Classical thin position for links is a relatively simple idea: Draw a link diagram on a piece of paper, then think of a horizontal red line sliding from above the kink to below, keeping track of the number of points of intersection at each time that it’s transverse to the link. To get to thin position, we want to redraw the diagram so that the number of intersections at any given point stays as small as possible. This perspective makes it seem like thin position is about isotoping knots around, but Scharlemann and Thompson noticed that it’s actually about surfaces: If you think of those red lines as the projections of horizontal spheres, then thin position is the thinnest way to push a sphere from above the knot to below it. Each time the sphere passes through the knot is defined by a bridge disk whose boundary consists of an arc in the knot and an arc in the sphere. If you get rid of the knot and replace those bridge disks with compressing disks, you get Scharlemann and Thompson’s thin position for 3-manifolds [1].

Hayashi and Shimokawa [2] took this a step further by defining a type of thin position in which one looks at surfaces in an arbitrary 3-manifold containing a link and considers both compressing disks and bridge disks. (The relationship between these different types of thin positions is the basis for the axiomatic thin position that I wrote about earlier, and which is now carefully written up in a preprint.) Maggy Tomova has generalized Hayashi-Shimokawa thin position even more (and used this type of thin position very effectively) by adding into the mix what she calls a cut disk – a disk whose boundary is in the surface and whose interior intersects the link in a single point. It has taken me a long time to get used to and appreciate this idea, but now that I realize how perfectly cut disks fit in with the general thin position machinery, I would like to try and explain it below.

There are two points that need to be made about cut disks. The first is that adding cut disks produces a version of thin position that still has all the properties one would want. The second is that it is worth doing, i.e. that it allows you to prove new things. For the first, note that a cut disk becomes a compressing disk if you ignore the link, and so cut compression is the same as compression along this disk. However, when you put the link back in, the resulting surface ends up with two extra punctures from the link. In thin position, you want every type of compression to decrease the complexity of the surface. Cut compression keeps the Euler characteristic (of the surface minus the link) the same, so this seems like a problem. But in fact, one can get around this by choosing a different complexity that puts more weight on genus than on the punctures (since cut compression decreases genus but increases punctures). I won’t go into detail about this, but the curious reader should be able to work it out (or look it up in one of Maggy’s papers!) So adding cut disks greatly changes the character of the thin position, but it still works.

As for a situation in which cut disks are necessary, let’s look at the first time Maggy employed cut disks. She and Martin Scharlemann have an early paper showing that if a Heegaard splitting has high distance then any other Heegaard splitting of that manifold is either high genus or is a stabilization of the original. They prove this by showing that any two strongly irreducible Heegaard surfaces can be isotoped so that they intersect in a collection of loops that are essential in both surfaces. One of the Heegaard surfaces then cuts the other into pieces that are properly embedded in the handlebodies bounded by the first surface. The existence of these surfaces with essential boundaries gives the desired distance bound on the first Heegaard splitting. Maggy used cut disks to generalize this to bridge surfaces for knots, something that cannot be done using standard thin position.

The problem is with the definition of essential. In the Scharlemann-Tomova proof, the fact that the loops of intersection are essential comes from the fact that a trivial loop of intersection in one surface would define a compressing disk in the other surface. Using Scharlemann-Thompson thin position, they can reduce to a situation where there are not disjoint loops bounding compressing disks on opposite sides of either surface, and this can be used (via the Rubinstein-Scharlemann graphic) to eliminate all trivial loops of intersection.

Thus essential is defined as not bounding a disk subsurface of the given surface. However, a bridge surface is not closed, so there are two types of trivial loops – those bounding disks and those that are parallel into punctures. The second of these types of loops bound cut disks! So to get rid loops of intersection bounding once-punctured disks in one of the surfaces, we need need to make sure that neither surface has two disjoint loops bounding cut disks. In other words, we want to allow once-punctured disks to play the role of compressing disks in the original theorem. By adding cut disks into our notion of thin position, Maggy found a very nice and very natural generalization of the Scharlemann-Tomova result to bridge surfaces.

[1] Scharlemann, Martin, Thompson, Abigail, Thin position for 3-manifolds. Geometric topology (Haifa, 1992), 231–238, Contemp. Math., 164, Amer. Math. Soc., Providence, RI, 1994.

[2] Hayashi, Chuichiro, Shimokawa, Koya, Thin position of a pair (3-manifold, 1-submanifold). Pacific J. Math. 197 (2001), no. 2, 301–324.

For various reasons I think it would be interesting to have a theory that allowed cut discs with multiple punctures and had some sort of indexing present so you could keep track of the maximum number of punctures present and perhaps a weighting so that you prefer to untelescope using cut discs having fewer punctures.

One (rather weak) reason for creating such a theory is that you could then slide a cut disc over another cut disc and end up with a cut disc having a higher weight. This sort of sliding is necessary if you want to adapt Casson and Gordon’s proof that weakly reducible and not reducible implies that there is an incompressible surface. (Nowadays, Scharlemann and Thompson’s approach is favored by most people over Casson and Gordon’s, but there may be times when Casson and Gordon’s is more useful.)

Comment by Scott Taylor — May 5, 2011 @ 10:12 pm |

This is a very interesting idea. In principle, it should be possible to define such a complexity, by first comparing the genera of two surfaces, then only looking at the intersection number if the genera are equal. It’s hard to say exactly what the implications would be. Essentially, it would separate the 3-manifold thinning from the link thinning, i.e. you would find thin position by first weakly reducing the generalized Heegaard splitting as much as possible, then isotoping it with respect to the link to thin that as much as possible.

Comment by Jesse Johnson — May 6, 2011 @ 9:01 pm |