If you do surgery on a knot, you get a dual knot in the resulting manifold. The original knot and the dual knot have the same complement. A Seifert surface for the original knot is then a (rational) Seifert surface for the dual knot. We can see this dual knot and corresponding Seifert surface before we do the surgery.

Take a knot with orange meridian and purple longitude.

For a null homologous knot, we may take the purple longitude to be the boundary of a Seifert surface. The orange meridian is the boundary of a meridional disk.

Let’s look at +4 surgery on the curve. The green curve will now bound a meridional disk, but we can’t see this disk fully until we reembed.

In the surgered manifold, the green curve now bounds a meridional disk. Since the surgery was integral, the orange curve is now a longitude. Since it was a +4 surgery, the purple boundary of our Seifert surface runs 4 times longitudinally.

Let’s push a copy of our new “dual” knot out of the surgery solid torus. We can make this copy parallel to the longitudinal orange curve.

We got this copy of the dual by a pushoff isotopy, so drag the purple Seifert surface along. The original dual we got from surgery now intersects the surface transversally once. The surface intersects the surgery solid torus in a single meridional disk. I shrunk the surgery solid torus.

Now let’s look back at the copy of the dual before we did surgery. Since it’s parallel to the orange curve, the dual is parallel to the meridian.

The Seifert surface of the dual can now be seen. It’s punctured once by the surgery solid torus, but that gets capped off by the surgery.

There’s a solid torus neighborhood of the surgery solid torus, the thickened dual knot and the parallelism of the dual knot to the orange meridian. Outside this, nothing has changed.

The (p,q)-torus knot is isotopic to the (q,p)-torus knot. Here we use the trefoil as an example. (Trefoil, always the trefoil…) It’s both a (2,3)-torus knot and a (3,2)-torus knot.

To see that these are equivalent, shift your point of view between inside and outside the torus on which the knot lies. After all, this Heegaard torus separates the 3-sphere into two solid tori. People don’t always find this approach satisfying.

Alternatively, we step between these two solid tori.

Let’s push these curves around to groom their alignment.

This is the Whitehead link.
It’s actually drawn thickened up a bit and made transparent since I really want to look at the space around the link, the Whitehead link exterior. Also shown is an axis for a 180 degree spin, an involution.

Quotient out by this involution and you get a tangle in .
The Whitehead link exterior becomes the and the involution axis becomes the strands of the tangle.

Taking the double branched cover gets you back to the space outside the Whitehead link.

Plugging up the outer sphere with a particular rational tangle makes the whole thing a rational tangle.
We’re inside the ball of the rational tangle and looking out to that small sphere:
Shrink the “ears” and slip off the spiral to see it unravel…

But this isn’t what’s so interesting.

There’s this thing called the Whitehead manifold. While asking Google to tell me the Wikipedia link again I noticed this post that appears to give a decent overview of the Whitehead manifold.

Basically -and omitting a few details- the Whitehead manifold is obtained by stacking together an infinite ray of Whitehead link exteriors, but starting the whole thing off with a solid torus. Let’s see the corresponding thing with tangles. (Note that I neglected to plug the beginning end up with the rational tangle.)

Here’s two stacked together.

Here’s a whole bunch more together.

You’ll notice the 90 degree rotation as they get stacked. That’s so if I were to plug up the beginning as before, stopping the stack at any finite depth would yield a rational tangle. Doing the rotating to the right or left at each step lets you make lots of different tangles of a given finite depth whose double branched covers will be the same stack of Whitehead link exteriors. Since we’re doing an infinite stack, this makes the Whitehead manifold double branch cover uncountably many different such tangles.

Okay, so maybe some of those statements should be checked… by someone else. And these are wild tangles rather than the usual sort of tangles. But that’s okay. Here are some more pics.

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.

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

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

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’

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.

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.

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.

Many of these pictures were created with Sketchup.
Some of these models and more are available at the 3Dwarehouse.
Lately most of them have been done with the beta version of Rhino OS X.