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I am looking for some article that gives an introduction to Heegaard Floer homology of knot.

I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any introductory article. (http://arxiv.org/abs/1411.4540 and http://arxiv.org/abs/1003.6041 are Greek to me!)

Can you point me to some introductory article? Could you also please give me some example of calculation of knot Floer homology, say for "small" knots like $3_1$ and $4_1$?

If possible, can you also tell me why Heegaard Floer homology does not determine the unknotting number of 8_10?

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(Since this is just a string of references, I do not believe this constitutes a 'real answer' but it is too long for a comment, so I'm placing it in the answer field. Editors, please feel free to correct my etiquette.)

As for a general introduction or survey article, you might also look at these:

  1. "An introduction to Heegaard Floer homology" by Ozsvath and Szabo. https://web.math.princeton.edu/~szabo/clay.pdf

  2. "A introduction to knot Floer homology" by Manolescu. http://arxiv.org/abs/1401.7107

Regarding the second part of your reference request, about the calculation of the knot Floer groups of the trefoil or the figure eight; well, these are alternating knots, and so their Floer groups are completely determined by their signature and Alexander polynomials (see Theorem 1.3). However, I think what you are asking for is an explicit calculation from a Heegaard diagram. In the paper "Holomorphic disks and knot invariants" by Ozsvath and Szabo, you can find such a calculation for the trefoil in Section 6.1. However, this is not an introductory article --- it is full strength. You may also benefit from this expository article (in PDF form) written by Andrew Manion. His exposition also contains examples of explicit calculations, especially in Section 3.

Sometimes the grid diagram approach to calculating knot Floer groups makes for a gentler introduction. For that, you might look at the paper "A combinatorial description of knot Floer homology" by Manolescu, Ozsvath and Sarkar. In Section 4 there are explicit calculations for the Hopf link and the trefoil.

Finally, for the third part of your reference request, I don't really understand what you mean by 'does not determine the unknotting number,' but I think you should look at the paper "Knots with unknotting number one and Heegaard Floer homology" by Ozsvath and Szabo, in particular Theorem 1.1 and Corollary 1.2. (The arXiv version is linked). They use the Heegaard Floer homology of the branched double cover of a knot to give an obstruction to that knot having unknotting number one.

They apply their obstruction (the symmetry condition of Theorem 1.1) to show that the alternating knot $8_{10}$ does not have unknotting number one. I am under the impression this knot was already known to have $u(K)\leq2$, therefore they conclude it has unknotting number two.

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    $\begingroup$ This it totally on-topic, so you should not feel bad at all for posting it. It's a great and useful answer! $\endgroup$ – Andy Putman Jun 1 '15 at 17:22

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