At the AMS Sectional in Iowa City a few months ago, there were a number of talks about circular thin position (i.e. circular generalized Heegaard splittings). This is an idea that was introduced by Fabiola Manjarrez-Gutierrez for studying knot complements , though as she notes, it can be applied to any 3-manifold with infinite first homology. Alexander Coward gave a talk about using these ideas to study knots with unknotting number one (i.e. knots that become the unknot after a single crossing change) and he pointed out a difference between circular thin position and standard thin position that really blew me away: There are infinitely many circular generalized Heegaard splittings for the unknot that come from stabilizing the minimal thin position exactly once. Below the fold, I’ll give a brief description of circular thin position, then explain this surprising phenomenon.
First we need a slightly more general than usual definition of a compression body: A compression body is a manifold that you get by taking for a compact, orientable surface (possibly with boundary), then attaching some 1-handles to . The vertical boundary is , the negative boundary is and the positive bondary is the rest of the boundary. A circular generalized Heegaard splitting for a knot complement is a decomposition into compression bodies that intersect alternately along their positive and negative boiundaries and whose vertical boundaries make up the boundary of the knot complement.
Here’s how you should think about this: Start with a Seifert surface for a knot , and just for fun assume not the leaf of a surface bundle. Then you can’t isotope the surface around the knot and back onto itself, but you can do something almost as good. You can attach some tubes to , creating a higher genus surface that cobounds a compression body with the original. Then you can compress this surface on the other side to create another compression body. If you repeat this enough times, you will eventually make your way around the knot and back to the original surface . The trail of compression bodies defines a circular generalized Heegaard splitting.
To get to the idea that blew me away, I want to work in a slightly different setting, where the pictures are easier to draw. Consider a solid torus containing an unknotted core for a point in the disk . This is shown on the left in the figure below. A meridian of will intersect in a single point. If we take the double branched cover over , we’ll get a new solid torus, which I want to think of as the complement of the unknot. The meridians of lift to a family of spanning disks for the unknot that define its disk bundle structure.
Now lets add a kink to , as in the middle figure on the left. That is, I want to isotope it so that instead of going monotonically around the solid torus, it changes direction and backtracks for a short while, then changes back to its original direction and completes the circle. If we take the double branched cover again, we still get the unknot. But now a number of the meridians of intersect the branch set in three points instead of one, and just lift to punctured tori instead of disks. Plus there are exactly two disks that are tangent to . I’ve mentioned before that when you lift a bridge surface for a knot to the double branched cover, the resulting surface will bound a handlebody on either side. Similarly, if we lift one of the one-intersection meridians and one of the three-intersection meridians to the double branched cover of , the resulting surfaces will split the knot complement into two compression bodies, defining a circular generalized Heegaard splitting.
What we just did corresponds to a stabilization, since we added a one-handle and a canceling 2-handle in the generlized (circular) Heegaard splitting. But the interesting part comes next. Drag those two points where is tangent to the meridian disks away from each other and around until they pass again on the other side. This is shown on the right in the Figure above. Now they intersect each meridian in either three points of five points. So in the double branched cover, half the meridians lift to once-punctured tori and the other half lift to once-punctured genus-two surfaces. But this still determines a circular generalized Heegaard splitting for the same reason as before. So we’ve created a new circular position for the knot without stabilizing it further. And, of course, we can do this again to get the meridians to lift to genus two and three surfaces, and so on.
So, what we find is that there are circular generalized Heegaard splittings for the unknot with a single pair of handles such that the thin and thick surfaces have arbitrarily high genus. In standard thin position, we can’t do this because there’s no way to send the handles around the back, like we do here.