This post comes along with a more detailed PDF version.

Kent Orr at Indiana University taught me a gorgeous, intrinsic (but not well-known) homological proof due to Pierre Vogel that the Alexander module of a knot is annihilated by an Alexander polynomial. The only fact it uses about a knot is that its complement is a homology circle. The starting point is:

Low-dimensional topologist’s definition of Cohn localization:
Let be rings with unity, let be a
ring homomorphism, and let be the chain complex

be a chain complex such that is a finitely-generated free module for all . The Cohn localization of is the initial ring with the property

An explanation of relevant terminology, alongside a proof that this definition of the Cohn localization is equivalent to the algebraist’s definition, is to be found in the PDF version of this post.

The goal of this post is to show that is -torsion, where denotes the complement of a knot , and and is the augmentation map, sending to .

Let be a meridian of and let be the infinite cyclic cover of in which (a loop) lifts to (an infinite line). Consider the chain complex of free modules. Then
by Alexander duality or Mayer-Vietoris (the point here is that
is a homology circle generated by ). The following purely algebraic fact is given without proof.

Fact:
The Cohn localization of is . It is a flat module.

Now by the low-dimensional topologist’s definition of Cohn localization

Since is contractible, we get that is torsion, which is QED.

Simple, elegant, and general!

There is a point I’m uncomfortable with. You see, is strictly larger than the set of all Alexander polynomials. This is not surprising, because the proof is purely homological, and thus works for CW-complexes which may not be manifolds, and where Poincare duality may not hold. But it is still disturbing.

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?

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.

However, it appears that their analysis missed a case.  The gap in their proof and a class of examples that fall outside the original classification were discovered by John Berge, using double-primitive knots.  The present preprint [2] by Berge and Scharlemann presents these examples and repairs the proof.  There are two interesting properties of the new examples.  First, some of them have distance three (the originals only had distance two) and second, it is not clear that the new pairs become isotopic after a single stabilization (it was for the originals).  For all the examples, new and old, it is unknown when the pair of Heegaard surfaces are actually non-isotopic.  (They only prove that if a pair of genus two splittings are not isotopic then they fall into one of these cases.)  If one could show that the distance three Heegaard splittings are in fact distinct, this would be the first example of a distance three Heegaard splitting that is not unique.  (All the examples of distinct Heegaard splittings where the distance is known have distance two, though there are other examples where the distance has not been calculated.)  If one could show that the new examples do, in fact, require more than one stabilization then this would be the first example where the stable genus of two Heegaard splittings is equal to the sum of their genera.  Either of these results would be very interesting.

Two more preprints that caught my attention are a pair of papers by Scott Taylor and Maggy Tomova [3],[4] that generalize Hayashi-Shimokawa thin position and Tomova’s c-disk thin position to graphs.  They use this to consider the problem of leveling an edge of a graph into a Heegaard surface.  This is related to a paper by Goda, Scharlemann and Thompson showing that given a tunnel-number-one knot in minimal bridge position in the 3-sphere, the unknotting tunnel can be isotoped while fixing the knot, so that the tunnel is contained in the bridge sphere [5].  (I wrote about this previously.)  Cho and McCullough, for example, use this result to bound the bridge number of a knot in terms of its position in the tree of unknotting tunnels [6].  The proof by Goda, Scharlemann and Thompson uses the classical version of thin position in which every level surface is a single sphere.  The more recently developed forms of thin position, in which spheres can be compressed and cut-compressed, have proven much more natural, as seen for example in [7].  So I have high hopes that this new approach will prove useful for applications such as, for example, leveling an unknotting tunnel with respect to a higher genus Heegaard surface for the 3-sphere.

Finally, I wanted to point out that Brandy Guntel has posted a preprint [8] with examples of knots that have distinct double-primitive and primitive/Seifert positions with the same surface slope in a genus two Heegaard splitting of the 3-sphere.  (I wrote about some of her examples previously.)  The knots are all twisted torus knots and the (computationally intense) argument is based on calculating the action of the Goeritz group on the homology of the Heegaard surface.

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.

If you tried SnapPy under Snow Leopard, leave a note in the comments with the results — neither Marc nor I have upgraded yet…

(I realize this is a bit lame for post, but I’m under the weather and it’s all I got.)

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.

The first set is for communicating images while talking on the phone (or Skype, etc.)

• Web whiteboards allow several people to sketch things with the mouse so that all can see the results. The one I use, at Jordan’s suggestion, is Vyew. Personally, the problem is that I can’t draw fluently with a mouse. I have a Wacom tablet, but I don’t use it enough to get past the look-at-the-screen-not-my-hand issue. These things work much better, I’m told, with a tablet computer or a Cintiq, but these aren’t cheap and I’ve never sprung for one.
• Do a video chat, draw on a piece of paper and hold it up to the camera. Tara Brendle says that this work ok.
• In Dublin, Noel Brady mentioned eBeam, which turns an ordinary whiteboard into a giant electronic slate. (They also have a version that works in conjunction with a projector.) Not cheap, starting at about $800.

For emailing pictures, one can always draw on a sheet of paper and then scan it in. This works pretty well; my department’s photocopiers have a neat feature where they’re email you a PDF instead of making a paper copy. The results look good and the file size is very small, so much so that I’ve started scanning all my lecture notes.

Finally, if you use a Mac, you need Skitch. It’s a little program that allows you to effortlessly grab anything off the screen, annotate it with some simple drawing tools, and export as PNG, PDF etc. Rather that try to describe why it’s just so useful, watch the 3 minute demo.

The topology groups at Iowa and LSU apparently have a regular electronically joint seminar, using some software created by the NSF. Some day I’ll have to see it in action.

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.

Indeed, in the modern age, I’m not sure why people write conventionally published textbooks at the advanced undergraduate or graduate level. You can just throw a PDF on the web, and people can use e.g. Lulu to get printed copies (though within in a decade, I’d guess students will all prefer to have things on their Kindle). I suspect that many more people would use a text distributed for free, and the ego boast from this would more than make up for the very modest loss in revenue from book sales. (Note the dominance of Hatcher’s Algebraic Topology, which is freely downloadable, though of course that’s a fantastic book on it’s own merits, and so it might have achieved that status anyway.)

Of course, a key feature of the open source approach is allowing others to create their own versions of one’s book at will, which does lead to a certain loss of control that some might find unappealing. Still, I think it could be very good for the community if people went this route.

(As a side note, like all good math bloggers Emmanuel occasionally posts on 3-manifolds.)

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.

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…

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:

Kirby I

Stabilization

Kirby II

Handle-slide

Kirby III

Circumcision

If then it is easy to show that Kirby III can be realized using Kirby I and Kirby II moves.
Kirby calculus is the art of manipulating framed links using Kirby moves. By the way, Akbulut has an interesting objection to the term “Kirby calculus”.
The Kirby Theorem’s importance in quantum topology is that it provides a way of proving that some map from surgery presentations of 3-manifolds to something simpler (integers, rational numbers, polynomials…) is a 3-manifold invariant. Typically, we know the value of for some basic 3-manifold (usually ), and we have some formula for how changed under surgery around a framed link (in terms of some variant of the linking matrix of , for example). We verify that for any other framed link related by Kirby moves, the values of on and on coincide. Such proofs appear in quantum topology papers over and over and over again…
The problem with the Kirby Theorem is that Kirby moves are quite violent, in that they make a mess of the link. Say you have a surgery presentation for a 3-manifold which has some nice property (such as: has linking matrix with unit determinant, belongs to the Johnson kernel of the mapping class group of some Heegard splitting, has unknotted components, or whatever), Kirby moves will lose these properties. If one wants to find a set of moves between surgery presentations with nice properly then, you have to do something extra. It’s also unknown how to determine whether or not two surgery presentations present the same 3-manifold.
I’ve been interested for a long time in understanding the proofs of Kirby Theorem in order to prove a relative version, where the nice properly I want to preserve is the existence of a representation of the fundamental group of the complement of in onto some fixed group. Here’s a micro-outline of the main proofs of the Kirby Theorem.
The Kirby Theorem was proved by Rob Kirby [1]. The idea of his proof was to view 3-manifolds as boundaries of 4-dimensional handlebodies (a 4-ball with 2-handles attached), and, interpreting Cerf theory using handles, to show how two Morse functions on the same manifold can be related to one another. The problem is that Cerf theory is a awful huge machine, and the proof is too difficult more most people to understand (certainly I can’t make head or fishtail of it, pardon the stupid joke).
Next, Fenn and Rourke (and later Roberts [2]) substantially simplified Kirby’s proof. Modulo a few Cerf theory “black boxes”, a useful insight which emerges from their proof is that the key point is that in their approach one needs a certain class in to vanish, where denotes the fundamental group of the cobordism between and . The class itself seems impossible to calculate directly. In every attempt to generalize the Kirby theorem which I have seen, itself vanishes, and the problem is side-stepped. In the setting I am interested in, I have no real control over
Anyway, pretend you didn’t hear that, and let’s move forward. The Kirby Theorem is a 3-dimensional statement and should have a 3-dimensional proof. This was provided by Ning Lu in 1990 [3](Matveev and Polyak give a similar proof [4]), who showed that your favourite finite presentation of the mapping class group implies the Kirby Theorem (he chose Wajnryb’s presentation [5], but Gervais’s presentation [6] – which didn’t exist at the time of course- works just as well if not better) . It would be wonderful if one could prove the opposite implication, but remember… Kirby moves are tremendously violent. Handle-slides in particular play such mayhem with generators of the mapping class group that I cannot imagine using them to find a presentation of the MCG. This proof is beautiful because each step in it is completely elementary (Wajnryb’s presentation is a deep result, but it’s completely elementary). I can’t see how to generalize it at all though, because I don’t know how to find finite presentations for quotients of the MCG.
How else could you imagine proving the Kirby Theorem?

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:

1. MATH 518 Differentiable Manifolds I
2. MATH 519 Differentiable Manifolds II
3. MATH 521 Riemannian Geometry
4. MATH 522 Lie Groups and Lie Algebras I
5. MATH 523 Lie Groups and Lie Algebras II
6. MATH 524 Linear Analysis on Manifolds
7. MATH 525 Topology (really, this is basic Algebraic Topology)
8. MATH 526 Algebraic Topology (really, this is more advanced Algebraic Topology)
9. MATH 527 Homotopy Theory
10. MATH 533 Fiber Spaces and Char Classes
11. MATH 535 General Topology

In contrast, my last university had just 4 or 5 such classes, and that was with the quarter system, so that’s only about 3 semester’s worth of such classes!

So I’d be curious to know how many such classes are there at your university (or any other university that you’re familiar with, e.g. where you went to grad school). Please post your data in the comments!