Berge Tangle

Here’s a tangle in S^2 x I.

One may show that it’s not equivalent to the trivial product tangle.

Here’s another view that shows off a 3-fold symmetry.

We’re gonna fill the “inner” sphere with a trivial tangle in three different ways.

These are each trivial (2-strand tangles in the ball) because the arcs can be pushed onto the boundary, fixing their endpoints.

With the fixed endpoints, they are what we call rational tangles. And we can see that the three are “mutually distance 1”.

These filling tangles also respect that 3-fold symmetry.

So here’s the three fillings…

And of course these three filled tangles are related by that 3-fold symmetry.

The deformation retraction of Bing’s house.

Last time we looked at how Bing’s house with two rooms is contractible. Been so long you might not remember, but I promised to show how it deformation retracts down to a point.

I found it tricky to satisfactorily sketch the full thing, but what I’ve got should do. I’ll show off the key move here, give one collection of pics of the entire deformation, and then point y’all to the Flickr for other viewing angles. If any of y’all other modelers/illustrators/mathematicians make a slicker version, let me know. I’d be happy to share it here.

Okay….

So what makes deformation retractions of 2-complexes easy — or at least gives you a place to start — is the existence of a “free edge”, an edge of the 1-complex of valence 1, a 1-cell run along just once by the 2-cells. Bing’s house doesn’t have any. The edges of its 1-complex shown in black (as in the previous post, though I’ve drawn the house blue and more rounded here) all have valence 3.

Let’s focus on those little loops bounding disks that form interior walls upstairs and downstairs. Here’s an upstairs one. If we were just contracting, we could begin something like this:

Smush the disk down to make a fissure and then pull it apart. Now we see a free edge and can start pushing it in. That’d be great for the deformation retraction… but for a deformation retraction, everything has to stay within the original complex. The pulling apart (well, the smushing too) takes it off its original structure.

Since a deformation retraction is a homotopy, we can retain the memory of the topology of the house as we start squishing it through its own walls. So when we smush the disk, we can get as much slack from the tubes to its sides to just pull it down through the disk. I’ll show this slightly separated like a fissure again, but the sides of the new cleft should actually occupy the same space where the original disk was. Then keeping these in along the disk we can start pushing in a hole… the image of the house under the homotopy at this stage now has a free edge.

Here’s the same sequence drawn on top of the original complex (left) and within a thin neighborhood of the original complex (right).

So we do this to both disks to get some free edges and then we start pushing.

Bing’s House

So there’s this example of a 2-complex that’s contractible, but not obviously so. Well actually, once you see it, it’s not too hard to see. Bing’s house with two rooms.

Blah. It’s not so apparent what’s going on. It’s a 2-complex, so let’s draw in the relevant 1-complex.

You can see two vertices and four edges. The surfaces of this 2-complex are all disks, and they make threefold incidences to the edges. The two loop edges bound disks, but they don’t show up since they’re the same translucent color as everything else. And all the corners can be somewhat misleading… Here’s a slicker picture with those two disks colored.

There has recently been a few words about it at MathOverflow where it’s pointed out that the contractiblity of Bing’s house is explained in Hatcher’s text and Cohen’s text. In this post, let’s see how this contraction works.

Hopf Stabilization

Here are a few pictures suggesting the change a Hopf stabilization makes to a fibered knot. Thought I’d share.

Basically I took 10 fibers, sliced them at angle, and made half of the slice translucent. It gives a glimpse of the dynamics involved.

Of course I had to play around with the colorings and renderings.

Rendering with glass looks neat, but it kinda distorts the innards a bit much.

There are a few more on the Flickr account. Just click on any of the above pictures to go there.

A (reverse) rational circle?

In this week’s This Week’s Find in Mathematical Physics (Week 286) Baez describes a construction of a rational homotopy circle. Thought it would be interesting to take a look at a the beginnings of the simplest “huge nightmarish space”. Edit: But as Josh clued me in, this isn’t exactly it.

I’ll mainly just show some pictures and direct you to Baez’s post for details. His construction is just above the pirates. In red, orange, and green are the first, second, and third cylinders. This is an abstract space, so the particular immersions shown are rather irrelevant. It’s the attaching that matters. Edit: The attachings shown here are reversed from the ones Baez describes.

Cabling a knot’s surface

Cabling a knot isn’t so tricky to imagine. Let’s just consider a tubular neighborhood of the knot.

Glue the top to the bottom. On the left we have the knot, on the right we have a (3,1)-cable… using the straight vertical framing.

Here’s a (3,5)-cable.

It’s not too bad to think about how a Seifert surface extends across the cable.
For a (p,q)-cable, we’ll take p parallel copies of the Seifert surface outside the tubular neighborhood and q copies of the meridional disk in the tubular neighborhood. The signs of p and q tell you the orientations you want on these pieces. Then you attach them together with |pq| twisted bands, twisted in the appropriate direction.

Let’s do this with the (3,1)-cable.

That pretzel knot

That pretzel knot P(-2,3,7) is a mischievous fella.

(Edit: Even more mischievous than I had reckoned for! Big thanks to Hyun-Jong Song for pointing out that I had the 18-fold cover wrong…. and the 19-fold cover too! The braids were correct in my notes, but they sure weren’t entered into KnotPlot correctly. Then I noticed the first picture of that alleged pretzel knot was actually of P(2,3,7) for cryin’ out loud!

I’ve redone the pictures correctly now, I hope. It’s somewhat reassuring that the relaxed versions of the covers now suggest the right symmetries for the lens space covers. For fun I’ve added some animations of these spinning. Braid words for the covers are at the end of the post as are pics of how they were derived.)

One of its famous tricks is that both 18 and 19 surgeries yield lens spaces. Since lens spaces are covered by the 3-sphere, the associated knots in these lens spaces lift to knots in the 3-sphere.

Starting from the grid number one descriptions of these associated knots in their lens spaces we can obtain grid diagrams (of grid numbers 18 and 19) for the lifts of these two knots. From grid diagrams we obtain braid descriptions that are more easily thrown into KnotPlot. We then let KnotPlot do its thing to obtain some “relaxed” pictures.

Here’s the 18-fold cover as the input closed braid with some views of its relaxation.

And here’s the 19-fold cover as the input closed braid with some views of its relaxation.

Note: I’m making no claims about orientations. Maybe either or both of the braids should have been mirrored.

Contact Heegaard Splittings

It was asked how one might see the contact Heegaard splitting associated to the JVHM open book on T3. Two pages of an open book form a Heegaard surface that is convex with respect to the induced contact structure and the binding is the dividing set.

(Recall that the presentation being used for T3 is a hexagonal prism with opposite sides identified.)

We can squish it down to one side to make one of the handlebodies more apparent. The binding goes to the red dividing curves.

Squishing it down to the other side would’ve given the same picture as this last one, but with a half rotation around the horizontal hexagon.

So what makes this a contact Heegaard splitting rather than just a splitting with convex Heegaard surface?

Fibered Cable

I was looking at the fibration of the (2,1)-cable of the core of a solid torus the other day…

Click on it for a larger version.

Here‘s a set showing the individual pages.

Two Balls

I used SketchUp for parts of a general audience talk last spring. Here’s one where I tried conveying the idea that the 3-sphere can be viewed as the one-point compactification of ordinary 3-space and as the union of two balls.

Of course I first walked the audience through the analogous constructions of lower dimensional spheres, but this one’s more fun. The animated gif above doesn’t capture it that well. Download the model and play with it yourself.