Surgery diagrams from double branched covers

Here’s a neat trick I learned a few weeks ago from Jonathan Bloom at Columbia University.  Let’s say you have a description of a 3-manifold  as the double branched cover of a link  in the 3-sphere, but you would like a description of M as a Dehn surgery on a link in the 3-sphere.  (This sort of thing comes up, for example when comparing different types of Floer and Khovanov homologies.)  If  were the unknot then  would be the 3-sphere and everything is easy.  If  is more complicated than the unknot, then things are a bit more difficult, but here’s what we can do:

Massive, online collaboration – A case study.

For the last six weeks or so, Tim Gowers and Terry Tao have been using their blogs (here and here) to carry out a very interesting experiment in large scale collaboration.  Gowers chose an open problem and created a system for organizing comments that allowed anyone to contribute to the discussion.  Now they seem to have solved the problem and Gowers has posted a blog entry with his thoughts on how the project went and what he learned from the experience.  Following the whole project requires a lot of combinatorics background, and would probably require a good amount of time even for an expert (there are over 1000, mostly long, comments in the discussion.)  But I think anyone who’s interested in using the web to enhance math research should take a look at his discussion of the experiment.

Update: Advanced knots in Heegaard surfaces

A couple of months ago I wrote about advanced positions of knots in Heegaard surfaces:  Embeddings of a given knot into a Heegaard surface (usually in the 3-sphere) such that the knot is not primitive in either of the two handlebodies that make up the Heegaard splitting.  (Recall that a knot in the boundary of a handlebody is primitive if it intersects the boundary of a properly embedded disk in a single point.)  Today, I want to mention two updates on the subject.

Thin position and graphs

Today I want to write about an application of axiomatic/iterated thin position that connects thin position very strongly to the theory of normal/almost normal surfaces.  This is the application that actually got me started thinking about thin position, and which I’m writing up in paper-form as I write these posts.    This will probably be my last post about thin position for a while, so before beginning my next post on axiomatic thin position, here’s a quick synopsis of what I’ve covered so far:

Building non-Haken 3-manifolds from geometric pieces.

I asked Ian Biringer to write a few words about his recent work with Juan Souto [1], showing that for any given bound, there are  finitely many “pieces” from which all non-Haken manifolds with bounded injectivity radius can be constructed.  Here’s what he wrote:

The project evolved from some work that Ian Agol did for closed hyperbolic 3-manifolds that have rank = 2 and injectivity radius bounded below by some constant (The rank of a 3-manifold will always here refer to the rank, or minimal number of generators, of its fundamental group.) In any such manifold, there is actually a base point at which the fundamental group can be generated by two loops with length bounded above by some constant depending only on Agol showed that if one takes a Gromov Hausdorff limit of a sequence of such manifolds using the base points above, the limit will be a genus 2 handlebody with a degenerate end. In fact, pulling some compact core of the limit back into the approximating manifolds gives a Heegaard splitting as long as you’re far enough down in the sequence. After some formalism using the fact that there are only countably many closed hyperbolic 3-manifolds, this proves that there are only finitely many closed hyperbolic 3-manifolds with rank = 2 and injectivity radius bounded below that do not have Heegaard genus 2.  (Editor’s note: Ian never wrote down this proof formally.)

Thin position and essential disks/spheres

I want to continue my series of posts about axiomatic thin position by considering what it means for a path in the complex of surfaces S(M) (or any complex satisfying the same axioms) to represent an index one vertex in the complex of paths in S.   (If you’re just tuning in now, you may want to first look at parts one, two, three and four.)  Recall from the last post that a vertex in P(S) represents an equivalence class of oriented paths in S, i.e. a path in S, modulo face slides that do not change the complexity of the path.  Edges in P(S) correspond to weak reductions/vertical face slides of paths in S.  Faces in P(S) come from pairs of weak reductions that commute with each other.  (Since this is a blog post, I will be vague about what that last sentence means.)

The complex of paths P(S) satisfies the three axioms that I defined in my second post so when we consider slender paths in this complex, we will be interested in index-zero and index-one vertices.  Last time, I pointed out that index zero vertices in P(S) correspond to slender paths in S and that a path in S corresponding to an index-one vertex in P(S) will have exactly one weakly reducible maximum and the rest will be strongly irreducible.  Today, I’d like to determine what we can say about the one weakly reducible maximum.