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I am extremely sorry if this is not the right place for this kind of question.

I have studied some knot theory, quantum invariants and would like to study more about categorification of knot invariants.

I have studied Khovanov homology (including D. Bar Nathan work on Khovanov homology for tangles) and H-F homology. But regarding more general approaches to categorification I do not know where to start. I have read L. H. Robert and E. Wagners work on combinatorial approach to categorification, which was great but it did not answer all of my questions.

To be more precise I would like to study works of M. Khovanov and B. Webster (or papers alike) on cateogrification of quantum knot invariants. But I do not know what one should study beforehand or at least not to what extent. My questions are:

  1. How much representation theory, algebraic geometry, homological algebra is needed to study those texts? Is it okay to just pick up definitions along the way or should I spend more time on some topics?

  2. What other papers on the topic are worth the read? Do there exist more suitable papers than those of Khovanov / Webster?

  3. Is there any paper which talks about different approaches to categorification and how do they fit together (some of this is covered in lecture notes by Q. You available on Khovanovs website)?

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