# Late night math thoughts April 14, 2010

David Gay

These are some thoughts on contact filling and packing questions, based on conversations with Olga Buse.

First observation: rather than embedding balls, as in the symplectic case, we try to embed solid tori (or higher-dimensional equivalents), i.e. (where means the open ball of radius in ).

So first we define the standard contact structure on by and

where is the coordinate (it stands for longitude'') and are polar coordinates on the 'th in .

Then, given a contact -manifold we are interested in finding embeddings for large values of . As we shall see when looking at the case , i.e. , this question probably only makes sense when we fix the smooth isotopy class of , and perhaps we also need to fix the ambient contact isotopy class of . (In dimension , fixing the ambient contact isotopy class of and the smooth isotopy class of means fixing a transverse isotopy class of framed transverse knots.)

Given , we can attempt some definitions as follows:

1. The supremum over all such that there exists an embedding . I claim that, at least in dimension , this is stupid because this value is always for the following stupid reason: Every transverse knot in has a neighborhood contactomorphic to for some small , and there exists the following embedding :

The image is , and if you iterate this embedding you can embed into for any . Also note that this definition would be stupid even if we fixed the ambient contact isotopy class of . But the point is that this embedding is not isotopic to the identity, since it changes the framing of .

2. Fix a transverse isotopy class of framed transverse knots , with being the knot and being the framing, and let be the supremum over all such that there exists an embedding with transverse isotopic to and with the framing coming from equal to . I claim that this is a little stupid, but not as stupid, for the following reason: First, recall that two framings of the same knot differ by an integer, but that in general there is no preferred 0 framing; i.e. the set of framings is an affine . Then observe that the observations in the preceding item show that either:
1. For all framings , , or
2. There exists some framing such that and such that, for all , .
Furthermore, in the second case, given any framing , is completely determined by according to some formula which can be derived from the explicit description of the embedding above.

3. Thus a better capacity is, given a transverse isotopy class of transverse knots , let , which is in .
4. What is the analog in higher dimensions? There are no framings now, and there is the funny contact squeezing that is isotopic to the identity. There does exist a definition, but I haven't looked at it carefully or internalized it.

So now on to packing questions:

1. Full Packing: Given closed and , does there exist a collection of radii and a collection of embeddings , with , with disjoint images, such that is the closure of the union of the images. First case should be .
2. Full Packing fixing isotopy classes: Same question as above, but fix the smooth isotopy classes of the embeddings in advance, and perhaps fix the ambient contact isotopy class of the images of the cores . This is presumably harder than the general full packing problem. But it would be interesting if, for example, there exist full -packings in dimension for some transverse knot types but not for others.
3. Packings without worrying about fully filling: This one I can only make sense of in dimension , I think. Pick a collection of transverse knots for which these framings mentioned above do exist, with for each . Can you disjointly embed copies of as neighborhoods of the 's realizing framings ? By choosing radius , we sidestep the individual capacity issues.

OK, its late now, so next time I'll write about possible strategies to either construct or obstruct, in analogue with blow-ups, etc. in the symplectic world. (Surgeries and handle attachments.)

Also, I want to write down the idea about possibly getting interesting symplectic embeddings of in by using knottedness, or by using -handle attachments.

David Gay 2010-04-15