## January 14, 2007

### Khovanov Homology

#### Posted by Urs Schreiber

The workshop is over and finally there is some time to let all that information sink in.

One thing I wanted to learn, and which I learned now, are some basic ideas of Khovanov homology.

On my flight to Canada I looked at

Dror Bar-Natan
Khovanov’s Homology for Tangles and Cobordisms

and

Aaron D. Lauda, Hendryk Pfeiffer
Open-closed TQFTs extend Khovanov homology from links to tangles
math.GT/0606331.

This morning Aaron gave a nice talk on this work. I am far from having absorbed everything, but here are some notes.

A (plane diagram of a) link $L$ is something you obtain by drawing a couple of closed lines on a piece of paper, such that each time there would be a crossing you make a decision if you want to pass under or over.

Khovanov homology is the name for an assignment of a chain complex to any such link, such that this assignment is invariant under deformations of the link - known as Reidemeister moves – up to chain homotopy.

In other words, the homology of this chain complex is a strict invariant of the link. This invariant generalizes the Jones polynomial invariant of links in that the latter is reobtained as the (graded) Euler characteristic of the Khovanov chain complex.

As Lauda and Pfeiffer write:

This construction can be seen as a categorification of the unnormalized Jones polynomial, replacing a polynomial in one indeterminate $q$ by a chain complex of graded vector spaces. The coefficients of the polynomial arise as the dimensions of the homogeneous components of the graded homology groups of the chain complex in such a way that the degree corresponds to the power of $q$.

The nice thing about the precise procedure for computing the Khovanov homology of a given link – as far as currently understood – is that it can be understood as coming from a couple of “correlators” that a certain closed string 2-dimensional topological field theory assigns to a certain collection of worldsheets that are determined by the given link.

But it turns out that not every 2d TFT will do. We need one that satisfies a certain normalization condition (determining the partition function of the sphere and of the torus) and a certain funny extra invariance condition known as the four tubes law.

That four tubes law relates four different ways to propagate two closed strings to two closed strings. It is not all that involved, but for exactly which reason it is this law that ensures that we get a link invariant - and not some other law - remains conceptually a little unclear, as far as I understand.

The way one obtains a worldsheet from a link is rather nice: we form the cylinder over the entire link, except that in each neighbourhood of a crossing we “smooth out” that crossing by letting the corresponding surface be of saddle shape.

Since there are two different ways to insert a saddle for each crossing, there will be a total of $2 ^n$ different surfaces associated to a link this way.

An edge in the Khovanov chain complex associated to the link is obtained from taking certain subsets of these surfaces, computing the correlator of the given TFT on them and adding up the resulting linear maps in a certain way.

Unfortunately, the precise prescription for doing this seems to have little a priori motivation so far, except for the empirical fact that it happens to yield a link invariant in the end.

Anyway. If Khovanov homology is so closly related to closed topological strings, one is lead to wonder if there is a generalization to open/closed topological strings. This is exactly what Aaron Lauda and Hendryk Pfeiffer investigated.

A link in wich not only closed curves but also open arcs are admitted is known as a tangle. There is a more or less obvious generalization of the precription for obtaining surfaces from links to tangles which naturally leads to worldsheets with physical boundaries: “D-branes”.

Aaron and Hendryk carefully generalize every step in the construction of Khovanov homology to this more general case of tangles. They do indeed obtain an assignment of chain complexes to tangles that is invariant (up to homotopy) under deformations of tangles.

Posted at January 14, 2007 4:54 AM UTC

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### Re: Khovanov Homology

Unfortunately, the precise prescription for doing this seems to have little a priori motivation so far, except for the empirical fact that it happens to yield a link invariant in the end.

As I understand it, Khovanov homology (the version people talk about now) is a stripped-down version of a fancier construction involving category O for sl_2. This comes from his studies under Frenkel which (here my details get hazy) tie into Witten’s program and Geometric Langlands somehow.

Anyhow, my colleague Josha Sussan has just put his dissertation up on the arXiv: Category O and sl(k) link invariants. The goal here is to extend what Khovanov did for Jones (really the bracket, but that’s my own pet peeve) to the HOMFLY polynomial. There should be a more stripped-down combinatorial version eventually, which will make it accessible to those of us who aren’t up on the details of category O.

At the risk of creating job competition for myself, I’ll advise people to go to the arXiv and check it out. Josh does some really interesting work, even if he isn’t the biggest self-promoter out there.

Posted by: John Armstrong on January 14, 2007 8:38 AM | Permalink | Reply to this

### Re: Khovanov Homology

In that article Sussan writes:

In order to get a categorification of the quantum group, we consider the above categories to be a category of graded modules.

Perhaps John Baez had better broadcast his, Dolan and Trimble’s categorified quantum group theory.

Posted by: David Corfield on January 14, 2007 1:53 PM | Permalink | Reply to this

### Re: Khovanov Homology

We will. Lots of people are working on this stuff, but our approach is so different that I’m not worried much about ‘priority’ — even apart from the fact that I’ve just about given up on the overall rat-race of ‘priority’.

Posted by: John Baez on January 14, 2007 7:45 PM | Permalink | Reply to this

### Re: Khovanov Homology

No need, then, to give you the “Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard” lecture.

Of course now we know that, as regards violets, this is wrong. I haven’t even seen so much as a frost, let alone a snowflake, since Christmas.

Posted by: David Corfield on January 14, 2007 8:36 PM | Permalink | Reply to this

### Re: Khovanov Homology

More perfect timing: Khovanov and Rozansky have posted the SO(2N) one-variable specialization of the Kauffman polynomial (not to be confused with the Kauffman bracket) on the arXiv.

Posted by: John Armstrong on January 15, 2007 3:24 PM | Permalink | Reply to this

### Re: Khovanov Homology

Let me also point out Cautis and Kamnitzer’s work giving a geometric (rather than category O) origin for Khovanov homology.

It’s based on convolution structures on varieties coming from the loop Grassmannian. That’s the space Bott used to prove Bott periodicity, fifty years ago, and it still has much to teach us.

I asked Joel Kamnitzer about the relation of the SL(k) version of his stuff to Sussan’s. He’s not totally sure but believes there’s a GL(m) vs. GL(n) duality involved.

Posted by: Allen Knutson on January 17, 2007 7:02 PM | Permalink | Reply to this

### Re: Khovanov Homology

I remember Catharina Stroppel mentioning something about this when she was here last semester. I didn’t catch the full details, but the rough sketch seems to be that Category Os (“Categories O”?) are equivalent to certain categories of sheaves over certain topological spaces. Does this sound familiar?

Posted by: John Armstrong on January 17, 2007 8:00 PM | Permalink | Reply to this

### Re: Khovanov Homology

Oh, and I should also mention that the same sort of charges – “yes it gives a link invariant, but what’s the motivation?” – were levelled (and still are) at the Jones polynomial. From a classical knot theorist’s point of view there’s just no topology to be seen in Jones. The fact that Khovanov homology has settled some outstanding conjectures classical knot theorists had stated yields grudging admittance that the post-Jones branch of knot theory might be useful for something, but still where’s the topology?

Posted by: John Armstrong on January 14, 2007 8:45 AM | Permalink | Reply to this

### Re: Khovanov Homology

yes it gives a link invariant, but what’s the motivation

Personally, I don’t need further motivation for link invariants. I would just like to better understand why the precise way that Khovanov homology is computed, which looks kind of involved and ad hoc to the uninitiated eye, yields an invariant.

For instance, there are alternatives to Bar-Natan’s three conditions (sphere and torus normalization, as well as the four tubes condition) which one can use and still get invariants. Different invariants. So where do all these rules come from? How does one guess them?

I presume the experts have a feeling for that, while I certainly don’t.

Of course I gather that it is expected that in the end Khovaov homology will have to be understood as the “Wilson surface” observables of a 4-dimensional quantum field theory.

To me, the Jones polynomial is “explained” by the fact that it is precisely the Wilson loop observable of $su(2)$-Chern-Simons theory.

It is clear that any such observable is a link invariant and hence I don’t mind if writing out that invariant in detail happens to be involved, since its relation to the TFT turns it from something mysterious into something that is tedious but straightforward.

I guess similar statements should apply to Khovanov homology. It should really be the Wilson surface invariant of some 4-dimensional TFT. The only natural candidate I see is some flavor of BF theory, actually #.

Posted by: urs on January 14, 2007 7:13 PM | Permalink | Reply to this

### Re: Khovanov Homology

To me, the Jones polynomial is “explained” by the fact that it is precisely the Wilson loop observable of su(2 )-Chern-Simons theory.

Actually that’s exactly where the whole category O bit comes from: trying to categorify that observable. So now you do know the motivation, if not the details.

Posted by: John Armstrong on January 14, 2007 7:44 PM | Permalink | Reply to this

### Re: Khovanov Homology

A workshop - Winter School on Knot Theory and Representations - at UT Austin has just ended today, which dealt with category O amongst other things.

Posted by: David Corfield on January 14, 2007 8:44 PM | Permalink | Reply to this

### Re: Khovanov Homology

Urs wrote:

To me, the Jones polynomial is “explained” by the fact that it is precisely the Wilson loop observable of $su(2)$ Chern-Simons theory.

John Armstrong wrote:

Actually that’s exactly where the whole category O bit comes from: trying to categorify that observable.

Indeed, Khovanov homology is just one aspect of a big dream that goes back to Crane and Frenkel’s 1994 paper — the first place the word ‘categorification’ appeared in print. I described this paper in August 1994, back in week38 of This Week’s Finds.

Crane and Frenkel’s dream amounted to no less than categorifying the whole theory of simple Lie groups and everything associated to it, including quantum groups, Chern–Simons theory, the associated tangle invariants, and more! In his 2005 research proposal, Dror Bar–Natan makes it clear this dream is alive and well:

I know my size. The big dream of categorifying all of quantum algebra is too big for me. I’ll be happy to wach and add my iota, but others will do the bulk of the work.

Crane and Frenkel’s paper was called Four dimensional topological quantum field theory, Hopf categories, and the canonical bases. The title pretty much gives the story away. If you categorify a quantum group, you’d better get a Hopf category. Vectors in the canonical basis of your quantum group had better correspond to isomorphism classes of objects in this category. But most importantly:

If you’re going to categorify the Chern–Simons tangle invariants and get invariants of 2-tangles, you’d better try to find a 4d topological quantum field theory $Z$ that reduces to Chern–Simons theory in 3 dimensions when you form $Z(- \times S^1)$.

I described some ideas for what this field theory might be, way back in week50, week51, and week52. The main clue back then was the relation to Donaldson theory.

But that was before Khovanov & Co. had worked out the 2-tangle invariants. And, it was before we knew that every simple Lie group has a 1-parameter family of categorifications. And, it was before we knew anything about how these 2-groups are related to field theory and string theory. So, maybe it’s time for someone to think about this stuff again!

One thing we can now guess is that Khovanov homology is related to Donaldson theory.

Posted by: John Baez on January 14, 2007 8:29 PM | Permalink | Reply to this

### Re: Khovanov Homology

…the first place the word ‘categorification’ appeared in print. I described this paper in August 1994, back in week38 of This Week’s Finds.

Do you know when ‘categorification’ was first coined? You discuss a draft of Crane and Frenkel’s paper in TWF2, that was January 1993.

Posted by: David Corfield on January 15, 2007 12:30 PM | Permalink | Reply to this

### Re: Khovanov Homology

David Corfield wrote:

You [JB] discuss a draft of Crane and Frenkel’s paper in TWF2, that was January 1993.

Well in that paper there is a reference to

L. Crane and I. Frenkel
Categorification and the Construction of Topological
Quantum Field Theory
,
to appear in Conference Proceedings AMS Conference
on Geometry Symmetry and Physics, Amherst, MA, Summer 1992

which I can’t get my hands on. Can anyone else pick up the trail?

Posted by: David Roberts on January 18, 2007 12:02 AM | Permalink | Reply to this

### Re: Khovanov Homology

David wrote:

Do you know when ‘categorification’ was first coined?

Hmm. I guess not. I’d forgotten about the 1992 Crane–Frenkel paper that David Roberts mentions!

Posted by: John Baez on January 19, 2007 4:18 AM | Permalink | Reply to this

### Re: Khovanov Homology

I know my size. The big dream of categorifying all of quantum algebra is too big for me. I’ll be happy to watch and add my iota, but others will do the bulk of the work.

The Vassiliev invariants can be defined in other classification theories, and the Kontsevich’s integral formulae for these invariants had related the subject to QFT, to the graphs combinatorics, to the $\zeta$-functions and to the number theory and to many other branches of the mathematics. One should not under-evaluate the contribution of D. Bar-Natan to the development of this theory.(p. 8)