Low Dimensional Topology

March 22, 2015

SnapPy 2.3 released

Marc Culler and I are pleased to announce version 2.3 of SnapPy. New features include:

  • Major improvements to the link and planar diagram component, including link simplification, random links, and better documentation.
  • Basic support for spun normal surfaces.
  • New extra features when used inside of Sage:
  • Better compatibility with OS X Yosemite and Windows 8.1.
  • Development changes:
    • Major source code reorganization/cleanup.
    • Source code repository moved to Bitbucket.
    • Python modules now hosted on PyPI, simplifying installation.

All available at the usual place.

March 9, 2015

Complex hyperbolic geometry of knot complements

Filed under: 3-manifolds,Hyperbolic geometry,Misc. — dmoskovich @ 3:41 am

This morning there was a paper which caught my eye:

Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold. (more…)

October 22, 2014

Understanding the anomaly

Filed under: 3-manifolds,Mapping class groups,Quantum topology — dmoskovich @ 11:48 am

I’ve recently been looking at the following paper in which 3+1-TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:

Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.

It makes me think a lot about just what the anomaly `actually means’… (more…)

June 12, 2014

A celebration of diagrammatic algebra

Filed under: 3-manifolds,Combinatorics,Misc.,Quantum topology — dmoskovich @ 5:24 am

Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:

K. Okazaki, The state sum invariant of 3–manifolds constructed from the E_6 linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it! (more…)

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.

(more…)

December 12, 2013

Banker finds a duplication in a 3-manifold table

Filed under: 3-manifolds,Triangulations — Ryan Budney @ 12:48 pm
Tags:

Daniel Moskovich recently wrote about the discovery by a lawyer of a duplication in the knot tables called the “Perko pair”.

Now a banker has found another duplicate in yet another table of 3-manifolds. This time it was Ben Burton, and the duplicate appears in the Hildebrand-Weeks cusped hyperbolic census.

(more…)

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
(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
(more…)

July 11, 2013

Smooth proof of Reidemeister-Singer

Every construction I know of 3-manifold invariants from Heegaard splittings factors through the Reidemeister-Singer Theorem:

Reidemeister-Singer Theorem: For any two Heegaard splittings H_1 and H_2 of a 3-manifold M, there exists a third Heegaard splitting H which is a stabilization of both.

This theorem is definitely part of the big story in 3-manifold topology, and is usually proven in the PL category, as for example in Nikolai Saveliev’s Lectures on the Topology of 3-manifolds. There is another nice PL proof due to Craggs, Proc. Amer. Math. Soc. 57, n 1 (1976), 143-147.

I think of a Heegaard splitting as being intrinsically a smooth topology construction (a level set of a Morse function), and so I would really like the proof of Reidemeister-Singer to live in the smooth category. I think that there should be consistent smooth and PL stories of 3-manifold topology living side by side. In the 1970’s, Bonahon wrote a smooth proof of Reidemeister-Singer, which uses Cerf Theory (naturally, because we’re investigating paths between Morse functions). Unfortunately, Bonahon’s proof was never published, and it is lost.

A year ago (but I only saw it this morning), François Laudenbach posted a smooth proof of Reidemeister-Singer to arXiv: http://arxiv.org/abs/1202.1130. I think that this is wonderful! There are too few papers like this- there is insufficient incentive to streamline the storylines of foundations. I am very happy to have found this proof, and I want such a proof to be a part of my smooth 3-manifold topology foundations.

Edit: Thanks to George Mossessian and to Ryan Budney, who point out in the comments that Jesse Johnson proved Reidemeister-Singer using Rubinstein and Scharlemann’s sweep-outs, which involves singularity theory which is much less sophisticated that Cerf Theory: http://front.math.ucdavis.edu/0705.3712
Perhaps that should be the “smooth proof from The Book” (or the “proof from The Smooth Book”)!

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