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.

Trivalent Cobordism

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.

Math in Design II

•March 17, 2009 • 2 Comments

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.

The square and granny tangles.

Sometime after tying your shoelaces and an overhand knot in a string you might have learned about the square knot:

And you might have tied the granny knot instead:

Well, if one were physically tying ropes together the ends might more naturally have gone out vertically up and down rather than off to the sides. The square knot is used to hold two ropes together. The granny knot is when you do it wrong… it’s more prone to slip. The funny thing is that the square knot can be closed up to form a split link whereas the granny knot can’t. Okay, so this is comparing physical versus topological properties.

Let’s look a bit more at the topology.
Continue reading ‘The square and granny tangles.’

A SketchUp Tool Box

•February 14, 2009 • 4 Comments

I’ve made a collection of some of the Ruby Scripts that I use with some frequency in SketchUp. The file SketchUpToolBox.zip includes:

• bezier.rb
• curvestitcherwithreverse.rb
• drawhelix13.rb
• formdraw.rb
• TubeAlongPath.rb
• weld.rb

With the exception of formdraw.rb, these are scripts created by others and offered up free around the ‘net with a couple of minor changes. Feel free to distribute/modify formdraw.rb too.

Read about installing and using Ruby Scripts in SketchUp here. Basically you’ll just place the contents of the file SketchUpToolBox.zip into the appropriate PlugIn directory, (re)start SketchUp, and select the new items from either the Draw menu or Plugins menu.

Below I’ll describe the basic work flow/issues in using some of these.
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Relative Open Books in T^2 x I

•January 13, 2009 • 4 Comments

Here are some relative open books in $T^2 times I$. They’re depicted in rectangular solids where you need to identify the left with the right and the top with the bottom. The pages of the open books meet the two torus boundary components (the front and the back) in curves. I’ve drawn “meridians” on them to suggest how to regard the monodromies as being the identity on the boundary of the page. Set in a bit from the boundary are marks that help to indicate the dynamics of the page. (Again, these are animated gifs.)

This first one is a relative open book for a “basic slice”. Refer to Section 4.2 and thereabouts of VanHorn-Morris’s dissertation (pdf). (This and the previous open book post take much inspiration from there.)
View the page as a once-punctured annulus with one positive Dehn twist around the puncture. The puncture becomes the binding.

Geometrically, notice that the pages here are all one single form that gets translated to the right. Not doing any Dehn twists along the annular boundary components means the marks on the pages follow the meridians.

It may seem odd that one of the annular boundary components is “vertical” while the other is “slanted”. But they both intersect the meridians once, the meridians are parallel, and they intersect each other once in the projection to the $T^2$ factor. Below it’s shown with another more symmetric presentation. The page is there too with the blue curve indicating a positive Dehn twist. (Identify the left and right to get the punctured annulus.)

Choosing different meridians on the boundaries is tantamount to doing Dehn twists along the boundaries. Below we have a once-punctured annulus page with a positive Dehn twist around the puncture and a negative Dehn twist around each annular boundary component. Notice how the marks move relative to the new meridians.

Next we’ll stack two together.

The JVHM Open Book

•November 24, 2008 • 11 Comments

The open book from the previous post wasn’t quite right. This became quite apparent when tried to color the two sides of a page. Along a “horizontal level” between each pair of binding components there needed to be three Dehn twists. Fortunately, the Dehn twists can be obtained through shearing.
A bit of reworking and rechoosing a fundamental domain gives a pleasant picture.

The three-torus may be viewed as a hexagonal prism with opposite faces identified. Here we show 10 pages of JVHM’s open book for the Stein fillable contact structure on the three-torus in this fundamental domain.

or this for a top view:

Yeah, these are animated gifs.

After the jump you can see more animations with these prisms tiled. You can also check out a few stills on my Flickr page. Or you can download the SketchUp model directly.

Lifting an open book to a cover

•November 7, 2008 • 1 Comment

In his dissertation Jeremy Van Horn-Morris describes a certain open book for the Stein fillable contact structure on $T^3$. Its page is a thrice punctured torus. It’s conjectured that this open book realizes the binding number of this contact structure, i.e. there’s no open book with twice punctured torus that supports the contact structure. (See Etnyre & Ozbagci – Invariants of contact structures from open books.) Anyway, I wanted to take a look at it. (Update: But this isn’t it. The top and bottom need to glue together with a half rotation. To get the JVHM open book for $T^3$ there needs to be some Dehn twists between the levels.)

This uses the standard presentation of $T^3$ as a cube with opposite faces identified. This only shows one page of the open book.

To get a better picture of all the connectivity, we can take covers of $T^3$ and lift the open book.

More pictures after the break.
Continue reading ‘Lifting an open book to a cover’

You can view $S^3$ as the set of points
${(x,v) | x in mathbb{R}^2, v in T_x mathbb{R}^2, ||x||^2 + ||v||^2 = 1}.$
So basically this would be the unit tangent bundle over the unit disk but you scale down the vectors as you get further from the origin. Over each circle about the origin of radius $r$ for $0 you get a torus separating $S^3$ into two solid tori as in an earlier post. At each $r=0$ and $r=1$ you get just a circle.

If you draw a properly immersed smooth 1-manifold in the unit disk (with only transveral self-intersections), then at each point away from the boundary and the origin its tangent line picks out two points in this model of $S^3$. At the boundary and the origin, it picks out just one point.

Since these points vary smoothly you get a 1-manifold in $S^3$, a knot or a link. The immersed 1-manifold is called a divide and the links of a divide have many nice properties as A’Campo has shown.

Blowing up an arc

•September 14, 2008 • 5 Comments

Lines intersect as they pass through a point, but you don’t want them to? Blow it up!

Remove a neighborhood of the point.

And sew back in a punctured projective space.

For a point on a surface like we have here, we sew back in a Mobius band.

Now cross this with $I=[-1,1]$.

Okay, so we sewed in a twisted $I$ bundle over the punctured projective space. But, c’mon, why not?

Now let’s round it out…

… and reinterpret. We blow up a trivial 1-strand tangle to get a solid torus, the complement of the trivial knot. What happens when we blow up a knotted 1-strand tangle?

Update: One more picture.