Surfaces in 3-manifolds up to positive Hopf plumbing

Let $Y$ be a closed 3-manifold. Let $S_Y$ be the set of compact, oriented, connected surfaces with boundary in $Y$. Let $\sim$ denote the equivalence relation generated by isotopy and positive Hopf plumbing, and consider the set $$\Sigma_Y= S_Y/\sim$$ of equivalence classes under this relation.

Problem: is there a natural geometric description of the set $\Sigma_Y$?

Let $S_Y^f\subset S_Y$ be the subset whose elements are fiber surfaces for fibered links in $Y$, and let $\Sigma_Y^f$ be the corresponding set of equivalence classes. Then the elements of $\Sigma_Y^f$ are in bijection with isotopy classes of contact structures on $Y$. The problem is asking about the geometric content of equivalence classes of non-fiber surfaces. 

Problem: define a Floer-theoretic invariant of elements in $\Sigma_Y$.

The Ozsvath-Szabo contact invariant assigns to each $\xi \in \Sigma_Y^f$ (aka each isotopy class of contact structures on $Y$) an element 

$$c(\xi)\in \widehat{\mathit{HF}}(-Y).$$

Perhaps there is a natural way to assign more generally to each $\xi\in \Sigma_Y$ a subspace

$$c(\xi)\subset \widehat{\mathit{HF}}(-Y).$$

For example, every $\xi\in \Sigma_Y$ is represented by some genus-$g$ surface $S$  whose boundary is a knot $K$. Let 

$$\mathscr{F}^S_{-g} \subset \mathscr{F}^S_{1-g} \dots \subset \mathscr{F}^S_i = \widehat{\mathit{CF}}(-Y)$$

be the Alexander filtration associated with $S$ and $K$. Is the image of the inclusion-induced map on homology $$i_*:H_*(\mathscr{F}^S_{-g})\to \widehat{\mathit{HF}}(-Y)$$ an invariant of $\xi$? (This is one way to define the contact invariant in the case that $\xi$ is really an element of $\Sigma_Y^f$.)

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