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I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. The definition involves counting the holomorphic representatives of $\phi \in \pi_2(x,y)$ when $\phi$ satisfies $\mu(\phi) = 1$.

First of all I do not understand how one might tell if $\mu(\phi) = 1$. The only definition of $\mu$ given in the paper is that it is the "expected dimension" of the moduli space of holomorphic representatives. I was wondering what is meant here? I have read other definitions of $\mu$ that define it as the index of an operator and while this may be precise I do not think it is what was intended to be used to work out the simple examples and exercises given in the paper.

Specifically, there are quite a few problems for the reader in the paper where a picture of the domain of $\phi$ namely $\mathcal{D}(\phi)$ is given and one is expected to compute $\mu(\phi)$ (see for instance page 17). I was wondering how one might even begin to do this?

For instance one can find a $\phi$ so that $\mathcal{D}(\phi)$ is the annulus given in figure $4$. We are later told in the paper that in this case $\mu(\phi) = 0$. How might one figure something like this out? In other examples $\mu$ is negative, how might one see this as well? Since the paper is for a beginner I am hoping there is an answer suitable for a beginner.

To sum up.

In what cases is it possible to just look at $\mathcal{D}(\phi)$ and use the picture to compute $\mu(\phi)$ and how does one go about it? I would be very happy if someone explained any of the examples given in the paper or simply point me to a reference. I feel like I am missing something simple and any help would be greatly appreciated.

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Look at later papers in which the definition is given combinatorially. I think one of these has Dylan Thurston as a coauthor. –  Scott Carter Mar 29 '12 at 9:33

2 Answers 2

up vote 6 down vote accepted

Robert Lipshitz has the nicest formula, described in Corollary 4.3 of this paper:

Robert Lipshitz, "A cylindrical reformulation of Heegaard Floer homology", Geometry & Topology 10 (2006) 955–1096, DOI: 10.2140/gt.2006.10.955, arXiv: math.SG/0502404

At this stage, I would not worry so much about the proof of the formula, although it's also not so bad.

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Thanks, this takes care of everything I asked and a whole lot more. I still have no idea how I was supposed to work out the examples in the original paper without knowing this formula for the index but I am not sure that question is useful. –  anonymous Mar 29 '12 at 13:21
3  
I'm not sure whether Ozsvath and Szabo knew this formula when they wrote the introduction you are reading. –  Dylan Thurston Mar 30 '12 at 9:41

Knowing that the maslov index is equal to the "expected" dimension of the moduli space of the disks is helpful too. For example in the figure on the right side of page 17, you can see that a holomorphic disk with boundary on $\alpha$ and $\beta$ can have "cuts" along the dashed lines, i.e. can send part of the boundary of the disk to the dashed lines. Since the length of these two "cuts" are variable, the moduli space is two dimensional and therefore the Maslov index is two. This also shows you that the moduli space is not compact. (Why?)

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