Joining two segments

•December 20, 2013 • 6 Comments

My how time flies…

The join of two topological spaces A and B is basically the space of all line segments between every pair of points.  As a nice embedded, visceral example of this, a tetrahedron may be viewed as the join of two skew line segments.

Indeed this is the illustration on Wikipedia.

It also arises naturally when you think about grid diagrams of knots.  Remember this from way back?

Side Trefoil Tetra Torus

Thinking along those lines, you might consider discretizing it a bit…  Instead of taking the join of two entire segments, just take the join of n points along each segment.

Tetrahedron Join

Here I’ve used n=7.   And that gives us the complete bipartite graph K_{7,7} naturally embedded as a subset of the tetrahedron.  Moreover, any grid number 7 knot is a cycle in this particular embedding of the graph.

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A Generalized Banding

•April 26, 2013 • 2 Comments

The preprint Band Surgeries and Crossing Changes between Fibered Links by Buck-Ishihara-Rathbun-Shimokawa caught my eye this morning. They describe a describe a generalization of the plumbing of a Hopf band. Like Hopf plumbing, this operation preserves fiberedness. But unlike Hopf plumbing which occurs in a neighborhood of a disk, it is non-local occurring in a neighborhood of an annulus.

I thought I’d show the product disk associated to the band. This lets one verify the persistence of fiberedness and work out the resulting monodromy (which I haven’t done myself yet).

banddiskcurves

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Twisting with Surgery

•November 2, 2012 • 1 Comment

TwistsAlongDiscAnnulus-06

Here’s a sequence of images I drew for a talk I gave about a month ago. I reckon there isn’t much more to say that’s not in the images. Hit flickr for larger versions.

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It’s full of surfaces!

•August 24, 2012 • 2 Comments

TrefoilFibrationSolid

That’s the exterior of the trefoil.   No, really.  I mean, well, it’s a torus embedded as the boundary of the exterior of a trefoil.

Okay, so it’s not how you’d probably choose to draw it.  It’s not how I’d first choose to draw it either.  Let’s see how I came to it.

Think about the space around the trefoil.  The thing’s hollow — it goes on (round and round) forever — and – oh my God

TrefoilFibrationSolidFull

it’s full of surfaces!   (That was terrible. My sincere apologies to Clarke.)

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Chains and Tangles

•February 1, 2012 • 3 Comments

These two links have homeomorphic exteriors.

chaintangles001 chaintangles002

They’re both strongly invertible. Let’s quotient the first link by its strong inversion to get a tangle. Then we’ll isotop that tangle around and eventually take its double branched cover to get the second.

chaintangles01

chaintangles02-1

chaintangles02

chaintangles03

chaintangles04

chaintangles05

chaintangles06

chaintangles07

chaintanglesA

chaintanglesB

chaintangles08

chaintangles09

chaintangles10

chaintangles11

chaintangles12

chaintangles13

chaintangles14

chaintangles15-1

chaintangles15

chaintangles16

Notice that this homeomorphism swaps the red and blue meridians and longitudes.

Hairy circle of spheres

•September 7, 2011 • 1 Comment

Well hairy isn’t too accurate unless there are hairs that are closed circles, but these are the concessions one makes for a dumb play on the Hairy Sphere Theorem. This post doesn’t really have much to do with that theorem.

What’s below are some pictures suggesting Seifert fibrations of S^1 x S^2, the circle of spheres. View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.

aaaaqqaq

A Seifert fibration of a 3-manifold is a filling of the 3-manifold with circles so that around each point a teeny tiny enough chunk looks like it’s filled with parallel lines. Here these circles get chopped into intervals. The interval going through the north pole connects up to give one circle, and so does the interval through the south pole. We call these circles the singular fibers.

All the other intervals connect up with a fixed number of others to form circles. These are the regular fibers. Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.

Here they have order 2.
2 with 5 copies origview

Here they have order 3.
3 with 4 copies origview

Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the S^1‘s of the S^1 x S^2.
1 with 5 copies origview

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The twice punctured torus

•July 29, 2011 • 1 Comment

For the conference honoring the 60th birthday of Caroline Series (only a German wiki?!?), I was one of a handful asked to contribute pictures inspired by her work. First up is my contribution followed by a description. After that are a few more.

Triangulation I, II, III, IV
(Hmmmm…. I guess I had transparency in the original background. Oh well.)

The projective measured lamination space of the twice punctured torus is homeomorphic to the three-sphere, \mathcal{PML}(\Sigma_2) \cong S^3. That’s a result of Thurston. Parker-Series [1], continuing work of Keen-Parker-Series [2], gives a triangulation of \mathcal{PML}(\Sigma_2) consisting of 28 tetrahedra, 56 faces, 39 edges, and 11 vertices. Each vertex may be represented by a particular (isotopy class of an) unoriented simple closed curve while each tetrahedron is spanned by the projective weightings of a Birman-Series \pi_1-train track [3].

Two of these vertices correspond to natural North and South Poles of the three-sphere as they never occupy the same tetrahedron and the remaining 9 vertices with the 14 faces among them form a polyhedral two-sphere separating these poles.
We are thus able to display this triangulation of \mathcal{PML}(\Sigma_2) \cong S^3 in \mathbb{R}^3 by omitting the North Pole, placing the South Pole at the origin, choosing an embedding of the polyhedral S^2 enclosing the origin, and extending the remaining edges and faces radially. (Actually when shown in these pictures I used a round sphere while the vertices of the polyhedral sphere defined the radial edges.) At each vertex other than the poles we center a copy of \Sigma_2 with its corresponding simple closed curve; at a representative point within each tetrahedron we center of copy of \Sigma_2 with its corresponding fundamental train track. To keep the parametrization of each copy of \Sigma_2 consistent we first view \Sigma_2 as being skewered along a radial axis with one puncture heading towards the North Pole and the other towards the South Pole. Then we ask the model (using the Face Me function in SketchUp… looks like it’s now called Always Face Camera) to rotate each copy of \Sigma_2 along its axis so that its “front” maximally faces the viewer. This allows us to dynamically rotate the model while inferring the parametrization of each copy of \Sigma_2 by its radial orientation.

[1] J. R. Parker and C. Series, The mapping class group of the twice punctured torus, Groups: topological, combinatorial and arithmetic aspects}, London Math. Soc. Lecture Note Ser. 311 (2004), 405–496.

[2] L. Keen, J. R. Parker and C. Series, Combinatorics of simple closed curves on the twice punctured torus, Israel J. Math. 112 (1999), 719–749.

[3] J. S. Birman and C. Series, Algebraic linearity for an automorphism of a surface group, J. Pure and Applied Algebra 52 (1988), 227–275.

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