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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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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. –  Ben Webster Mar 27 '10 at 19:02
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If you convince me it would interesting, I might be able to do a few small ones by hand. –  Ben Webster Mar 27 '10 at 19:03
    
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. –  Charlie Frohman Mar 27 '10 at 19:12
    
By rank, I mean tensor with the rationals and find the dimension of the corresponding vector space. –  Charlie Frohman Mar 27 '10 at 19:13
    
What is the status of Stephan Wehrli and Ania Beliakova's approach? –  Charlie Frohman Mar 27 '10 at 19:15
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2 Answers

up vote 3 down vote accepted

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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Any at all. I have a conjectural approach and I wanted to compare answers. I am looking forward to your paper. –  Charlie Frohman Jun 25 '10 at 12:45
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