It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus $\mathbb{T}^3$. In paritcular:  

It is true that every nice-enough 3-manifold admits a self-indexing Morse function $f:M\to[0,3]$ (with unique maximum and minimum), and from this we get a Heegaard diagram, with splitting surface $\Sigma=f^{-1}(\frac{3}{2})$. Now apparently, a converse also holds, so that given a Heegaard decomposition we can read off a self-indexing Morse function (*edit:* perhaps multiple Morse functions, as per Ryan Budney's comment).  

With the known Heegaard decomposition of $\mathbb{T}^3$ (handlebodies $N$ and $\mathbb{T}^3-N$ for a small neighborhood $N$ about the generating 1-complex $S^1\cup S^1\cup S^1\subset\mathbb{T}^3$, and splitting surface $\partial N$), **what is the corresponding Morse function?** I can't write one down.



Morse Function on 3-Torus from Heegaard Splitting
The Morse function on $(S^1)^3$ that gives the genus 3 Heegaard splitting (the standard one) is basically just a smoothed "distance from the 1-skeleton function". 


So if you think of $S^1$ as the unit circle in $\mathbb C$, then 


$$f : (S^1)^3 \to \mathbb R$$


is given by


$$f(z\_1,z\_2,z\_3) = |z\_1-1|^2 + |z\_2-1|^2 + |z\_3-1|^2$$


where we're taking the norm/modulus squared of vectors in $\mathbb C$ in the above formula. The critical points are 8 triples $(z\_1,z\_2,z\_3)$ of the form $(\pm 1, \pm 1, \pm 1)$, so you have the minimum $(1,1,1)$, maximum $(-1,-1,-1)$, index 1 critical points $(-1,1,1), (1,-1,1), (1,1,-1)$ and index two (the negatives of the index one critical points). 


If you really demand self-indexing then you'll need the function $\frac{f(z\_1,z\_2,z\_3)}{4}$


