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In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface for a Heegaard Diagram associated to a three manifold, $Y$. This is contrasted with Ozsvath and Szabo's original count in $\text{Sym}^g(\Sigma)$. One main advantage of this setting is that $\Sigma \times [0,1] \times \mathbb{R}$ is quite easy to draw and visualize, whereas $\text{Sym}^g(\Sigma)$ is not.

My question is as follows: how do we actually compute this? In Figure 1 of the linked paper, Lipshitz draws what a holomorphic curve might look like, but this is for a Heegaard surface with only one intersection point. Are there examples explicitly computed out there? I am having trouble determining how they should look in general and how we can easily tell when there is a curve connecting two intersection points. In the original formulation of HF, counting holomorphic disks was very straight forward, if annoying in higher genus, so I expect this to not be too complex either.

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    $\begingroup$ Have you looked at Section 13 in the "cylindrical reformulation" paper? The so-called "tautological correspondence" seems to be what you're looking for: you're literally counting the same objects. $\endgroup$
    – Marco Golla
    Commented Nov 28, 2023 at 0:32
  • $\begingroup$ @MarcoGolla Sorry for the delay! I have looked through it. I haven't fully internalized all the details, but I don't think this quite does what I was hoping for: I was looking for clarification on how the holomorphic curves should look in the cylindrical version. I suppose I could pull back the disks in OS' version using this correspondence, but that seems a bit complicated, especially considering how late in the paper it comes? $\endgroup$
    – semper-lux
    Commented Dec 4, 2023 at 19:49
  • $\begingroup$ Part of the point of that paper was to prove invariance independently of the original papers (and show that invariance is easier in this context), so it's not surprising that the comparison arrives so late in the paper. As far as "how the curves look", the way I think of them is in terms of their projection onto the Riemann surface (i.e. by looking at their associated domain), but this is also how I "see" things in the original theory. $\endgroup$
    – Marco Golla
    Commented Dec 4, 2023 at 20:13
  • $\begingroup$ @MarcoGolla To be clear: if they’re showing invariance and such, you’d assume they had already defined the objects well enough to draw pictures and see curves. That’s what I meant. Let me be a little more clear! Here is a stabilization of S^3 for simplicity. There are no disks starting at z in this picture. What is wrong with the “holomorphic curve” I drew in the corresponding cylindrical version? Picture $\endgroup$
    – semper-lux
    Commented Dec 4, 2023 at 21:03
  • $\begingroup$ It looks to me like the boundary of the "curve" you drew is not null-homologous, so there can be no such curve. The way I think of these curves is as graphs over their domains, with "spikes" at the intersections of $\alpha$- and $\beta$-curves. $\endgroup$
    – Marco Golla
    Commented Dec 4, 2023 at 22:49

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