The *join* of two topological spaces and 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?

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 points along each segment.

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

While it’s a rather simple object, I’ve nonetheless found it to be a rather pleasing one. Lots of emergent rhythms.

Kinda gotta hold it and spin it around to really appreciate it, so print one out if you’d like. I’ll talk about some of the others models there whenever I eventually get around to it…

Of course I would be amiss without plugging Segerman’s fantastic body of W O R K with 3D printing. If you haven’t seen it before, you’re missing out.

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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).

Let’s talk through the construction.

The generalized banding is defined by an arc in a Seifert surface with one transverse self intersection and meeting the boundary of the surface in its endpoints. A neighborhood of this blue arc is an annular chunk of the surface. The white tubes are the pieces of the link (the boundary of the surface). The rest of the boundary of this annulus continues on doing whatever it was already doing in the surface.

Now run a band across the surface following the arc, going over itself, and joining the boundary of the old fiber.

A spanning arc of the band gives rise to a product disk. If the side of the original surface we saw was “up” then here we have a product disk going from the red arc to the green one.

Taking the sutured manifold coming from this new banded surface and decomposing it along this product disk leaves us with the sutured manifold coming from the old surface. (Yeah, we could see explicit pictures of this… maybe in a future update.) So if the original Seifert surface was a fiber, then the banded surface will be a fiber too.

Here’s a few more pics of this disk on its own and with parts of the surface.

There’s a few more on the Flickr. Clicking any of these pics should take you there.

A couple of interesting things to note:

1) If the “hole” the original blue arc went around actually bounded a disk in the surface, then this gives us the standard Hopf banding. As I’ve drawn it here, this would result in the positive Hopf band.

2) If the hole doesn’t bound a disk, then it is not a Hopf band as the monodromy it offers is neither right-veering nor left-veering.

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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.

**Twisting along a disk.**

**Twisting along an annulus.**

Of course, you’ll probably want the annulus embedded so that its boundary is not a trivial link…

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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

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

Yeah. So…. The exterior of the trefoil is a fibration over the circle with once-punctured tori fibers. Above shows a set of eight fibers, then those fibers animated. Note: On Flickr you have to look at the “original” size to view animated gifs.

Let’s look at it without the torus boundary from some other angles.

Here’s a few of those again, but a bit slower.

You may be getting an idea how I drew this. In each fiber you can see 2 somewhat flat 3-pronged regions that go around in a circle. A bit more tricky to see are the 3 somewhat flat 2-pronged regions that go around in another circle. In fact those two circles link each other once… like the cores curves of a genus one Heegaard splitting, a Hopf link.

This might help you see them, though black background and the translucence perhaps wasn’t the best choice for the gif.

Here’s two views of them from eight fibers at once.

The exterior of any (p,q)-torus knot is fibered over the circle, and each fiber may be viewed as taking p q-pronged disks around one component of a Hopf link, q p-pronged disks around another, and then joining all those prongs with pq bands. Of course you still have to fuss with hooking them up correctly. Then for the further fibers you do it again, rotating those disks around the circles a bit each time.

I’ve done this here with p=2 and q=3 to get our trefoil exterior. I made sure each pronged disk clocked around an appropriate amount, used Rhino’s “Blend Crv” function to make the edges of the bands, and then the “Network Srf” to actually make the band. Really, someone with more time should be able to write a program that automates this construction for any (p,q) torus knot or link with however many fibers. Of course there’s a few degenerate situations and other annoyances to reckon with…

And I’d be amiss if I didn’t mention that one could instead view these fibrations as Milnor fibrations. Daniel Dreibelbis has some produced some lovely representations of these fibrations with Mathematica. Here’s his fibration of the trefoil:

Maybe someone could tinker with his code to move the point at infinity in the stereographic projection onto the trefoil. Then you could have Mathematica automagically fiber the trefoil exterior!

Totally let me know if you do!

Let’s round off this looooong overdue post with a bit more eye candy.

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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.

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

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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.

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 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*.

In the three examples above, red fibers are shown at regularly spaced latitudes (along fixed longitudes) going from the north pole to the south pole. The yellow fibers are copies rotated to other longitudes.

Let’s look at just the red ones as the orders of the singular fibers increase.

Neat, I guess.

While we’re here. Have some eye candy. There’s more in this set.

I think all these below come from the order 1.

(Oh, and these pictures are a result of experimenting with Grasshopper for Rhino and some non-photorealistic rendering.)

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(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, . That’s a result of Thurston. Parker-Series [1], continuing work of Keen-Parker-Series [2], gives a triangulation of 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 -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 in by omitting the North Pole, placing the South Pole at the origin, choosing an embedding of the polyhedral 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 with its corresponding simple closed curve; at a representative point within each tetrahedron we center of copy of with its corresponding fundamental train track. To keep the parametrization of each copy of consistent we first view 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 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 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.

Generic train tracks for the tetrahedra:

Simple closed curves for the vertices (sans the poles):

The four pics of the full triangulation that make up the first picture:

The list of tetrahedra and vertices labeled with the Parker-Series notation:

And of course clicking on the pictures will take you to Flicker where you can view somewhat higher resolution versions.

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Through the double branched cover, we get the fibration of the solid torus by disks.

Here’s a few more views of these.

Simple as they are, there’s a pleasant rhythm.

There are a couple more on the Flickr (just click any of these pics). You have to view the “original” size for the gifs to animate.

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Take a knot with orange meridian and purple longitude.

For a null homologous knot, we may take the purple longitude to be the boundary of a Seifert surface. The orange meridian is the boundary of a meridional disk.

Let’s look at +4 surgery on the curve. The green curve will now bound a meridional disk, but we can’t see this disk fully until we reembed.

In the surgered manifold, the green curve now bounds a meridional disk. Since the surgery was integral, the orange curve is now a longitude. Since it was a +4 surgery, the purple boundary of our Seifert surface runs 4 times longitudinally.

Let’s push a copy of our new “dual” knot out of the surgery solid torus. We can make this copy parallel to the longitudinal orange curve.

We got this copy of the dual by a pushoff isotopy, so drag the purple Seifert surface along. The original dual we got from surgery now intersects the surface transversally once. The surface intersects the surgery solid torus in a single meridional disk. I shrunk the surgery solid torus.

Now let’s look back at the copy of the dual before we did surgery. Since it’s parallel to the orange curve, the dual is parallel to the meridian.

The Seifert surface of the dual can now be seen. It’s punctured once by the surgery solid torus, but that gets capped off by the surgery.

There’s a solid torus neighborhood of the surgery solid torus, the thickened dual knot and the parallelism of the dual knot to the orange meridian. Outside this, nothing has changed.

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Now the strands fit in a small square x I of the sphere x I.

This way it’s easier to stack lots of them.

This stack has a quarter turns between them.

The top row smooths the strands out a wee bit:

It never ends.

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