Continuing musings on contact packing April 16, 2010

David Gay

First, about constructions and obstructions for contact embeddings of ( $ S^1 \times B^{2n}(r), \xi_0)$ .

By the way, here's an observation I forgot to make: Recall our standard contact structure $ \xi_0$ on $ S^1 \times \mathbb{R}^{2n}$ given by $ \xi_0 = \ker \alpha_0$ and $ \alpha_0 = d\lambda + r_1^2 d\theta_1 + \ldots + r_n^2 d\theta_n$ . Now consider two open subsets $ U, V \subset (\mathbb{R}^{2n}, \omega_0 = 2 r_1 dr_1 d\theta_1 +
\ldots + 2 r_n dr_n d\theta_n)$ , both diffeomorphic to $ B^{2n}$ , with a symplectic ambient isotopy $ \phi_t \colon\thinspace (\mathbb{R}^{2n},\omega_0) \to
(\mathbb{R}^{2n},\omega_0)$ taking $ U$ to $ V$ (i.e. $ \phi_1(U) = V$ ). Then the map $ \Phi_t \colon\thinspace S^1 \times \mathbb{R}^{2n} \to S^1 \times \mathbb{R}^{2n}$ defined by $ (\lambda,r,\theta) \mapsto (\lambda,\phi_t(r,\theta))$ is a contact isotopy for $ \xi_0$ , and therefore $ S^1 \times U$ is contact isotopic to $ S^1 \times V$ . So when $ n=1$ , we get that $ S^1 \times U \subset (S^1 \times \mathbb{R}^2,\xi_0)$ is contact ambient isotopic to $ S^1 \times B^2(r) \subset (S^1 \times \mathbb{R}^2,\xi_0)$ if and only if the area of $ U$ is the same as the area of $ B^2(r)$ .

Constructions: Some ideas:

  1. 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 $ 3$ , because it is only in dimension $ 3$ that the binding is $ 1$ -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.

  2. 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.

  3. If you care about packings, you can try a combination of the above.

  4. 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.

  5. 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.

  6. 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 $ (M^3,\xi)$ 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