This post is going to be a bit more technical than usual (though not necessarily any more coherent). As I’ve been working on porting thin position techniques to the analysis of large data sets and other arenas, I’ve had to spend a lot of time trying to understand how the fundamental ideas fit together, and one in particular is Scharlemann’s no-nesting Lemma. This Lemma says the following: Given a strongly irreducible Heegaard surface and an embedded disk with essential boundary in , you can always make the interior of disjoint from by isotoping away disks and annuli in that are parallel into . As I’ll describe below, it turns out that this Lemma in many ways encapsulates the fundamental properties of thin position.
Recall that a Heegaard surface is strongly irreducible if every pair of compressing disks on opposite sides of have non-trivial intersection (along their boundaries). If is not strongly irreducible then it’s called weakly reducible. It turns out that there is a converse to the no-nesting Lemma that implies that the no-nesting property completely characterizes strongly irreducible Heegaard surfaces. In other words, a Heegaard surface will be strongly irreducible if and only if for every embedded disk with essential boundary in , you can make the interior of disjoint from by isotoping disks and annuli in into and then across .
To prove this converse, I’ll use a trick that I learned from Yoav Rieck: If is a compressing disk for a handlebody and is a loop disjoint from then we can construct an essential annulus that has as one of it boundary loops by taking a small boundary parallel annulus along , then attaching it to with a band along . Yoav used this construction to characterize distance two Heegaard splittings, but we can also use it to characterize weakly reducible Heegaard surfaces. If is weakly reducible and are disjoint disks on opposite sides of then there is an essential annulus in the side that shares a boundary loop with . The union of this essential annulus and is a disk with essential boundary in whose interior cannot be made disjoint from by removing disks and annuli. Therefore every weakly reducible does NOT satisfy the no-nesting property.
Now, if you’ve read this far, then maybe you’ll follow me a little farther into obscure technical detail. If we want to generalize the Scharlemann no-nesting Lemma to other forms of thin position, we should translate it into the language of complex of surfaces and the disk complex. But the problem is that the disk complex only knows about actual compressing disks, and doesn’t see disks that intersect in their interior. But we can get around this by translating the no-nesting Lemma into an equivalent statement. (OK, both statements will be true and therefore equivalent, but the point is that the new formulation will follow relatively easily from no-nesting and vice versa.)
Here’s the equivalent statement: If is strongly irreducible and we compress one or more times to one side then the new surface will be incompressible to the opposite side. Note that this is almost the definition of strongly irreducible: If we compress along a disk on one side then any disk on the opposite side of will not be a compressing disk for the new surface, since its boundary will not be contained in the new surface. However, it’s conceivable that there will be a compressing disk for the new surface that wasn’t a compressing disk for the original surface. This is where Scharlemann’s no nesting Lemma comes in: If there were such a disk then it would be a nested disk for the original , so no-nesting tells you this can’t happen. Conversely, if you have a surface with a nested disk then compressing along innermost loops of intersection gives you one or more compressions on one side followed by at least one compression on the other side.
This new statement is about compressing disks in embedded surfaces, so this can be translated into the complex of surfaces, and thus imported into other forms of thin position. But in fact, this statement is equivalent (in the same sense as above) to the most fundamental property in standard thin position, namely the Casson-Gordon axiom: If is a Heegaard surface for a three-manifold with compressible boundary then is weakly reducible. I’ll leave it as an exercise for the reader to figure out how you get between the two statements. But the important point is that the Casson-Gordon axiom is essentially equivalent to Scharlemann’s no-nesting Lemma, so any version of thin position in which the Casson-Gordon holds will also have no-nesting.
(Update 11/1/12) Trent Schirmer just pointed out to me that the second formulation – That compressing a strongly irreducible surface to one side makes it incompressible on the other side – is equivalent to Jaco’s two-handle addition Lemma. This Lemma states that if you have a 3-manifold with compressible boundary and you attach a two-handle to the boundary along a loop that intersects every compressing disk, the resulting manifold has incompressible boundary. I’ll leave it to the reader to determine how they’re equivalent.