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In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly from the diagram, but I have not seen it spelled out completely explicitly, which is, I am afraid, the level needed for me.

I feel like the definition I can understand the best is the one given in Ozsvath-Szabo's 'Holomorphic triangles and invariants for smooth four-manifolds', page 382 (but 57, really). But still, could someone explain it on the level 'you see the diagram, you extract these numbers from it and combine them in such and such a way to get the dual spider number'? Perhaps also with an example? It seems to me, that to calculate this dual spider number, given a triply periodic domain and a triangle (is a triangle given by its domain?), one should put g (for genus) points on the surface in the complement of the attaching curves, then connect them to some intersection points by some arcs and count some intersection numbers with the translates (in some suitable direction) of the attaching curves.

In the paper I mentioned there is a formula right underneath the discussion of the triple spider number. What is meant by $\# \partial (\mathcal{P})$ in it? In fact, what I really want to understand, is how to count the first chern class of a triangle (given its domain) from the diagram.

Maybe I could also ask a small related question - in the formula for the Euler measure just above in the same paper, is it calculated by just summing the Euler characteristics of the regions minus a quarter of the number of the corners of each region? I cannot understand the presence of (#corner points of $F$) (with multiplicities) in it, i.e. its interpretation in terms of the diagram.

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