## It’s full of surfaces!

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.

This is so beautiful!

Pierre Dehornoy said this on August 30, 2012 at 9:43 pm |

mindblown O_O

Lulu said this on December 3, 2012 at 5:52 am |