15
$\begingroup$

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?

For concreteness' sake, let's agree to represent the given contact three-manifold via an open book decomposition, though any other combinatorial encoding (e.g. a contact Heegaard splitting, a sequence of bypass attachments, etc.) would be equivalent from the standpoint of the question of decidability of tightness.

Note that overtwistedness is semi-decidable, meaning that if there exists an overtwisted disk, we can find it by simply running a naive search. The problem is that we don't know when to stop searching, so if the manifold is tight we will just keep searching indefinitely and never know the answer. Thus to solve the problem, it is sufficient (and in fact necessary) to devise some way of "certifying" tightness (by exhibiting some finite amount of data) which applies to every tight three-manifold. Is such an equivalent characterization of tightness known? I imagine that the important paper of Colin--Giroux--Honda (arxiv) must contain something close to this, but unfortunately I have not read it in enough detail to say for myself.

Note that the existence of a symplectic (semi-)filling implies tightness, but the converse is false by Etnyre--Honda (arxiv).

$\endgroup$
  • 1
    $\begingroup$ Here is a paper by Conway--Kaloti--Kulkarni which shows that the contact invariant in Heegaard Floer homology does not detect tightness: arxiv.org/abs/1409.4055 $\endgroup$ – John Pardon Aug 31 '15 at 6:58
  • 4
    $\begingroup$ I think that Andy Wand's work goes in this direction. At a first glance, arxiv.org/abs/1404.1705 does not quite contain what you want, but maybe the paper he has in preparation does. $\endgroup$ – Marco Golla Aug 31 '15 at 7:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.