When are two hyperbolic 3-manifolds homeomorphic?

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?’

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

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

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.

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.

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:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

SnapPy 1.6: Now with more links and precision!

Marc Culler and I have released version 1.6 of SnapPy.  There are two sets of new features:

1. Creating links formulaically, e.g. via combining tangles algebraically.  See our page of examples.
2. Arbitrary precision calculation of certain things (e.g. tetrahedra shapes) and finding associated number fields, a la Snap.   Very basic at this point compared to what Snap can to, but here are examples of what we have so far.  To use this, you need to install SnapPy in Sage, which should be easy.

Regina on Windows and MacOS

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.

The geometry of neural networks

For the next few parts in my continuing series on the geometry/topology of high dimensional data, I plan to write about some of the classical approaches and algorithms for analyzing large data sets, with an emphasis on the explicit and implicit geometric/topological aspects.  I’ve chosen topics based on what I’ve heard/read most about, so I hope that these will be relatively representative topics, but I make no guarantees. Later, I’ll switch to recent approaches that explicitly employ ideas from topology. I’ll start, in this post, with artificial neural netorks, which were one of the earliest approaches to machine learning/artificial intelligence. This algorithm is a good representative of a gradient method, i.e. an algorithm that iteratively improves a model for the data by following the gradient of some function.

Kuperberg proves that knottedness is in NP

Gil Kalai, my old Graph Theory professor at Hebrew University, and a great mathematical inspiration, who won the Rothschild Prize a few weeks ago (congratulations Gil!), wrote a very nice blog post about another massive recent result in low dimensional topology.

Big Data and the Topologist

As I mentioned in my previous post, I plan to write a series of posts about study of large data sets, both the ways that high dimensional data has traditionally been studied and the topology that has recently been applied to this area. For anyone who has experience thinking about abstract geometric objects (as I assume most of the readers of this blog do) the concepts should seem pretty straightforward, and the difficulty is mostly in translation. So I will start with a post that focusses on defining terms. (Update: I’ve started a second blog The Shape of Data to look into these topics in more detail.)

ICERM program: Low-dimensional Topology, Geometry, and Dynamics

ICERM is a new math institute at Brown focusing on computational and experimental research in mathematics. As one of the conspirators behind this, I’m pleased to say that the semester program for Fall 2013 will be on low-dimensional topology, geometry, and dynamics. There will be three associated week-long workshops that are still being planned, but the basic topics are listed on the program web page.