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.



Toroidal Heegaard splittings
In Hempel's "[3-manifolds as viewed from the curve complex](https://arxiv.org/abs/math/9712220)," 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.


