curious little things

I'm teaching a differential geometry course this semester and having fun with it. This is kind of over-the-top...

Recently a bunch of nice classic topology videos have been appearing in various locations. I thought I'd put together a little list of some of them.

Thurston's "Not Knot", Part 1 and Part 2. These videos describe some connections between knot theory and geometrization, in particular hyperbolic geometry.

Thurston's "Outside In" video. This video covers Thurston's technique for turning the sphere inside-out, the original proof of which goes back to Smale.

Thurston talks about his motivations and the timeline for geometrization here. There is a moment mid-way through the video where Thurston talks about when he thinks geometrization might be proven in general. This is about one year before Perelman's papers started appearing on the arXiv.

Milnor's lectures on Differential Topology. These took place at the Statler Auditorium at Cornell in 1965. Perhaps this is an early Cornell Topology Festival event? If any readers know I'd like to know. Video 1, Video 2, Video 3. The first two videos are basic manifold theory background, the 3rd video gets at his exotic sphere constructions. These are fairly light videos, similar to Milnor's "Topology from a differentiable viewpoint" lecture notes.

Kirby's Edinburgh Lectures on topological manifolds and smoothing theory.

Mike Freedman's lectures on 4-manifolds. This is a part of MPI's semester on 4-manifold theory.

I've been enumerating vertex-normal 3-manifolds in triangulated homotopy 4-spheres recently. I thought I'd put a list of the most commonly found 3-manifolds somewhere. This seems like a good a place as any.

In my search, what I do is run through all homotopy 4-spheres that can be triangulated (in the unordered delta complex / semisimplicial / Thurston sense) with 6 or less 4-dimensional simplices. There's millions of those, but from a sampling (of about 1000) of those I do a more or less random walk in Pachner space and generate a list of roughly 120,000 triangulated homotopy 4-spheres that are Pachner-equivalent -- roughly it's a little neighbourhood of the original 4-sphere in the Pachner graph. Typically I explore triangulations up to about 14 or 16 pentachora.

The 3-sphere is by far the most commonly found 3-manifold, since the link of any vertex is a normal 3-manifold. Next most common is connect-sums of S^2 x S^1. Roughly 8 or less summands is fairly typical when searching triangulations with 14 or 16 pentachora. S^3/Q8 where Q8 is the quaternionic group Q8 = \pm { 1,i,j,k } is fairly common as well, as well as that direct-sum up to three copies of S^1 x S^2. The oriented S^1 bundle over a Klein bottle is fairly common. L_{3,1} connect sum its mirror image is fairly common as well, this one sometimes appears connect sum up to about 4 copies of S^1 x S^2. The Seifert-fibred manifolds over a torus with one singular fiber of order n/1 occurs quite often as well for n <= 3. These manifolds also tend to come with a few S^1 x S^2 summands.

Relatively rare but occuring manifolds tend to be Seifert-fibered manifolds over S^2 with three and four singular fibers. There are also a few hyperbolic manifolds that occur frequently, primarily the hyperbolic manifold that's the 0-surgery on the 2-component link 7a_6, giving in my `Embeddings of 3-manifolds...' preprint (with Ben Burton now, who wrote much of the code I'm using for these computations).

More rare still are manifolds with slightly more complicated JSJ-decompositions, things like the figure-8 knot complement with an identification on the torus cusp that turns it into a Klein bottle.

These manifolds with complicated JSJ-decompositions are what I'm most curious in at present. These manifolds tend to have fairly complicated surgery descriptions so traditional handle type constructions tend to be difficult to use for these manifolds.

One experimentally re-occuring theme that comes up is that if a manifold M # (S^1 x S^2) embeds in a homotopy 4-sphere, then it is `nearby' an embedding of M. Likely one is just the other with a 1-handle attachment but my software does not allow for checking this. I think it's reasonable to assume that this is fairly typical behaviour but it's interesting so far that there's been no deviations from it. The 4-manifolds literature does not have a theorem that would predict this -- some kind of generalized Dehn Lemma for 2-spheres in 3-manifolds embedded in 4-manifolds.

Next up I'll try exploring larger homotopy 4-spheres.

I've been kicking around a project, trying to discover which 3-manifolds embed in the 4-sphere for five years now. This preprint represents the state of the art. V5 should appear on Thursday, Sept. 27th. There's two big changes in the preprint from V4.

1) Andrew Donald read it, and found a mistake. The version of the paper he was working off was the arXiv version 4. In that paper, the mistake appears in section 5, item 37. The manifold is SFS[S^2 : 1/2, 1/2, 1/2, -5/3]. I miscomputed a characteristic link, resulting in a miscomputed mu invariant. So now (V5) the manifold is listed in Section 4, item 23, as Donald shows it embeds in S^4. Donald also found several other embeddings in his preprint.

2) Ben Burton and I have been finding embeddings of 3-manifolds in 4-spheres via a new technique. What we're doing is the analogue of normal surface theory in triangulated 3-manifolds. But we're computing normal 3-manifolds in triangulated 4-manifolds. The idea is we take a triangulation of a (homotopy) S^4, and search for 3-manifolds that appear to be linear in each top-dimensional simplex and transverse to the triangulation. This is an integer linear programming problem. The setup in 4-manifold theory is quite similar to the 3-manifold theory formalism. In 3-manifold theory you have normal triangles and quadrilaterals. Normal 3-manifolds in triangulated 4-manifolds consist of tetrahedra and "prisms" -- things that look like triangles cross an interval. Interestingly, Ben and I recover most of Andrew Donald's embeddings this way, as well as most of the embeddings appearing in our paper that were constructed via other techniques. At present we've found two completely novel embeddings. These are 3-manifolds that have one incompressible torus, separating the 3-manifold into two torus knot complements. In one situation it's two trefoil complements, and the 3-manifold is a homology sphere. In the 2nd case it's a (3,5)-torus knot complement glued to a Seifert fibered manifold SFS[D^2 : 1/2, 1/2], and this 3-manifold has homology (Z_2)^2. See the preprint for details.

At present I've only verified that these embeddings are into homotopy 4-spheres. In the next few months I hope to confirm these are triangulations of the standard S^4. I'm also hopeful that we'll find plenty of other new and interesting 3-manifolds in the 4-sphere via these techniques. But the techniques are fairly heavy handed. As I write this, a cluster of 96 CPUs is busily hunting for 3-manifold embeddings around-the-clock.

A classical conjecture in knot theory says that when you have a connect-sum of knots, you obtain a minimal-crossing planar diagram for it by taking the minimal-crossing planar diagrams of the prime summands, and take the connect-sum of their diagrams. Said another way "crossing number of knots is additive". That got me to wondering, are there similar possibilities for other operations on knots and links, related to crossing number?

The above photo is a planar diagram for a two component link. There's 40 crossings in the diagram. If additivity of crossing numbers were to generalize to JSJ-decompositions, I suspect there should be no diagrams for this link with less than 40 crossings. What do you think?

More here.

The figure-8 knot is a lovely example of a knot in 3-dimensions. It's one of the simplest knots you can draw. The complement admits a hyperbolic structure of finite volume. And the hyperbolic structure is very nice -- it is the union of two regular ideal tetrahedra, moreover, the n-th face of tetrahedron 1 is glued to the n-th face of tetrahedron 2, with two of the gluing maps by right-handed 2pi/3 twists, and two by left-handed 2pi/3 twists. The complement of the figure-8 knot (thought of as a knot in the 3-sphere) fibers over the circle, and the fibre is a once-punctured torus. The monodromy is one of the simplest Anosov maps you can write down.

In a recent paper with Ben Burton and Jon Hillman, we find an analogous object in 4-dimensions. It's the complement of an embedded 2-sphere in the 4-sphere -- the knot is called a Cappell-Shaneson knot. So like the figure-8 knot, it fibers over the circle. It also has fiber a once-punctured product of circles, and the monodromy is a readily-described linear automorphism. The gluing instructions for this triangulation are a little less friendly to describe -- for those I suggest looking at the paper but they're reasonable.

The figure-8 triangulation has two edges, 4 triangles and two tetrahedra, and of course the one (ideal) vertex. The 4-manifold triangulation has similarly has one ideal vertex, but it only has one edge, 4 triangles, 5 tetrahedra and 2 pentachora. During the early drafts of the paper, I computed the edge link, it's the induced triangulation of the 2-sphere normal to the edge. It has 20 triangles. It's dual decomposition (depicted) has six hexagons and six squares.

A key point of interest about this paper is that we only prove that this triangulation is homeomorphic to the complement of the Cappell-Shaneson knot. We still hope to find a PL-equivalence, but that's a task for another day.

One of the things I hadn't noticed about Victoria until recently, is that it's easy to find snakes and some small lizards here in the summer months.

Photo by David Budney.

My father and I found this little one while walking around my neighbourhood. I haven't caught or identified any of the lizards, yet.

Some photos of my cousin working.

The most frequently asked question I get after a presentation in seminars is "how did you draw those pictures?"

For example, the main image in this slide

was created with x-fig, which is free software that comes with most distributions of Linux. x-fig allows relatively easy constructions of vector graphics, stored in postscript files. This is what it looks like when it's being created in x-fig.

The little square boxes are control points. That image consists only of "approximated spline" types in x-fig, but in varying widths and styles.

Most sketches start off as hand-sketches. I can't find the original hand-sketch the above was based on, but an earlier closely-related sketch is here.

In this case, I first created an image like the above hand sketch. I then re-created it in x-fig. I think I first used the ellipse tool but then created a duplicate ellipse out of splines, since you can't crop an ellipse in x-fig. I copied the image three times and modified the copies appropriately (mostly by rotation) to create the final image.

To ensure the splines are appropriately smooth and convincing I use the "zoom" feature. It helps to have a nice mouse and/or a big monitor to work with.

I also like to use the (La)TeX package dvips to have TeX insert appropriately typeset text into the images. So while I'm editing the image I tend to make frequent save/export-to-postscript cycles, together with recompiling the .pdf file from the .tex file to see how the final presentation works out.

A complaint I often hear is that x-fig is difficult to use. I think maybe x-fig is a little bit of a pain to use for the first couple of days, but once you get the hang of it, it's quite friendly. It does have some pesky input routines -- perhaps the "grouping" device is a little on the clumbsy side. I wrestled with x-fig the first week of writing my dissertation, but after that it's been mostly smooth-sailing since.

Sometimes I'm interested in more vivid life-like images. Like this

The above image was created in using a more elaborate process. Given a knot in the 3-sphere (or Euclidean 3-space) parametrized "nicely" I have some code that searches for things like quadrisecants and 6-tuples of points on the knot that sit on a common round circle. Once this data is computed, the surface of the knot is spit out as a triangulation. The circles and/or quadrisecants are also spit out. A white "backing wall" is inserted into the data. This data is packaged into a PoVRay source file, which is then rendered in PoVRay. I then take the resulting image, adjust the colour levels appropriately and add the text to the background, and it's done. So this can take some time! Modern computers make rendering this kind of image far more pleasant than say, in the mid 90's.

Explicit parametrizations for knots that are useful for such renderings can be non-trivial to construct. In principle you think it shouldn't be all that hard -- perhaps by the use of a constructive Stone-Weierstrass theorem applied to the derivative ? but there are practical matters, since knots are usually given by planar diagrams. The above parametrization came from a Fourier representation of the knot.

One of the things I find pretty neat about the splicing operad is how much goes into it. It's a relatively complicated structure and I'm a little surprised there aren't many similar, related structures in the literature.

I suspect one of the main reasons the splicing operad wasn't discovered earlier is that it isn't some variant on a common construction in one field. The motivation for the splicing operad construction was twofold: the action of the cubes operads on long embedding spaces. Moreover, this action does not see everything about the homotopy-type of embedding spaces. In dimension 3 splicing or the JSJ decomposition was the key other ingredient to the homotopy-type. That gave the desire to see splicing fit into a higher-algebraic framework.

This post is about how the idea of the splicing operad fails if one perturbs it just a little bit. An alternative title for this post could be Is there an operad of trivial links? I know I'm not the only one interested in such an operad.

As in my previous post, there is a splicing operad for every embedding space EC(j,M) where j is an integer and M is a compact manifold. When M is a point or a closed manifold this splicing operad isn't a big innovation. The splicing operad is mostly interesting when M is a manifold with non-empty boundary. It's particularly interesting to me when M a disc. At level k of the splicing operad for EC(j,M), an element is represented by a symbol L. L is a (k+2)-tuple L=(L_0,L_1,...,L_k,s), where L_0 is an element of EC(j,M), and the other L_i's are "hockey pucks", and s is a permutation of the set {1,...,k}. So given L in the splicing operad you can "forget" about L_0. That gives you an object (L_1,...,L_k,s) which, for lack of a better word, you could say is an object that belongs to a "space of trivial links". For several reasons you could hope for an operad structure on this "space of trivial links" making this map a map of operads. Unfortunately, that can't be done.

There a few good reasons for this. For one, the structure maps in the splicing operad make essential use of the L_0 part, even if all you care about is the pucks. But you could step further back, and ask what would happen if you simply got rid of all the L_0 parts and defined the structure maps like in the operad of overlapping cubes/discs. i.e. simple composition of functions -- no elaborate conjugations like in the splicing operad. So for example, given (L_1,...,L_k,s) and (J_1,...,J_l,t), the o_1-operation applied to these two would be defined to be (L_1 o J_1, ..., L_1 o J_l, L_2, ..., L_k, the corresponding permutation). This definition leads to problems, too. The big one being that the disjointness one requires for the "annular part" of the hockey pucks is not preserved under composition. It happens in very simple cases, for example when k=2, L_1=L_2=the identity "puck", taking only two pucks as input, and simply choosing them to have non-disjoint annular parts. So I'm talking about the operad structure map O(2)xO(1)xO(1) --> O(2).

My suspicion is that there is some natural higher algebraic structure for spaces of trivial links but it seems like the splicing operad isn't quite the right direction to go. In particular I think if one wants any kind of efficient description of the homotopy and homology of spaces of trivial links, you're going to want to give these spaces a suitable higher algebraic structure. Exactly what that is, I'm not so sure. On any type of space of trivial links in R^n there is an action of the operad of framed n-discs, but this gives a relatively shallow perspective on the homotopy-type since pulling one component of a link through another is not represented by this action. Presumably there is some better way to study these spaces? That's one hope.