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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?

flag	asked Mar 27 2010 at 17:50 Charlie Frohman 2,395●5●15

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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 2010 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 2010 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 2010 at 19:12

By rank, I mean tensor with the rationals and find the dimension of the corresponding vector space. – Charlie Frohman Mar 27 2010 at 19:13

What is the status of Stephan Wehrli and Ania Beliakova's approach? – Charlie Frohman Mar 27 2010 at 19:15

Ben · Accepted Answer · 2010-03-28 04:10:23Z UTC

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.

Catharina · Answer 2 · 2010-06-24 06:57:57Z UTC

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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

answered Jun 24 2010 at 6:57

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 2010 at 12:45

Computations of the Link homology categorifying the second colored Jones polynomial

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