Archive for April, 2009

Software for Pretty Diagrams

I wanted to make some pretty pictures for posts on this blog, and so I had to decide how to go about making them.  I like the hand drawn look, but it’s also nice to be able to edit using the computer.  In the end, I settled on scanning hand-drawn images, editing them in Gimp a bit, and converting them to SVG to edit more in Inkscape.

Since this is a lot of work, I have tried to automate it to the extend possible.  The result is a tiny program which works, somewhat, sometimes.  It makes diagrams that look a bit like whiteboard drawings, so I’m calling it whiteboard.  It’s badly written, badly documented, and it might set your computer on fire.  If that doesn’t scare you away, please use it for whatever you like, and if you do something cool I’d love to hear about it.

Knot Exteriors

Topologists often study knots by looking at what remains when the knot is removed from S^3.  I suppose they’re looking (somewhat ironically) at everything but the knot.   Hmm.  In any case, it works, so why don’t we let K be a knot in S^3 and define the knot exterior to be E(K)=\overline{S^3\setminus N(K)}, the closure of S^3 minus a regular neighborhood of the knot.

Two knot exteriors, with knotOne way to visualize knot exteriors is to take the projection of S^3 to \mathbb{R}^3 where the point at infinity is chosen to lie inside N(K).  In this case, the knot exterior looks like a 3–ball minus a knotted arc inside.  Two such exteriors are drawn at the right.  I’ve added the knot (a neighborhood of which we removed, remember) in blue.  The first is the exterior of the unknot, and the second is the exterior of a trefoil knot.    It’s pretty easy to see in the first case that the exterior is a solid torus.

The converse of this fact, that if E(K) is homeomorphic to a solid torus then K is the unknot, is also true, because a compressing disk for the solid torus gives a Seifert surface for the knot.

Some Simple Heegaard Splittings

Three simple Heegaard splittings

Consider picture I, in which we have two balls with blue circles on their boundary.  It’s not too hard to see that if we glue these two balls together so that the blue curves on their boundaries are identified, we obtain the space S^3.  In picture II we have two solid tori with blue curves, and again, if we identify the boundaries of these two in such a way that the blue curves are identified, we obtain S^3.

This second one might be slightly harder to see, so let’s look at it another way: take the first solid torus, and glue a thickened disk along its edge to the blue curve.   We consider the thickened disk to be a neighborhood of the disk bounding the blue curve in the second solid torus.  What’s left after we’ve glued this disk to the first torus?  A 3-ball.  Glue that sucker in, you get S^3.

In picture III we ‘ve changed things up a bit.  We still have to solid tori glued together, but this time the blue curves are different.  It’s an exercise for the reader to determine this space.  As a hint, it might be easier to see if you cut both solid tori along the disks bounding the blue curves.

l23_traced

An even more complicated (and difficult to visualize) example is pictured here.  In the top line I’ve written it as a union of solid tori, and the second line it’s written as a thickened disk attached to the trefoil knot in the boundary of the second solid torus.

These are all examples of decompositions of 3-manifolds called Heegaard splittings.  The first two are genus zero splittings of S^3, III is a genus one splitting of the mystery manifold M, and the last is a genus one splitting of a thingy called a “lens space.”

The point of all this is to see how a knot in bridge position gives a Heegaard splitting for a manifold associated to the knot, which I will write about next time.