Genus 2 Heegaard decompositions of rational homology 3-spheres I'm trying to build an example of a rational homology 3-sphere $M$ (that is not an integral homology 3-sphere) with an irreducible genus 2 Heegaard splitting so that $M$ is not a lens space or connect sum of lens spaces.
I haven't been successful trying to draw a Heegaard diagram for such a 3-manifold, so I suspect the problem is overdetermined. Is there an obvious reason for why this is?
Also, I suspect that some small Seifert fibered spaces might be a good place to look, but I haven't yet understood how to generally construct a Heegaard diagram for a general Serifert fibered space.
 A: Suppose that $K$ is the figure eight knot in the three-sphere $S^3$.  Let $n(K)$ be a small open neighbourhood of $K$.  Let $X = S^3 - n(K)$.  You can find a Heegaard diagram for $X$ as follows.
Let $N = N(K)$ be a closed neighbourhood of $K$, slightly larger than $n(K)$.  Draw both in a knot diagram of $K$.  Add to $N$ a small neighbourhood $P$ of an "unknotting arc" (also called a "tunnel").  So $V = N \cup P$ is a handlebody, and $V' = N \cup P - n(K)$ is a compression body. Note that $W = S^3 - \mathrm{interior}(V)$ is a handlebody as well.
So $(V', W)$ is a genus two Heegaard splitting of $X$. It is an exercise to give a diagram for this splitting.  (Hint: The diagram has only three curves, not four.  To find it, start by drawing a knot diagram for $K$.  Now add the unknotting arc $\alpha$ along the core of $A$.)  Any non-longitudinal Dehn filling of $X$ gives a rational homology three-sphere.  Most of these are not integral homology spheres.  The filling slope gives the fourth curve in the desired Heegaard diagram.
Note that something similar works for all two-bridge knots and links (and tunnel number one knots, in general). Doing this for torus knots will yield many Seifert fibered examples - the examples above are mostly hyperbolic.
