A (reverse) rational circle?

•December 10, 2009 • 10 Comments

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.

Picture 16

Picture 17 Picture 18
Picture 19 Picture 20

Picture 21

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Cabling a knot’s surface

•November 18, 2009 • 7 Comments

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.
onecable
Here’s a (3,5)-cable.
biggercable

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.
31cablesfce

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That pretzel knot

•October 20, 2009 • 4 Comments

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.)

P-237

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.

18foldliftofP-237

18foldliftP-237rotate

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

19foldliftofP-237

19foldliftP-237rotate

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

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Contact Heegaard Splittings

•September 27, 2009 • Leave a Comment

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.

JVHM-T3-OpenBook-ContactHeegaardSplitting0011

(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.

JVHM-T3-OpenBook-ContactHeegaardSplitting0014

JVHM-T3-OpenBook-ContactHeegaardSplitting0015

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?
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Fibered Cable

•August 3, 2009 • 2 Comments

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

2,1 Cable of the solid torus

Click on it for a larger version.

Here‘s a set showing the individual pages.

Two Balls

•July 25, 2009 • 5 Comments

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.

The 3-sphere as 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.

sclduggery

•July 24, 2009 • 1 Comment

Here’s a picture of a once-punctured genus 1 surface.

Picture 14

You can put it in a genus two handlebody. View the handlebody as corresponding to the free group on two generators a and b. Here I’m showing them as the yellow and blue cores of two handles.

Picture 9 Picture 10

Then the boundary of this once-punctured genus 1 surface may be viewed as representing an element of this free group.

Picture 12

Using the correspondence we can write the boundary of the surface as the product a b a^{-1} b^{-1} which is also denoted [a,b] and known as the commutator of a and b.

Those of y’all who have learned a bit about fundamental groups know that the boundary of a (compact, orientable) once-punctured genus k surface can be expressed as the product of k commutators of curves on the surface. Y’all also know that I’m being loose with basepoints, curves, and group elements.

At the Georgia Topology Fest this past May, Calegari spoke about scl, where this mix of the algebra and geometry of this can lead. He discusses it in greater detail in a recent entry of his blog. I’ll tell you a bit and then show off a fun fundamental example.

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