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).
