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.
