A SketchUp Tool Box

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.

Relative Open Books in T^2 x I

Here are some relative open books in . 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 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

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.

But that’s hard to see…. How about this:

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

In his dissertation Jeremy Van Horn-Morris describes a certain open book for the Stein fillable contact structure on . 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 there needs to be some Dehn twists between the levels.)

This uses the standard presentation of 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 and lift the open book.

More pictures after the break.

The Link of a Divide

You can view as the set of points

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 for you get a torus separating into two solid tori as in an earlier post. At each and 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 . At the boundary and the origin, it picks out just one point.

Since these points vary smoothly you get a 1-manifold in , 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

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 .

Okay, so we sewed in a twisted 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.

A Product Disk for the Hopf Band

We have the Hopf link.

With an annular Seifert surface, we have a Hopf band. Choosing the surface as shown orients the link to give the positive Hopf link.

Now we take an arc (blue) on the band, push it off in the positive direction (sequence of yellows), and lay it back down on the backside of the band (red).

The endpoints of these arcs sweep out a meridian (gray) on each component of the Hopf link. But we’ve offset them a bit to emphasize the direction of sweep and that these arcs sweep out a product disk.

Here’s the disk on its own.

Ignore the disk and look at the initial blue arc and final red arc on the surface. With the positive side of the surface up, we see that the final red arc veers to the right from the endpoints.

The monodromy of the positive Hopf link is right veering.

Toroidal Coordinates on S^3

We’re looking at what amounts to being the genus 1 Heegaard sweep-out of S^3. There are a few concentric tori, two core curves, and two intersecting meridional disks. Of course above we’re looking at an odd perspective from the level of an orangish torus.

Pulling back, just looking at the two core curves and a couple of disks they bound, we see the following.

Since is the unit sphere in , we can think of these two disks as the unit –disk and the unit –disk. You might think of each of these in terms of polar coordinates: . Then the angles and are independent while the two radii are bound by . In some sense this gives what one might call toroidal coordinates on .

Ko Honda’s Octahedral Relation

This morning at Kirbyfest (Happy Birthday!) Ko Honda gave a talk about his work on developing a contact category. You’ve got an oriented surface (with boundary) together with dividing curves as your objects. Morphisms arise via bypass moves. Ko noticed that the octahedral relation is satisfied:

We see four arcs of the Red dividing curves in a White disk of the surface.

The Green triangles are the exact triangles. The Yellow triangles are the commuting triangles.

You can get the SketchUp model here. The disks remain facing the camera in the horizontal direction by using the “face me” function of a component.

The Lantern Relation III: Eye Candy

So this is a bit of fluff… mainly exercises with the drawing tools. I’ve had these models sitting on my computer for a while and thought I’d toss up a quicky while at Kirbyfest.

We’re looking at one half of the lantern relation. The four vertices on the sphere are the four punctures. Each picture shows us doing the three twists along curves that separate the vertices in pairs. See the earlier lantern relation posts for details.