TITLE: Hakenness of Heegard splitting
QUESTION [8 upvotes]: This is somewhat related to this previous quesiton. Suppose I give you a Heegard splitting of $M^3$ of genus $g$ with a gluing map $\phi.$ Is there some condition on $\phi$ which would guarantee that $M^3$ was Haken?
EDIT of course, there are conditions which tell you that $M^3$ has nontrivial rational homology, but I am looking for something more...

REPLY [10 votes]: Yes, there is a well-known condition of Casson & Gordon, that a Heegaard splitting which is weakly reducible, but not reducible, gives a manifold which is Haken. In terms of the curve complex, this states that the Heegaard splitting has distance precisely $=1$. Distance 0 means reducible, distance 1 means weakly reducible, where we are considering the distances between the sets of meridians for the two handlebodies in the curve complex of the Heegaard surface.