Timeline for Khovanov homology and Crane-Yetter TQFT

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Jun 28, 2017 at 20:25	comment	added	Pavel Safronov		Yes, but the Crane-Yetter theory by itself is not very interesting (e.g. it's invertible). For instance, on a closed 4-manifold the invariant is expressible in terms of the Euler characteristic and the signature. I don't know if the CY invariant with a surface defect has been computed in the literature.
Jun 28, 2017 at 14:53	comment	added	Satoshi Nawata		@Pavel Thanks for your comment. So do you mean that, roughly speaking, $Z_{CY}(M)=Z_{CS}(\partial M)$? In that case, if we consider a defect supported on knotted surface in the Crane-Yetter theory, is it probable that it gives ``quantum'' surface knot invariant depending on $q$?
Jun 28, 2017 at 11:42	comment	added	Pavel Safronov		Categorification of Chern-Simons would be some 4d TFT $Z_{Kh}$ such that $Z_{CS}(\partial M)$ (which depends on $q$) is the character of $Z_{Kh}(\partial M)$ (the latter theory has no $q$-dependence). In the Crane-Yetter case (which depends on $q$) there is a map $f\colon Z_{CY}(\partial M)\rightarrow \mathbf{C}$ and you can recover the partition function $Z_{CS}(\partial M)$ as the image of $Z_{CY}(M)\in Z_{CY}(\partial M)$ under $f$. This looks different from what $Z_{Kh}$ is supposed to be.
Jun 28, 2017 at 10:04	history	asked	Satoshi Nawata	CC BY-SA 3.0