What’s below are some pictures suggesting Seifert fibrations of S^1 x S^2, the circle of spheres. View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.

A Seifert fibration of a 3-manifold is a filling of the 3-manifold with circles so that around each point a teeny tiny enough chunk looks like it’s filled with parallel lines. Here these circles get chopped into intervals. The interval going through the north pole connects up to give one circle, and so does the interval through the south pole. We call these circles the singular fibers.

All the other intervals connect up with a fixed number of others to form circles. These are the regular fibers. Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.

Here they have order 2.

Here they have order 3.

Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the S^1‘s of the S^1 x S^2.

In the three examples above, red fibers are shown at regularly spaced latitudes (along fixed longitudes) going from the north pole to the south pole. The yellow fibers are copies rotated to other longitudes.
Let’s look at just the red ones as the orders of the singular fibers increase.

Order 1

Order 2

Order 3

Order 4

Order 5

Order 6

Order 7

Order 8

Order 16

Order 25

Neat, I guess.

While we’re here. Have some eye candy. There’s more in this set.

I think all these below come from the order 1.

(Oh, and these pictures are a result of experimenting with Grasshopper for Rhino and some non-photorealistic rendering.)

(Hmmmm…. I guess I had transparency in the original background. Oh well.)

The projective measured lamination space of the twice punctured torus is homeomorphic to the three-sphere, . That’s a result of Thurston. Parker-Series [1], continuing work of Keen-Parker-Series [2], gives a triangulation of consisting of 28 tetrahedra, 56 faces, 39 edges, and 11 vertices. Each vertex may be represented by a particular (isotopy class of an) unoriented simple closed curve while each tetrahedron is spanned by the projective weightings of a Birman-Series -train track [3].

Two of these vertices correspond to natural North and South Poles of the three-sphere as they never occupy the same tetrahedron and the remaining 9 vertices with the 14 faces among them form a polyhedral two-sphere separating these poles.
We are thus able to display this triangulation of in by omitting the North Pole, placing the South Pole at the origin, choosing an embedding of the polyhedral enclosing the origin, and extending the remaining edges and faces radially. (Actually when shown in these pictures I used a round sphere while the vertices of the polyhedral sphere defined the radial edges.) At each vertex other than the poles we center a copy of with its corresponding simple closed curve; at a representative point within each tetrahedron we center of copy of with its corresponding fundamental train track. To keep the parametrization of each copy of consistent we first view as being skewered along a radial axis with one puncture heading towards the North Pole and the other towards the South Pole. Then we ask the model (using the Face Me function in SketchUp… looks like it’s now called Always Face Camera) to rotate each copy of along its axis so that its “front” maximally faces the viewer. This allows us to dynamically rotate the model while inferring the parametrization of each copy of by its radial orientation.

[1] J. R. Parker and C. Series, The mapping class group of the twice punctured torus, Groups: topological, combinatorial and arithmetic aspects}, London Math. Soc. Lecture Note Ser. 311 (2004), 405–496.

[2] L. Keen, J. R. Parker and C. Series, Combinatorics of simple closed curves on the twice punctured torus, Israel J. Math. 112 (1999), 719–749.

[3] J. S. Birman and C. Series, Algebraic linearity for an automorphism of a surface group, J. Pure and Applied Algebra 52 (1988), 227–275.

Generic train tracks for the tetrahedra:

Simple closed curves for the vertices (sans the poles):

The four pics of the full triangulation that make up the first picture:

The list of tetrahedra and vertices labeled with the Parker-Series notation:

And of course clicking on the pictures will take you to Flicker where you can view somewhat higher resolution versions.

Through the double branched cover, we get the fibration of the solid torus by disks.

Here’s a few more views of these.

Simple as they are, there’s a pleasant rhythm.

There are a couple more on the Flickr (just click any of these pics). You have to view the “original” size for the gifs to animate.

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.

Now the strands fit in a small square x I of the sphere x I.

This way it’s easier to stack lots of them.

This stack has a quarter turns between them.

The top row smooths the strands out a wee bit:

It never ends.

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.

Once groomed, it becomes more simple to see the annulus that traces their isotopy.

More pics are here.

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.

Turns out there are four.

The red is orientation preserving and non-separating.

The green is orientation preserving and separating.

The blue and yellow are each orientation reversing and non-separating.

Let’s put them with the foliations shown earlier.

Let’s animate some of these things too.

Here’s another take on the one above.

The curves are shown like ribbons since I was also thinking about the twisted interval bundle over the Klein bottle.
This makes the blue and yellow into Mobius bands. And thus we can see this space as also the twisted circle bundle over the Mobius band.

The Flickr set has many more pictures.

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.

We’re gonna make a genus 2 Heegaard surface from two fibers with the knot sitting on one of them.

Billow out the surface, and carry the knot along.

Then billow it out in the other direction too and we’ll have a nice genus 2 Heegaard surface with the knot on it.

Rather than looking for the primitivizing disks, we’ll find compressing disks for the handlebodies on each side of the Heegaard surface that are disjoint from the knot. See, the Dehn surgery on the knot transforms the handlebodies into two solid tori and these two disks will become meridional disks.

We’ll start by finding an essential arc on the blue surface that is disjoint from the knot. (If we wanted primitivizing disks, we would start with an essential arc that crossed the knot once.)

Now we’ll sweep this arc through the fibers to the red surface, though the two handlebodies. The easy way is through the billowing we did.

This gives one meridional curve.

To get the other… well, going through the fibers outside the billowing, we apply the monodromy of the trefoil (a Dehn twist along two curves) to the arc on the red surface. Together with the arc on the blue surface we obtain this loop:

If you start by pulling the bottom red bit down and unhook it from the blue twisting, you might be able to convince yourself that this is indeed an unknotted loop. With a bit more work you could even convince yourself that it actually bounds a disk disjoint from the surface. Drawing that disk would’ve made quite a mess. Here’s the curve as it sits on the surface.

Now we can look at both these curves on the Heegaard surface together.

Count that they intersect 19 times. You see 18 in the middle here. The last comes from the red arc they have in common, but that can be perturbed to give a single transverse intersection.

One may count that the twisting is 7 — plus or minus, and perhaps the inverse mod 19 depending on how you’re counting. Starting off with the common red arc of the two meridians as 0, then count the intersections in order along the light blue meridian and again in order along the dark blue meridian. The torsion is the number that you multiply by the light blue count of an intersection to get the dark blue count of the same intersection, mod 19. A portion of this counting is shown here:

There’s an intersection that the light blue counts as 2 and the dark blue counts as 5. So if 2q=5 mod 19 then we may take q=12. Since 7=-12 mod 19 we’ve got the lens space claimed (up to homeomorphism).

Here are those two meridians again with the knot.

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.