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Has anybody done computations of such a theory? Is there a place I could look up and see what the answers are for low crossing knots?

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    $\begingroup$ My suspicion is no. At the moment, there doesn't seem to even be consensus on the right way to do the categorification. Obviously, I have some ideas about the right way to do this, and I know Frenkel, Stroppel and Sussan are working on a representation theoretic qpproach which will hopefully be the same (I haven't seen a draft of their paper yet), but there's also things like Khovanov's paper on the subject, which I suspect is not. $\endgroup$
    – Ben Webster ♦
    Commented Mar 27, 2010 at 19:02
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    $\begingroup$ If you convince me it would interesting, I might be able to do a few small ones by hand. $\endgroup$
    – Ben Webster ♦
    Commented Mar 27, 2010 at 19:03
  • $\begingroup$ For instance I conjecture that the total rank of the homology of the trefoil corresponding to the second Jones Wenzl idempotent, normalized so that the unknot has invariant [3] is 9, and for the figure eight it is 15. $\endgroup$
    – Charlie Frohman
    Commented Mar 27, 2010 at 19:12
  • $\begingroup$ By rank, I mean tensor with the rationals and find the dimension of the corresponding vector space. $\endgroup$
    – Charlie Frohman
    Commented Mar 27, 2010 at 19:13
  • $\begingroup$ What is the status of Stephan Wehrli and Ania Beliakova's approach? $\endgroup$
    – Charlie Frohman
    Commented Mar 27, 2010 at 19:15

2 Answers 2

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Slava Krushkal and I have an alternative approach set inside of Dror Bar-Natan's universal construction. It should agree with results obtained by Webster and Frenkel, Stroppel Sussan. Computations are reasonable in our setting. We hope to place the paper on the arxiv shortly.

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Hi Charlie,

I did some calculations, but they are hard. Already for the unknot one gets an interesting, but infinite complex with cohomologies in all degrees!

We are just finishing a paper on this which hopefully will appear at the end of next week. Are there any specific knots you are interested in?

Catharina

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  • $\begingroup$ Any at all. I have a conjectural approach and I wanted to compare answers. I am looking forward to your paper. $\endgroup$
    – Charlie Frohman
    Commented Jun 25, 2010 at 12:45

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