I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these kinds of things have been known since the '60's and '70's. Instead, I mean two kinds of things.
- Questions that make no reference to Heegaard-Floer theory that can be answered using Heegaard-Floer theory (and, preferably, cannot be answered in other ways).
- Things about knots or 3-manifolds that can be computed with Heegaard-Floer theory for which algorithms did not previously exist.
I'm asking this question here in response to a large number of talks about Heegaard-Floer theory I've attended over the years. It seems like a hot subject and lots of talented young people are working on it, but most of the talks I've attended about it addressed what seemed to me to be technical questions internal to the subject. And when I've asked the speakers this question, I never seem to get a good answer. But since it is such a hot subject, I assume there must be some killer applications.