Curves on a Klein Bottle

•October 22, 2010 • 7 Comments

Perhaps you’ve seen a Klein bottle before.
Klein Bottle000

But perhaps you’ve not counted the curves on the Klein bottle before. I’m talking about unoriented essential simple closed curves, considered up to isotopy. How many are there?

On the torus (the orientable double cover of the Klein bottle) there are infinitely many such curves, parametrized by the rational numbers and 1/0. However, on the Klein bottle there are…. not so many. I’m not going to give a proof here, and I’ll spoil the fun after these two pictures and the jump.

Klein Bottled5

Klein Bottlec9

Continue reading ‘Curves on a Klein Bottle’

Advertisements

That pretzel knot again

•September 2, 2010 • 4 Comments

Remember P(-2,3,7), that pretzel knot? (Yeah, big edit there. Yikes!)

Here it is again.
P-237onfiber-knotflat
Doesn’t quite look like it, I know. But I drew it that way because this is a trefoil with its fiber:
P-237onfiber-trefoil
and that pretzel knot sits nicely on the fiber:
P-237onfiber-knotonflatfiber

Because it sits on the fiber of a genus one fibered knot (and thus belongs to a family of Berge’s doubly primitive knots), it has a lens space Dehn surgery along the framing the fiber induces. Let’s get a glimpse of this.
Continue reading ‘That pretzel knot again’

Tomography

•August 20, 2010 • 2 Comments

One way of understanding a set of points in space is to pick a direction, slice it by planes orthogonal to that direction, and see what’s in the planes. In the real world, this practice is called tomography. One of the more delicious and enlightening uses of this that’s been of percolating through the net this summer has been the blog Inside Insides.

Here’s a fun one. (Edit: Just linked to that blog entry directly.)

Notice how the chosen direction of slicing from end to end reveals a symmetry of the guts of the watermelon that wouldn’t have been apparent if it were sliced from side to side.

Taking an object in 3D and slicing along a direction it gives a sequence of objects in 2D, like with this watermelon. Show them one after another and you’ve got a movie. We can do the same one dimension higher. Take an object in 4D and slice it along a direction to obtain a sequence of objects in 3D. And make that into a movie.

Like our knots of 1-spheres in 3D, there are knotted 2-spheres in 4D. So what happens when we do our tomography of knots? Slicing our usual knots and links in 3D, each slice gives points in the plane. For knotted 2-spheres in 4D, our slices are knots and links in 3D. Making movies, the slices of points in the plane and the slices of knots in 3D dance around and interact.

The other day Ayumu Inoue posted on the arXiv a description of some of these tomography movies of the “n-twist spun trefoils” which are knotted 2-spheres in 4D. What makes his movies interesting is that, like that of the watermelon, they reveal a symmetry that previous movie descriptions didn’t. Moreover, he made some really neat animations.


On the right hand side is a “diagram” of the 2-twist spun trefoil. Doing the tomography trick on this diagram he is able to effectively obtain the movie of the tomography of the knotted sphere in 4D on the left.


This tomography is slightly different as the slicing rotates along an axis, but it gives a clue as to why we refer to this knotted sphere as a 2-twist spun trefoil. (Wikipedia doesn’t have an entry yet for spun knots, let alone twist spun knots. The Mathworld descriptions are lacking…)

Check out the rest of his work. Inoue used Blender, a free and open source program, to create these. I might have to give Blender another try.

Berge Tangle

•July 17, 2010 • 3 Comments

Here’s a tangle in S^2 x I.

BergeTanglea4

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

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

BergeTanglea1

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

BergeTangleR BergeTangleG BergeTangleB

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

BergeTangleG-2

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

BergeTangle-RGB5 BergeTangle-RGB

These filling tangles also respect that 3-fold symmetry.
BergeTangle-RGB4

So here’s the three fillings…

BergeTangler2

BergeTangleg2

BergeTangleb2

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

BergeTangler4

BergeTangleg4

BergeTangleb4

Continue reading ‘Berge Tangle’

The deformation retraction of Bing’s house.

•June 23, 2010 • 14 Comments

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.

Picture 16

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:

Picture 13

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.

Picture 10

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

Picture 12 Picture 11

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

Continue reading ‘The deformation retraction of Bing’s house.’

Bing’s House

•March 25, 2010 • 17 Comments

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.

Picture 12

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

Picture 13

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.

BingsHouse-smooth

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.

Continue reading ‘Bing’s House’

Hopf Stabilization

•January 23, 2010 • 5 Comments

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

Picture 7

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

Picture 5

Picture 9

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

Picture 12

Picture 26

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

Picture 25

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