used by Manifold. More precisely, numbers used in ManifoldHP have 212 bits for the mantissa/significand (roughly 63 decimal digits) versus 53 bits with Manifold.

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November 26, 2013

What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”.  It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

Conference banner
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October 13, 2013

A noteworthy knot simplification algorithm

Filed under: Computation and experiment,Knot theory — dmoskovich @ 8:26 am

This post concerns an intriguing undergraduate research project in computer engineering:

Lewin, D., Gan O., Bruckstein A.M.,
TRIVIAL OR KNOT: A SOFTWARE TOOL AND ALGORITHMS FOR KNOT SIMPLIFICATION,
CIS Report No 9605, Technion, IIT, Haifa, 1996.

A curious aspect of the history of low dimensional topology are that it involves several people who started their mathematical life solving problems relating to knots and links, and then went on to become famous for something entirely different. The 2005 Nobel Prize winner in Economics, Robert Aumann, whose game theory course I had the honour to attend as an undergrad, might be the most famous example. In his 1956 PhD thesis, he proved asphericity of alternating knots, and that the Seifert surface is an essential surface which separates alternating knot complements into two components the closures of both of which are handlebodies.

Daniel Lewin is another remarkable individual who started out in knot theory. His topological work is less famous than Aumann’s, and he was murdered at the age of 31 which gives his various achievements less time to have been celebrated; but he was a remarkable individual, and his low dimensional topology work deserves to be much better known. (more…)

October 2, 2013

Regina 4.94

Filed under: 3-manifolds,Computation and experiment,Triangulations — Benjamin Burton @ 4:00 pm

It’s the season for it!  For those of you who work with normal surfaces, Regina 4.94 also came out last week.  It adds triangulated vertex links, edge drilling, and a lot more speed and grunt.

Take the new linear/integer programming machinery for a spin with the pre-rolled triangulation of the Weber Seifert dodecahedral space.  Regina can now prove 0-efficiency in just 10 seconds, or enumerate all 1751 vertex surfaces in ~10 minutes, or (with a little extra code to coordinate the slicing and searching for compressing discs) prove the entire space to be non-Haken in ~2 hours.

Read more of what’s new, or download and tinker at regina.sourceforge.net.

September 30, 2013

SnapPy 2.0 released

Marc Culler and I pleased to announce version 2.0 of SnapPy, a program for studying the topology and geometry of 3-manifolds. Many of the new features are graphical in nature, so we made a new tutorial video to show them off. Highlights include
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May 31, 2013

The algorithm to recognise the 3-sphere

Filed under: 3-manifolds,Computation and experiment,Triangulations — Ryan Budney @ 10:48 am
Tags: ,
The purpose of this post is to convince you the 3-sphere recognition algorithm is simple.  Not the proof!  Just the statement of the algorithm itself.  I find in conversations with topologists, it’s fairly rare that people know the broad outline of the algorithm.  That’s a shame, because anything this simple should be understood by everyone.   

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April 23, 2013

When are two hyperbolic 3-manifolds homeomorphic?

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry — Henry Wilton @ 7:46 am

A preprint of Lins and Lins appeared on the arXiv today, posing a challenge [LL].  In this post, I’m going to discuss that challenge, and describe a recent algorithm of Scott–Short [SS] which may point towards an answer.

The Lins–Lins challenge

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post).  But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

Lins-Lins manifolds

(I’ve taken the liberty of copying the diagrams from their paper.)

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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.

September 8, 2012

ICERM Fall 2013: Topology, geometry, and dynamics

I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics.   You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other.   The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.

There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:

  1. Exotic Geometric Structures. September 15-20, 2013.
  2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
  3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

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:

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