Which knotted objects are worthy of study?

Some people seem to rejoice in knotiness. To non-topologists, it’s not clear why anyone would care about even a plain old knot in (or a long knot), but to us it’s the most natural thing in the world. To them it would seem to specific, too specialized, not really interesting; but we know that they are wrong, right?
But what then, about links? High dimensional knots? Tangles? Braids? High dimensional links? Homotopy links? I’m sure we were a bit skeptical about the usefulness of these when we first saw them, but now we can just about accept them.
What about the next step? Knots and links in arbitrary manifolds? Singular knots? And then what about virtual knots? free knots? coloured knots? knotted trivalent graphs? What about these new objects of study like knotted handlebodies? Turaev’s topology of words (knotted words)?
How does one decide such a topic is interesting… why and when is extending a result about links in the 3-sphere to higher dimensional stuff or stuff in strange manifolds interesting? How does one become interested in it?
I know that I’m still a bit skeptical about virtual knots, for instance. But I’ve come to accept knotted trivalent graphs as natural… for rather strange reasons. How about all of you?

Genus two Heegaard splittings

I’ve been distracted away from blogging in the last few months, but there have been some recent additions to the arXiv that I couldn’t resist writing about.  The most recent reconsiders a paper by Rubinstein and Scharlemann [1] about genus two Heegaard splittings.  Rubinstein and Scharlemann applied their double sweep-out/graphic method to show that two distinct Heegaard splittings of the same 3-manifold can be made to intersect in a relatively simple manner, which allows them to characterize how the two Heegaard splittings are related.  In particular, the hyper-elliptic involutions on the 3-manifold determined by the two Heegaard surfaces  commute.

SnapPy 1.0.1 released

Marc and I have just uploaded a version 1.0.1 of SnapPy, which fixes bugs pointed out by Ryan Budney, Neil Hoffman, and Matthias Goerner (who even gave the solution!), and some others as well. If you already have SnapPy, you can upgrade it exactly the same why you installed it.

Communicating Topology in the 21st Century

Because of the importance of pictures in low-dimensional topology, communicating electronically with with collaborators, students, etc., has some special challenges. (Not that other mathematicians have it easy — I’d hate to have send lots and lots of equations via email.)

Here’s some useful tools/ideas for dealing with this, some of which I use myself, and others which I’ve only heard about.

An open source mathematics book

Number theorist Emmanuel Kowalski has an interesting post about a truly open source math book on algebraic stacks. Not only can you download the entire 1302(?!) page book as a PDF file, you can get the complete LaTeX source files, and the whole thing is kept under version control so people can submit changes, etc.

I’ve been thinking that a similar approach would be good for textbooks. When I teach a course, I’m often frustrated by being unable to find a text that has everything I need. Or I do find such a text, but it’s poorly written in places, or aimed too high or low for my particular students. Or maybe there are theorems in the text that I’d like to assign as homework instead of lecturing on them. In such situations, it would be great if there was a whole collection of open-source textbooks that I could cut and paste from, massage the result a bit, and end up with something closer to the “perfect” text for a course.

Hamilton Geometry and Topology Workshop

I’m currently in Dublin for the Hamilton Geometry and Topology Workshop. The theme this year is “Computational and Algorithmic Topology”, so I spoke yesterday (as the very first speaker!) on “Practical solutions to hard problem in 3-dimensional topology”. You can view my slides here.

Pseudo-Anosov automorphisms and curves over finite fields

During a recent visit, number theorist Jordan Ellenberg told me about a “time-worn analogy” between

(a) A pseudo-Anosov homeomorphism acting on a surface.

(b) The Frobenius automorphism of a smooth algebraic curve .

Jordan has two very interesting posts on this subject, one on what the dilatation should be in case (b) and a recent one where he discusses the finite field analogue of the following question related to the Virtual Haken Conjecture:

Conjecture: A hyperbolic 3-manifold which fibers over the circle has a finite cover with .

As I noted earlier, this is known when the fiber has genus two, or more broadly if the monodromy is hyperelliptic. Intriguingly, Jordan explains the analogous conjecture in the context of (b) is also known in exactly this case…

Some thoughts on the Kirby Theorem

I don’t think it would be too controversial to assert that the Kirby Theorem is an important theorem in low dimensional topology. Given a 3-manifold and an framed link (by “framed” let’s mean “integer framed”), let denote the 3-manifold obtained from by surgery around . The Kirby Theorem states that, given two framed links , the 3-manifolds and are homeomorphic if and only if and are related by a finite sequence of the following local moves:

Reader survey: graduate courses in topology and geometry

In my department, we’re considering whether we have too many basic graduate courses, by which I mean courses with (mostly) fixed syllabi aimed at first and second year graduate students, as opposed to advanced topics courses which never cover the same thing twice. In geometry/topology, we have not less than 11 such one-semester courses, of which two arguably belong more to algebra:

Topology fun and games

Jeff Weeks, the author of SnapPea, has written some new educational games this year that are highly worth trying out:

• A new version of his classic Torus Games for iPhone and iTouch.
• A new set of Hyperbolic Games, including a maze, pool (very challenging), and soduku. (For OS X and Windows)