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.

Here’s the knot with it’s Seifert surface hanging off. The surface continues off to the left, but we’re only looking near the knot.

Take a tubular neighborhood and restrict to looking outside it.

For the (3,1)-cable, we’ll take 3 pages and one meridional disk.

Then we’ll attach them with twisted bands so the resulting boundary can run along the cable.

And there you go. But it’s kinda nice to smooth it out.

Smoothing it out also has the benefit of helping you see how cabling extends the fibration of a fibered knot.

The way we’ve been looking a the knot, a fibration looks like this.

With the cable smoothed, we can see how the surface corkscrews upwards to fill in the fibration.

And we can even stack this before gluing the top to the bottom to get other (3,q)-cables. Here’s the fibration of the (3,5)-cable on its side.

You can get the SketchUp model here and take a closer look yourself.

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.

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?

A contact handlebody is a regular neighborhood of a Legendrian graph. A contact Heegaard splitting is a Heegaard splitting that divides the contact manifold into two contact handlebodies. (Definitions due to Giroux.) It turns out that given a contact manifold, a contact Heegaard splitting is essentially the same as a supporting open book.

Let’s look at the Legendrian graphs for our example.

The contact structure as viewed here has the contact planes go through one full rotation in the vertical direction while just translating in the horizontal directions. Each of the two Legendrian graphs is a vertical circle and three horizontal ones. In each graph of these pictures, the three vertical rods are all identified together, and so the three vertices of the graph are all valence 4.

Here’s a few more views. Or you can just check out the entire set of pictures.

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.

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.

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.

The elements of the commutator subgroup of a group can all be written as a product of commutators. That is to say if then there exists a finitely many (say, ) elements such that .

The commutator length, or simply cl, of a given is just the smallest number (the smallest ) of commutators (the ) needed to express as their product.

The stable commutator length, aka scl, of a given reflects the asymptotic behavior of cl as grows.

One’s first impulse is to say that cl is just cl. Take the example at the very top. Of course , but isn’t it obvious that you can’t do any better?

Here’s a few pictures even.

Go ahead. See if you can do better.
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Don’t wanna spoil the fun.

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Here’s a monkey washing a cat.

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That always gets my head a thinkin’.

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Yeah, so if by “obvious” you meant “wrong”, then you are correct.

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Thinking about Euler characteristics and irregular three-fold covers leads you to a genus 2 surface.

So cl. (Why is it not 1?) Hence cl

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.

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.

Note how the annulus runs through itself. These self-intersections are ribbon.

Snipping along a vertical arc of the annulus leaves an immersed disk. Its boundary is the connect sum of the knot and its mirror.

Some intersections may simply be slid away.

Others may not.

More images here.

Here’s a quickie — just checking out some Section Plane functionality of SketchUp. You can build a 2-complex and see how its slices evolve. This 2-complex is the same as the spine of a tetrahedron.

Get the model here.

Daniel Piker of Space Symmetry Structure has various investigations in design rooted in mathematical concepts. Let me highlight a couple.

Check out his medial surfaces between link components. Below is Piker’s image of the medial surface between the components of the Whitehead link.

Also an interesting read is his discussion of Rheotomic surfaces and structures arising from conformal maps. Here’s an image of Piker’s from that post.