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.

Long Disks and Lopsided Bigons

I’ve been working in a collaboration with Gordon and Luecke where we’ve been looking at thin positions of knots in manifolds that arise from non-integral, non-half-integral surgeries on knots in S^3 (with some other conditions). Fat vertexed graphs of intersection hijinks ensue.

A particular situation arises where we have a knot K that is 2–bridge with respect to a Heegaard surface F and on the graph from S^3 we see at a vertex a forked extended Scharlemann cycle (with a certain one of two possible labelings) with its trigon followed by a bigon or another trigon. These are on the left-hand sides of the figures below. The black and white (dark and light gray) faces are on opposite sides of the Heegaard surface F, the edges between them are curves on F. The “corners” are arcs that run along the boundary of the neighborhood of K from one intersection of K with F to another.

It so happens that the 41–edges of faces f and g cobound a disk d (disjoint from the rest of these faces and the knot). On the right-hand sides of the above images we reassemble the faces to form disks. The purple colored things run along the boundary of a neighborhood of K, between corners. The red shows the original K and these disks guide an isotopy of it to the salmonish colored arc.

The one on top is a “long disk” because it is like an extended bridge disk. Rather than guiding just one arc of K - F onto F, it guides three arcs. Consequentially this thins the knot to be 1–bridge.

The one on the bottom is a “lopsided bigon” because it has two edges on F and two corners along K, but one corner is made of three arcs and the other is made of one. This too gives a thinning since we can isotop three arcs of K down to two arcs on F and one arc on K.

It is perhaps not so obvious that these actually give isotopies. Fortunately the complex determined by these faces and the knot is, up to homeomorphism, unique in the top case and almost unique in the bottom case. Moreover these complexes can be embedded in R^3 and one can wrap the long disk or lopsided bigon around it to explicitly see the isotopy. I made a SketchUp model (get it here) to illustrate all this, though I cut it open leaving some identifications undone for better viewing.

Here’s one of complexes for the bottom.

Continue on for the construction of the top complex or just head over to my Flickr set. Or get the model.

Sutured Manifold Decomposition of 7_7

While packing for my move and sorting through old papers, I’ve come across several drawings from my ill spent youth (aka grad school).

Here we’ve done Gabai’s sutured manifold decomposition on the exterior of a Seifert surface for the knot 7_7 in Rolfsen’s table, thereby showing it is a minimal genus Seifert surface. The Seifert surface used is the one obtained from Seifert’s algorithm on the knot diagram shown. Note (1) our viewpoint is from the inside of the sutured manifold and (2) our first decomposition disconnects the sutured manifold to give the two sutured manifolds beginning the bottom two lines.

Math in Design

This morning I came across some work of students in Patricia Muñoz’s Morfologia class posted on YouTube (via).

Look through their gallery or check out the course abstract (en inglés).

The idea is to use the intersection of two “simple” objects as a primitive for design.
With the intersection, slight modifications are introduced achieve a sense of aesthetic… tending towards a minimum (or maximum, critical point) of some sort of aesthetic energy function on a configuration space, if you’ll indulge me. Exercise: Make this precise.

This seems to be a common sort of approach in design.  Begin with a core concept, and then tweak it / bend the rules…  Well I could go on and on about this, but perhaps it’s best left to future posts for parceling.

After the jump: I’ve picked out a couple of the videos, though you should check out all the others too.

Braids and the Configuration Space of Points

One can view a n string braid as a loop in the configuration space of n points in the plane. I made a quickie screencast to illustrate this.

Actually I was trying to see if I could get SketchUp to animate the moving points seen in the first half of the screencast. In the end I did it manually with the assistance of section planes. The second half however is done automatically. Get the model and click on the beginsweep/endsweep buttons after opening it in SketchUp.

Canceling Cones

I don’t think the reconciling of the non-smoothable cone points in that last post was terribly plain to see. Perhaps if you could manipulate the models yourself, it would be more apparent.

These two pictures show the essence of the move. Take a curly Q crossed with an interval. Shrink the middle loop down to a point. Then pull apart the resulting cones. Reverse to smooth a pair of cones.

Here’s the same guys bent a bit.

Continue on to see how this applies to that octaheralishy Boy’s surface.

Boy’s Surface

One of the things knocking around in my noggin has been sphere eversions, especially ever since Carter showed me his drawings at LSU. Of course there’s Thurston’s corrugation. And Morin’s surface.

But another method is to pass through a immersion of the projective plane, a non-orientable closed compact surface which the sphere double covers. So this man named Boy came up with a smooth immersion of the projective plane when Hilbert assigned the homework to show it can’t be done. Here are a few pictures of my attempts to make a model of it.

You might notice the 3-fold symmetry. You might compare that with the 4-fold symmetry of Morin’s surface.

There are many approaches to Boy’s surface. Kirby describes an interesting method (pdf). Let’s see how that works.

The Lantern Relation II: The Fake Lantern Relation

Gotta love orientations. Handedness, order of composition, sign conventions. Thought I’d put the end results of the Real and the Fake Lantern Relations side by side. Shown again is the four punctured sphere with three colored annuli on it.

Let’s say R, G, B denote right handed Dehn twists along the red, green, and blue curves and that dddd denotes left handed Dehn twists along the boundary components. We’re composing from left to right.

The Real Lantern Relation	The Fake Lantern Relation
RBGdddd	RGBdddd

In the last row, I picked out the arc that had the simplest looking image in the Fake Lantern Relation to compare with the corresponding one in the Real Lantern Relation. With a bit of eyeballin’ you can see the arc in the Real is isotopic to the identity. Here’s a simplified picture of the arc in the Fake.

Embedding Models? + El Thetarino

To paraphrase Kazez: Why look it up when I can just ask you?

So I could spend a lot of time googlin’ and researching and not coming up with anything, or I could just ask the good wholesome people of the internet. Anyone know a method of embedding models into this blog so that people can at least easily rotate and perhaps pan and zoom?

And just so there’s a picture. An order 2 Scharlemann cycle gets its brown “corners” identified to form a Mobius Band. An order 3 Scharlemann cyce gets its brown “corners” identified to form a Whatchamacallit Band. I’ve been calling it a Twisted Theta Band since the induced boundary graph is a theta-graph, but it’s not just a product theta-graph. Idunno: the Threebius? Tribius? Threeta? El Thetarino if you’re not into the whole brevity thing?