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...