Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev skein relation.

A universal finite type invariant can be obtained either from the Kontsevich integral, of from Chern-Simons theory. These invariants takes values in some space of Feynman diagrams (so-called chord diagrams) encoding some abstract Lie algebraic information. They are universal in the sense that every finite type invariant factors through them, up to invariants of lower order. In other words, every finite type invariant induces an invariant of chord diagrams (it is the easy part), and conversely any invariant of diagrams can be enhanced to a knot invariant (this is of course the hard part).

Kontsevich's construction can be extended to tangles, and arguably the better way of viewing it is as a functor from the category of tangles to the category of diagrams.

FInite type invariants are interesting because most of known numerical invariant are of finite type. In particular, setting $q=e^{\hbar}$ in any quantum invariant, one get a formal power series whose $n$th coefficient if a type $n$ invariant.

It turns out that many of these invariants were categorified. The main example is Khovanov's categorification of the Jones polynomial: so far I understand, it's a 2-functor from the category whose morphisms are isotopy classes of tangles and 2-morphisms are cobordisms, to a 2-category whose morphisms are certain chain complexes, and 2-morphisms are morphisms of complexes up to homotopy.

My first questions are:

Is there a widely agreed definition of what the categorification of a finite type invariant should be ? What about a universal one ? Is there an interesting 2-structure on the category of chord diagrams ? Or, as hinted here, is it better to categorify the spaces of diagrams themselves ?

Indeed, since this somehow amongs to construct a "universal homological invariant", one could imagine to replace the target category of diagrams by a category which is already a category of chain complexes, categorifying the 4T relation.

I googled some related buzzwords but did'nt find an answer so far. Of course, having the classical case in mind, a naive starting point is:

Does it happen something interesting if one extends Khovanov's construction to singular tangles ? Does it leads to an "homological invariant of chord diagrams" ?

Since Khovanov's construction categorifies the Kauffman bracket it's probably doable to see what happen when one flips crossings, but I'm not aware of a nice interpretation of the result.

Edit: To elaborate on my comment below. From a categorical point of view, finite type invariant are exactly the functors from the category of tangles to a "quantum" ribbon category $C$, by which I mean a graded, complete ribbon category whose braiding squares to a perturbation of the identity. Then the statement "every finite type invariant induces an invariant of chord diagrams" translates into "this functor induces a functor from the category of diagrams to the quasi-classical limit of $C$ (which is a so-called infinitesimal braided category)". It makes the relation with quantum groups rather obvious. Then Kontsevich's theorem, reformulated using results of Drinfeld, say something like "every infinitesimal braided category can be quantized". While I agree that categorifying one single finite type invariant hardly makes sense, I think that the above statements should admits nice 2-ification, which should be related to 2-Lie algebras.

Indeed, some authors are trying to categorify quantum groups themselves, among other things in order to recover Khovanov's construction by an analog of Reshetikhin-Turaev functor.

The paper pointed out by Chris is very interesting: it uses some notions I'm not familiar with, but gives a definition along the lines of: "an homological invariant is of finite type if its extension to a knot with enough singularity is acyclic" and prove that some perturbative expansion of Khovanov homology provide such invariants.

Edit 2: As a hint to a possible answer, let me mention this paper by Hinich and Vaintrob which gives a very nice "propic" construction of the space of chord diagrams out of the Lie operad, making rigorous the statement that it is universal among metrizable Lie algebras. I guess that their construction applied to the operad of 2-Lie algebras should provide a good candidate for the analog of chord diagrams in a 2-ification of the theory of finite type invariants.

  • $\begingroup$ I suppose I think of the "categorification" idea as only useful when there is some bigger structure involved that your numerical invariants are just slicing a piece off of. With Vassiliev invariants (as opposed to polynomial invariants) there's no clear categorical overstructure. But of course there's a clear bigger-picture: the Vassiliev filtration of the complement of the space of knots. Vassiliev invariants fit into that picture very naturally, that's how they first arose. $\endgroup$ – Ryan Budney Jul 28 '12 at 18:44
  • $\begingroup$ I agree that there is a point here, as it makes hardly senses to categorify one single numerical invariant. On the other hand, these numbers are in fact endomorphisms of the base field in some category, or better images of endomorphisms of the unit object in the monoidal category of diagrams (which are not numbers anymore) through some functor. $\endgroup$ – Adrien Jul 28 '12 at 19:42
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    $\begingroup$ Somehow, we can rephrase your objection by saying that categorification is clearly related to "non-perturbative" things (rational quantum groups, Jones polynomial, and Chern-Simons theory according to Witten) while finite type invariants are by nature perturbative things, related to perturbative expansion of CS theory, formal quantum groups and formal expansion of the Jones polynomial. $\endgroup$ – Adrien Jul 28 '12 at 20:08
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    $\begingroup$ I haven't actually read this paper, but it seems to give a nice categorification of Vassiliev invariants in terms of local systems on the space of knots: On the classification of Floer-type theories, Nadya Shirokova, arXiv:0704.1330. $\endgroup$ – Chris Brav Jul 29 '12 at 16:55

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