TITLE: Toroidal Heegaard splittings
QUESTION [9 upvotes]: Suppose I have a Heegaard splitting of a closed oriented irreducible 3-manifold $M$, defined by the Heegaard diagram $(\Sigma_{g},\{\alpha_{1},\dots,\alpha_{g}\},\{\beta_{1},\dots,\beta_{g}\})$. Are there any obvious sufficient or necessary conditions for the attaching curves for when $M$ is toroidal (or atoroidal)?
Any sort of lead would be helpful.

REPLY [3 votes]: I have found the paper `The disjoint curve property and genus 2 manifolds' by Abigail Thompson, where she proves that if a Heegaard splitting for a 3-manifold $M$ does NOT have the disjoint curve property, then $M$ is atoroidal.
Let $M_{1}\cup_{\Sigma}M_{2}$ be a Heegaard splitting for $M$. Then the Heegaard splitting has the disjoint curve property if there exist essential simple closed curves $c, a$ and $b$ on $\Sigma$ where $c$ is disjoint from $a$ and $b$ on $\Sigma$, and $a$ bounds a disk in $M_{1}$, and $b$ bounds a disk in $M_{2}$.