An introduction to Khovanov homology, Heegaard-Floer homology

Question

I am interested in knot theory and low dimensional topology. I would like to start studying Khovanov homology and Heegaard-Floer homology.

I (partially) read the original paper of Khovanov and then watched an online lecture on Khovanov homology. I noticed that the lecture deals Khovanov homology more categorically. I think after the original work of Khovanov, people refined and generalized the definition or method of Khovanov homology.

So I would like to know how people deal Khovanov homology recently. Is there a standard textbook for graduate student on Khovanov homology?

(Also, I would like to learn Heegaar-Floer homology too. So if there is a standard text book fot this, please let me know.)

Answer 1

I highly recommend that you begin reading about Khovanov knot homology from the works of Dror Bar Natan.

In particular,

• On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2-16 (2002) 337-370.
• Khovanov's Homology for Tangles and Cobordisms, Geometry and Topology 9 (2005) 1443-1499.

His exposition is clean, intuitive, and motivated by the geometric/cobordism perspective. I dare say it's fun to read.

Regarding, Heegaard-Floer Knot Homology, I'd go to the source: Ozsváth and Szabó.