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.
In Hempel's "3-manifolds as viewed from the curve complex," one of the main theorems is a necessary criteria for being toroidal. In particular, he shows that if a 3-manifold is toroidal then all of its Heegaard splittings, $\Sigma$, have $d(\Sigma) \leq 2$, where $d$ is the Hempel distance. Therefore, following the construction in the same paper for high distance Heegaard splittings gives you a large collection of atoroidal manifolds to work with.
Going the other direction, here is a way to obtain tori in a 3-manifold given by a Heegaard splitting (checking if these tori are essential is less clear to me). Suppose that $c_1$ and $c_2$ are curves on $\Sigma$ which bound an annulus in the handlebody determined by your $\alpha$ curves, as well as an annulus in the handlebody determined by your $\beta$ curves. Gluing these annuli together gives a torus in your 3-manifold, intersecting the Heegaard surface in $c_1$ and $c_2$.
In practice, a way to construct such $c_i$ is to start with $c_1$ and "slide" it over your $\alpha$ curves so that this sliding sweeps out the desired annulus. One can also generalize this construction to collections $c_1 ... c_n$ where $c_1$ and $c_2$ bound an annulus in the $\alpha $ handlebody, $c_2$ and $c_3$ bound an annulus in the $\beta$ handlebody etc.
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 $M_{2}$.
