A few days ago, two people named Joshua (one Howie and one Greene) independently posted to arXiv a similar solution to an old question of Ralph Fox:
Question: What is an alternating knot?
The preprints are:
This post will briefly introduce the problem; I look forward to reading the solutions themselves! (more…)
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.
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.
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.
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.
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!). (more…)
This past week, Ciprian Manolescu posted a preprint on ArXiv proving (allegedly- I haven’t read the paper beyond the introduction) that the Triangulation Conjecture is false.
-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture.
This is big news. I feel it’s the last nail in the coffin of the Hauptvermutung. I’d like to tell you a little bit about the conjecture, and about Manolescu’s strategy, and what it has to do with low dimensional topology. (more…)
A few posts back, I defined normal loops in the triangulation of a surface and said I would use this idea to define train tracks on a surface. The key property of normal loops is that the normal arcs form parallel families and we can encode the topology of the curve by keeping track of how many parallel arcs are in each family. Train tracks encode loops in a surface in a very similar way. A train track is a union of bands in the surface (disks parameterized as ) with disjoint interiors, but that fit together along their horizontal sides. In other words, the top and bottom edges of each band are contained in the union of the horizontal edges of other bands. A picture of this is shown below the fold.
I plan to write a post or two about normal surfaces and branched surfaces in three-dimensional manifolds, but I want to warm up first, with two posts about the two-dimensional analogues of these objects. Train tracks play a huge role in the approach to the topology of surfaces initiated by Nielsen and Thurston, for understanding mapping class groups, Teichmuller space, laminations, etc. They organize the set of isotopy classes of simple closed curves in a surface in a way that allows one to take limits of infinite sequences of loops. (The limits are called projective measured laminations.) In this post and the next, I will discuss train tracks from a rather unusual perspective, via normal loops in a triangulation of the given surface.
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:
- Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert: Gluing equations for PGL(n,C)-representations of 3-manifolds.
- 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.