I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this [expository paper](http://www.math.princeton.edu/~szabo/clay.pdf) 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.



Maslov index and Heegard Floer homology
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](https://doi.org/10.2140/gt.2006.10.955), arXiv:[math/0502404](http://arxiv.org/abs/math/0502404).


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


