Hopf Stabilization

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

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

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

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

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

A (reverse) rational circle?

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.

Cabling a knot’s surface

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.

Here’s a (3,5)-cable.

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.

That pretzel knot

That pretzel knot P(-2,3,7) is a mischievous fella.

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.

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

The Flickr set has more pics of these.

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

Also I nudged the relaxations a bit to help coerce it along… slight chance that a crossing change occurred. Here are the input files I used for the KnotPlot relaxations.

Contact Heegaard Splittings

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.

(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.

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?

Fibered Cable

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

Click on it for a larger version.

Here’s a set showing the individual pages.

Two Balls

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.

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

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

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

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

Using the correspondence we can write the boundary of the surface as the product which is also denoted and known as the commutator of and .

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 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.

Cables and torus knots

So it’s high time for another post. And while a raytrace of a reflective chrome torus hovering over a chessboard is tempting, how about a torus knot on glass?

Think of this green torus as surrounding a meridian of another solid torus.

Then this torus knot will “interpolate” from one cable to another.

You can make sense of “interpolate” homologically: OuterCable-InnerCable=TorusKnot.

This was all quickly whipped up using Rhino. There’s bit of experimenting some of the materials of their Toucan rendering. The glass distorts stuff a bit too much sometimes. Below are a selection of more pics. Click on any to go to my Flickr and see yet more.

Mirrors and Ribbons

The connect sum of a knot and its mirror is the basic example of a slice knot.

Put one above the xy-plane and mirror it below.

The mirroring sweeps out an immersed annulus.