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.)

The next big thing in quantum topology?

The place to be in May for a quantum topologist is Vietnam. After some wonderful-sounding mini-courses in Hanoi, the party with move to Nha Trang (dream place to visit) for a quantum topology conference.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is -Efficient triangulations and the index of a cusped hyperbolic -manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!).

New connection between geometric and quantum realms

A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970

This is big news!!

Wise’s CBMS Lecture Notes on Cube Complexes

This fall, the topology group at OSU is reading through Dani Wise’s lecture notes on cube complexes, based on his series of talks at the CBMS-NSF conference back in 2011.  Henry Wilton and Daniel Moskovich have written on this blog about Wise’s work and its role in the proof of the Virtual Haken Conjecture. This is just a quick note to say how  impressed I’ve been with the lecture notes. They start from the very beginning, include a lot of good examples and have proved to be very accessible for all of us non-experts (which includes me).

It’s just too bad that Wise’s notes are no longer available on the conference web page. (Now that a paper copy is available from the AMS, the PDF file has been replaced with a note saying that the editor insisted they be taken down.)  You can still e-mail Dani Wise to request a copy, but I expect that some people (such as beginning graduate students) might be reluctant to e-mail someone they don’t know like this. I can assure you, he was very gracious when I asked him for a copy and seems to be very eager to distribute the notes widely. But, if you have any thoughts on how the PDF file could be distributed more efficiently, I would love to hear about it in the comments.

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.

The virtual Haken conjecture

Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.

Agol’s work on the Virtual Haken Conjecture

Over at Geometry and the Imagination, Danny Calegari is reporting live from Paris on talks by Agol and Manning on the announced proof of the VHC:  Part I, Part II, Part III.