Hey Everyone!
So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain homology theory. My question is, why do this? I mean this in the sense that why is it important/interesting to have this "quantum invariants" $\rightarrow$ "homology theories" transformation? If this question is perhaps to basic and /or unsuited for the site I would appreciate some bibliography on the matter as Khovanov's "A categorification of the Jones polynomial" article doesn't seem to answer this question for me.
Thanks!



Why "Categorify"? Relating to link/knot homologies...
A good reason is that categorified invariants are usually more subtle than the uncategorified ones, and their additional structure gives more information about the knot/link. For example there are rather simple knots with the same Jones polynomial that are distinguished by their Khovanov homology. Furthermore, Kronheimer and Mrowka have recently proved that the Khovanov homology detects the unknot, while it is an open problem to determine if the Jones polynomial does or not.


Another very interesting categorification is the Knot Floer homology (defined by Ozsvath and Szabo, and independently by Rasmussen), whose graded Euler characteristic is the Alexander polynomial. This categorified invariant is also much subtle, for example (as proved by OS) it detects the genus of a knot, while the Alexander polynomial gives only bounds on it, and KFH detects fibred knots (Ni) while the Alexander polynomial gives obstructions about it.


