I am trying to make sense of what is going on in [Cas16] in terms of diagrams. Let me sum up the construction a bit, where $n\leqslant k$ are integers and $b\geqslant 1$ as well.
$C_{k,b,n}$ denotes the standard compression body, obtained from the thickened surface of genus $k$ with $b$ boundary components $\Sigma_{k,b}\times I$ and attaching $n$ 2-handles along non-separating essential curves on $\Sigma_{k,b}\times\{1\}$. $Z_{k,b,n}$ denotes $C_{k,b,n}\times I$. We see that: $$\partial C_{k,b,n}=(\Sigma_{k,b}\times\{0\})\cup(\Sigma_{k-n,b}\times\{1\})\cup\underset{=A}{\underbrace{(\partial\Sigma_{\bullet,b}\times I)}},$$ with $A$ a collection of annuli. Therefore, we can decompose $\partial Z_{k,b,n}$ as: $$\begin{align} \partial Z_{k,b,n}&=(C_{k,b,n}\times\{0\})\cup(\partial C_{k,b,n}\times I)\cup(C_{k,b,n}\times\{1\})\\&=\underset{=\partial^{inn}Z_{k,b,n}}{\underbrace{(C_{k,b,n}\times\{0\})\cup(\Sigma_{k,b}\times I)\cup(C_{k,b,n}\times\{1\})}}\cup\underset{=\partial^{out}Z_{k,b,n}}{\underbrace{(A\times I)\cup(\Sigma_{k-n,b}\times I)}}, \end{align}$$ where we respectively call $\partial^{inn}Z_{k,b,n}$ and $\partial^{out}Z_{k,b,n}$ the inner and outer boundaries of $Z_{k,b,n}$.
What I am interested in is the inner boundary. We can decompose it as follows: $$\partial^{inn}Z_{k,b,n}=\underset{=Y_{k,b,n}^-}{\underbrace{(C_{k,b,n}\times\{0\})\cup(\Sigma_{k,b}\times[0,1/2])}}\cup\underset{=Y_{k,b,n}^+}{\underbrace{(\Sigma_{k,b}\times[1/2,1])\cup(C_{k,b,n}\times\{1\})}}.$$
Supposedly, the decomposition $\partial^{inn}Z_{k,b,n}=Y_{k,b,n}^-\cup Y_{k,b,n}^+$ is the standard Heegaard splitting of some handlebody. My question is: What is this Heegaard splitting exactly?
I can see that the thickened surface $\Sigma_{k,b}\times I$ is diffeomorphic to $\natural^{2k+b-1}({\bf S}^1\times{\bf D}^2)$, and as such $C_{k,b,n}$ and $Z_{k,b,n}$ are handlebodies of genus $2k-n+b-1$. Now, if I follow along the construction, I see that $$\partial^{inn}Z_{k,b,n}\cong\Sigma_{k-n,b}\times I\cong\natural^{2k-2n+b-1}({\bf S}^1\times{\bf D}^2),$$ that $$Y_{k,b,n}^\pm\cong C_{k,b,n},$$ and that $$Y_{k,b,n}^-\cap Y_{k,b,n}^+=\Sigma_{k,b}\times\{1/2\}.$$ Therefore, am I supposed to see that the decomposition of $\partial^{inn}Z_{k,b,n}$ is the standard genus $k$ splitting with $b$ boundary components of the genus $2k-2n+b-1$ handlebody? If so, What is a Heegaard diagram of that splitting? It should have some set of parallel curves, some set of dual curves, and some holes of the surface left untouched:
The question is: how many are there in each of those three sets, and why?
I can find about no litterature on Heegaard splittings where the surface is allowed to have boundary components (what I suspect is called sutured Heegaard splittings, is that true?). The only thing which is clearly written is about Heegaard splittings with compression body (but the Heegaard surface is still closed).
However, in the papers, they seem to be comfortable with the notion, which makes things a bit difficult to work out the details. This, in particular, completely loses me whan it comes to comparing the different definitions of relative trisections; in [GK16] or [Cas16], they do not specify that the sectors should have genus $k$, but in [CGP18] they do, which makes things work differently when it comes to diagrams.
[Cas16] N. A. Castro. “Relative Trisections of Smooth 4-manifolds With Boundary”. PhD thesis. University of Georgia, 2016. Direct link.
[CGP18] N. A. Castro, D. T. Gay, and J. Pinzón-Caicedo. “Diagrams for Relative Trisections”. In: Pacific Journal of Mathematics 294.2 (2018), pp. 275– 305. arXiv link.
[GK16] D. T. Gay and R. Kirby. “Trisecting 4–manifolds”. In: Geometry & Topology 20.6 (2016), pp. 3097–3132. arXiv link.
