This is somewhat related to [this previous quesiton][1]. 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?


  [1]: http://mathoverflow.net/questions/136352/sufficient-conditions-on-non-haken-manifolds