TITLE: Background needed to understand modern research on knot homology theories
QUESTION [5 upvotes]: I am a student of mathematics, and have some background in 

Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), 
Differential Geometry (Lee, Kobayashi-Nomizu), 
Riemannian Geometry (Do Carmo), 
Symplectic Geometry (Ana Cannas da Silva) and 
Differential Topology (Hirsch, Minor (Morse Theory), Milnor (h-cobordism theorem)). 

I would like to learn Floer homology and Khovanov homology. 
Q1. Is my background sufficient to learn these topics right now? What are the standard first-level and second-level sources for these topics? 
Q2. At the research level, will I need to know any other areas to work on these topics? In particular, would I need to know algebraic geometry and to what extent? To give you an idea of my present knowledge, I currently know absolutely no algebraic geometry, and even my commutative algebra background has several gaps, especially in parts about DVRs.
Q3. Once I finish books recommended in Q1, what are some good papers to start reading on these areas? 
I would greatly appreciate reasoning behind any comments that aren't strictly factual and are opinions of the writer.
Thank you in advance! :)

REPLY [4 votes]: I'm not an expert in Floer homology or Khovanov homology, but if that's your goal I don't think you need quite as wide a background as suggested in the other current answer (though admittedly that answer was written when the title of the question was much broader). For example, in the talks I've seen about these things, I don't think much in the way of spectra comes up, or even homotopy theory at the level of May. My more modest suggestion is first to acquaint yourself with some more classical knot theory. Rolfsen is a very good start for the really classical stuff, while there are now many places to learn about the more modern skein-type invariants. My personal recommendation would probably be Lickorish, but there are several good books and surveys now about that material. After that, some symplectic topology might be useful, though you say you've already got some of that. Beyond that background, I think that this is an area that does not yet have many secondary textbook sources, but there are very many survey articles and I think the primary material tends to be written fairly well compared to some other areas of mathematics. Just Googling "Introduction to Khovanov homology" and "Introduction to Floer homology" brings up a lot of surveys and lecture notes. I would suggest diving into that stuff and then referring outward once you hit something specific that you don't understand or want more background about. Based on what you wrote in your question, I think your broad topological background is already in pretty good shape to get started.