**David Gay**

First, about *constructions* and *obstructions* for contact embeddings
of (
.

By the way, here's an observation I forgot to make: Recall our standard contact structure on given by and . Now consider two open subsets , both diffeomorphic to , with a symplectic ambient isotopy taking to (i.e. ). Then the map defined by is a contact isotopy for , and therefore is contact isotopic to . So when , we get that is contact ambient isotopic to if and only if the area of is the same as the area of .

*Constructions:* Some ideas:

- Take an open book supporting the contact structure and try to
fill the manifold with a standard neighborhood of the binding. This
is only right in dimension
, because it is only in dimension
that the binding is
-dimensional. (Although, in higher
dimensions, ostensibly the open book has pages which are Weinstein
manifolds, so the binding is contact, so the binding again has an
open book, etc... and you might get something useful by going all
the way down to the binding of the binding of the binding ...) To do this, it
seems to me you want to fill as much of the page as possible with a half-open
annulus, with boundary on the binding, and with the monodromy restricted to the
annulus equal to the identity. This monodromy should be the actual return map
coming from some Reeb vector field (transverse to pages) for some contact form
for this contact structure. Topologically the monodromy factors as
a product of Dehn twists, and Dehn twists are supported inside arbitrarily
small neighborhoods of curves, so you might think that you can get very large
neighborhoods this way, but you have to be careful because the return map
should really come from the contact form, not just any smooth return map
representing the given element of the mapping class group. If you do succeed in
doing this, then the thickness of the neighborhood is more or less the ratio
between the length of the binding and the area of the annulus, I think.
- Another open book idea is to try to fill the manifold starting with a
braided transverse knot for an open book. You want to see the knot as being a
fixed point of the monodromy, or a collection of points which are permuted, and
then you want to fill as much of the page as possible with a disk around
this point which is fixed under the monodromy (or a collection of disks around
the various points, which are permuted by the monodromy). Again, one expects
problems making the disks very big, and in particular I am imagining some kind
of pseudo-Anosov issue coming up, where a disk is forced to be stretched a lot
in one direction and shrunk a lot in a transverse direction.
- If you care about packings, you can try a combination of the above.
- There should also be constructions for algebraic knots coming from
algebraic geometry, although I'm curious how you can use algebra to understand
thickness, maybe that's not possible.
- Another idea is to do some efficient packings with many solid tori in
some other manifold and then do surgeries along some but not all of those tori
to get back to the contact manifold you care about.
- Can't really think of any other useful constructions, maybe there's
something very differential geometric in flavor where you just expand out from
the knot by some kind of exponential map until you are forced to crash into
yourself, then you stop. But that's probably not the kind of thing I know how
to do.

Still need to comment on obstructions, and to talk about polydisk embeddings.

Here's one thought on obstructions: If is overtwisted then there exist transverse knots which are very far from fully packing, in the sense that they have a standard overtwisted neighborhood and the standard tight neighborhood looks like it can never fill more than that. But I'm not sure how to make that precise.

More later...

David Gay 2010-04-16