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?

**EDIT** of course, there are conditions which tell you that $M^3$ has nontrivial rational homology, but I am looking for something more...


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