The twice punctured torus

•July 29, 2011 • 1 Comment

For the conference honoring the 60th birthday of Caroline Series (only a German wiki?!?), I was one of a handful asked to contribute pictures inspired by her work. First up is my contribution followed by a description. After that are a few more.

Triangulation I, II, III, IV
(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, \mathcal{PML}(\Sigma_2) \cong S^3. That’s a result of Thurston. Parker-Series [1], continuing work of Keen-Parker-Series [2], gives a triangulation of \mathcal{PML}(\Sigma_2) 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 \pi_1-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 \mathcal{PML}(\Sigma_2) \cong S^3 in \mathbb{R}^3 by omitting the North Pole, placing the South Pole at the origin, choosing an embedding of the polyhedral S^2 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 \Sigma_2 with its corresponding simple closed curve; at a representative point within each tetrahedron we center of copy of \Sigma_2 with its corresponding fundamental train track. To keep the parametrization of each copy of \Sigma_2 consistent we first view \Sigma_2 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 \Sigma_2 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 \Sigma_2 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.

Continue reading ‘The twice punctured torus’

Rational Tangle Fibration

•June 6, 2011 • 1 Comment

Here’s a couple of rational tangles with what one might consider a fibration. They’re actually homeomorphic. One’s got an extra half twist to it.

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

TwoTanglesA

Here’s a few more views of these.

TwoTanglesB

TwoTanglesOtherC

TwoTanglesE

TwoTanglesF

Simple as they are, there’s a pleasant rhythm.
Continue reading ‘Rational Tangle Fibration’

Seifert surfaces of dual knots

•April 21, 2011 • 1 Comment

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.
knot with basis

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.
knot with basis3

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.
knot with basis and surgery curve3

knot with basis and surgery curve5

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.
knot with basis and surgery curve - postsurgery7

knot with basis and surgery curve - postsurgery6

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.
knot with basis and surgery curve - postsurgery - parallelpushoff

knot with basis and surgery curve - postsurgery - parallelpushoff6

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.
knot with basis and surgery curve - postsurgery - parallelpushoff4

knot with basis and surgery curve - postsurgery - parallelpushoff7

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.
knot with basis and surgery curve - parallelpushoff

knot with basis and surgery curve - parallelpushoff2

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.
knot with basis and surgery curve - parallelpushoff4

knot with basis and surgery curve - parallelpushoff3

knot with basis and surgery curve - parallelpushoff6

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.

Once again, Whiteheadtangletangletangletangletangletangletangletangletangletangletangletangletangletangletangletangletangletangletangletangltangletangletangletangletangletangletangle

•April 2, 2011 • 1 Comment

I squished around that Whitehead tangle.

Whiteheadtangle-squarespherebig2

Whiteheadtangle-squarespherebig

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

Whiteheadtangle-squareSingle

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

This stack has a quarter turns between them.

Whiteheadtangle-squareF2

The top row smooths the strands out a wee bit:
Whiteheadtangle-squaretanglestackwsmoothed

It never ends.
Whitehead flythrough

p q is q p

•March 28, 2011 • 8 Comments

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.

23-torusknot 32-torusknot
23-torusknot-tilt 32-torusknot-tilt

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.

23-32-together

23-32-together4

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

torusknots-bothsides2

torusknots-bothsides1

32-torusknot-thin1 23-torusknot-thin

Once groomed, it becomes more simple to see the annulus that traces their isotopy.
Continue reading ‘p q is q p’

Whiteheadtangletangletangletangletangletangletangletangleadnauseum

•March 22, 2011 • Leave a Comment

This is the Whitehead link.
WHlinkwinvol4
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 S^2 \times I.
WHtangle-top
The Whitehead link exterior becomes the S^2 \times I 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.
whouterclosure
We’re inside the ball of the rational tangle and looking out to that small sphere:
whouterclosure2
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.
WHdouble

Here’s a whole bunch more together.
WHBig2

WHdeepinside7

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.

WHdeepinside3

WHv2-tangle

WHv2-tangle

WHv2-tangle3

WHv2-tangle-bigblackbackground

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’

 
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