Morse functions inducing Heegaard diagrams Let $(\Sigma, \alpha, \beta)$ be a Heegaard diagram for a 3-manifold $M$, corresponding to a Heegaard splitting $M = H_1 \cup_\Sigma H_2$. There may be many self-indexing Morse functions $f: M \to \mathbb{R}$ with only one maximum and one minimum such that $\Sigma = f^{-1} (3/2)$. Intersecting the stable and unstable manifolds of $f$ with $\Sigma$ yields a Heegaard diagram $(\Sigma, \alpha',\beta')$ for $M$, and one can construct a Morse function $f$ such that $(\Sigma, \alpha, \beta) = (\Sigma, \alpha',\beta')$. I have the following questions:

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*Are any two Heegaard diagrams obtained from such self-indexing Morse functions slide-equivalent?

*Is there a unique (up to isotopy) Morse function $f$ such that $(\Sigma, \alpha, \beta) = (\Sigma, \alpha',\beta')$?

*If $\phi: M \to M$ is a diffeomorphism taking $(\Sigma, \alpha, \beta)$ to $ (\Sigma, \alpha',\beta')$, and $f : M \to \mathbb{R}$ is a Morse function producing $(\Sigma, \alpha',\beta')$, is $f \circ \phi$ a Morse function producing $(\Sigma, \alpha, \beta)$?

 A: *

*Are any two Heegaard diagrams obtained from such self-indexing Morse functions slide-equivalent?

Yes. Suppose that $V$ is a genus $g$ handlebody with boundary $S$.  Suppose that $\alpha$ and $\alpha'$ are "defining curve systems".  That is, both systems have $g$ curves, all curves bound disks, and $V$ cut along either set of curves yields a three-ball.  Then there is a sequence $(\alpha_i)$ of such systems so that $\alpha_0 = \alpha$ and $\alpha_n = \alpha'$ and $\alpha_i$ differs from $\alpha_{i+1}$ by a handle slide.  (This can be proven by induction on $|\alpha \cap \alpha'|$ and a surgery argument.)
Finally, any defining curve system is induced by some Morse function.


*Is there a unique (up to isotopy) Morse function $f$ such that $(\Sigma, \alpha, \beta) = (\Sigma, \alpha',\beta')$?

Yes, but the proof will be a bit painful.  Perhaps someone else has a reference for this.


*If $\phi: M \to M$ is a diffeomorphism taking $(\Sigma, \alpha, \beta)$ to $ (\Sigma, \alpha',\beta')$, and $f : M \to \mathbb{R}$ is a Morse function producing $(\Sigma, \alpha',\beta')$, is $f \circ \phi$ a Morse function producing $(\Sigma, \alpha, \beta)$?

Yes.  This is an exercise in the definitions.
