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May 26, 2007

Link Homology and Categorification in Kyoto

Posted by John Baez

Aaron Lauda points out an interesting conference:

You can see notes for some of the talks!

Categorifying knot invariants is all the rage, as these talks show:

The last talk here hints most strongly at the big picture. But, it’s best to look at all the talks, including ones I haven’t listed here, if you want to understand what’s really going on. It’ll take a lot of work unless you’re pretty familiar with quantum invariants of knots, since the notes are quite terse.

Maybe some people who went to the conference can say a bit about the high points?

Posted at May 26, 2007 6:38 PM UTC

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Re: Link Homology and Categorification in Kyoto

For those who are interested in this stuff, I’m sure there will be plenty of it at the Faro conference, which I’ll be talking about on my own weblog while I’m there in early July.

Posted by: John Armstrong on May 26, 2007 7:54 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

As Aaron is not in the same position as Derek and Jeff, could he be persuaded to tell us what he thought of this conference, or perhaps something on whether he feels ‘categorification’ means something different to those working around Khovanov homology?

Posted by: David Corfield on May 27, 2007 12:12 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

I can say for a fact that many people working on knot homologies from the knot theory side think it means something different. Here’s a rough sketch of the evolution as filtered through my experience.

“Categorification” in the sense we know it is introduced. Khovanov, as a student of Frenkel, brings representation theory to bear on certain categorifications and develops a homology theory whose Euler characteristic is the Kauffman bracket (called “Jones polynomial” in the usual confounding of the terms). Knot theorists’ only exposure to the term “categorification” is through this homology, and they identify the term as specifically referring to this sort of construction.

This became apparent to me at the AMS meeting in Oxford, OH this past March, when a number of the participants in the quantum topology session were confused by my use of the term “categorification” over dinner one night. I was referring to the possibility of different categorifications of the bracket that might have nothing to do with homology theories. Since then I’ve become excited about categorifications by anafunctors, and I’m hoping to talk about them with Dr. Przytycki and his group at George Washington over the summer to help bring them into the wider world of categorifications.

Posted by: John Armstrong on May 27, 2007 12:58 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

I wouldn’t have characterised knot theorists’ understanding of categorification in this way. It may be true that there are many people interested in Khovanov homology who don’t know about the wider picture, but that’s not universal, and certainly not the case amongst most of the people at the Kyoto conference, for example.

Perhaps a better way to put it is the following! This is perhaps the “big question” about Khovanov homology for the “Baez school”. :-)

“Why are triangulated categories unreasonably effective in producing good categorificiations?”

(Saying triangulated categories here is perhaps just a fancy way of saying ‘theories involving homology’.)

To back up this claim of unreasonable effectiveness, let me quickly (?) describe the Bar-Natan model of Khovanov homology, in a way that emphasises how things “get better” when we pass to a triangulated category (that is, switch to thinking about complexes up to homotopy). This will follow my introductory talk in Kyoto, slides at http://tqft.net/kyoto1, especially pages 3,8,10-12.

The Bar-Natan model produces Cob(su2), a 2-category. Its objects are ‘points on a line’, its 1-morphisms ‘Temperley-Lieb diagrams’, but with no relations between them, and its 2-morphisms are (linear combos of) surfaces between Temperley-Lieb diagrams, modulo some relations. (The surfaces have to have ‘vertical boundary’ matching the the sources and targets of the source and target 1-morphisms.)

The relations are ‘sphere = 0’, ‘torus = 2’, and ‘neck cutting’, which says ‘cylinder = 1/2 ((punctured torus and disc) + (disc and punctured torus))’.

Let’s now pass to the ‘matrix category’, allowing formal direct sums of objects and matrices of surfaces. Now these relations allow you to prove an isomorphism: the circle is isomorphic to the direct sum of two empty diagrams. If you go back through everything above and introduce gradings the right way, you’ll in fact see that these two empty diagrams are shifted up and down in grading by one.

What does this say? That the Grothendieck group (we better take the split group here, as we’re not in an abelian category), is exactly the usual Temperley-Lieb 1-category. (Actually, from what I’ve said here, it might just be a quotient, but this is easy to patch; see, for example, math.GT/0612754.)

Thus we can say: “Cob(su2) categorifies TL, as a tensor category”. Of course, TL is a braided tensor category! How do we see Cob(su2) as a categorification of TL like that? The answer turns out to be “switch to complexes over Cob(su2), up to formal homotopy”. The content of the proofs of the invariance of Khovanov homology then translates to: “Kom(Cob(su2)) decategorifies (via a triangulated Grothendieck group) to TL as a braided tensor category.”

And essentially this story is repeated all across knot homology. We make some construction, which decategorifies to some familiar gadget (for example, the MOY skein, for sun polynomials), but there’s no sign of the braiding until we pass to a category of complexes. Even further afield, the work of Stroppel et. al, and of Kamnitzer and Cautis (see the conference notes, linked earlier in the thread), are finding braidings in algebraic geometry, but in every case you have to insert the words “derived category of” in the right place to make things work.

I don’t know the answer to this, but perhaps you guys do!

Posted by: Scott Morrison on May 28, 2007 2:03 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Scott,

That’s a fascinating question about braidings. I guess in representation theory one finds braid groups appearing on the derived level because the fundamental braid symmetries in Lie theory factor through Weyl group or Hecke algebra actions when one factors through cohomology or K-theory.

I think the same comment applies in mirror symmetry - e.g. the constructions of braid group actions through spherical twists (which if I understand is what Cautis-Kamnitzer use?) factors through a reflection group action on K-theory.

I wonder if there is any general result along these lines?

I guess this doesn’t explain why you need derived as opposed to abelian categories though… my favorite answer for that is that derived categories are “function spaces” in the sense that you can do harmonic analysis on them — e.g. you have good pullbacks and pushforwards and integral kernels, and functors between them are given by correspondences — like K-theory or cohomology. But I’d be curious to hear a more pertinent answer :-)

Posted by: David Ben-Zvi on May 28, 2007 4:52 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

But I’d be curious to hear a more pertinent answer

You mean it’s not because you’re all secretly doing 2D TFT?

Posted by: Aaron Bergman on May 28, 2007 6:39 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Scott said, “Why are triangulated categories unreasonably effective in producing good categorificiations?”

Scott, do you mean categorifications in general? Or those specific to knot theory? If the latter, I have absolutely nothing to say, but I can take a stab at the general question. Beware that I’m not at all an expert.

A derived category is in many ways a linear analogue of the stable homotopy category. So the nonlinear, unstable analogue of your question would be Why are homotopy categories unreasonably effective in producing good categorifications?

My feeling is that question is a little like the question of why the number 1 is so useful in counting. An nn-category ought to give rise to a “shadow” mm-category by forcing the kk-isomorphisms for km+1k\geq m+1 to be equalities. In the case n=0n=0, we get the set of isomorphisms classes. In the case n=1n=1, I think we ought to get the homotopy category.

Now if categorification does anything, it makes nn-categories for n1n\geq 1. In particular, taking the m=1m=1 shadow, it makes homotopy categories. So a homotopy category really ought to be the first thing you get to when categorifying.

Put another way, the reason why theories involving homology are useful in categorification is simply that homological algebra is the linearization of simplicial algebra, and simplicial algebra is essentially what categorification is about.

Perhaps people who have written papers on these things can correct me now.

Posted by: James on May 28, 2007 10:07 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Let me rephrase Scott’s question from the higher category theory perspective.

Why do we like Khovanov homology? Because it gives invariants of 2-tangles! (The sign problem having been resolved in math.GT/0701339.) What’s “the right way” to have an invariant of 2-tangles? According to the periodic table philosophy, it’s to have a 2-monoidal 2-category with duals (that is, a 4-category with only one 0-morphism and only one 1-morphism) as explained in math.QA/9811139. (From here on in every time I say “category” I’ll leave “with lots of duals” implicit.)

From this perspective what does the sentence “Khovanov homology is a categorification of the Jones polynomial” mean? Well “decategorification” means take (split) Grothendieck group. What does this do on the periodic table? If you have a 1-monoidal 1-category, then its Grothendieck group is a ring, which is a 1-monoidal 0-category (everything here is enriched over Vect). Similarly if you have a 2-monoidal 1-category (a braided category) its decategorification is automatically abelian, that is it is a 2-monoidal 0-category.

So, when we take the 2-monoidal 2-category KHOV and we decategorify it we should get a 2-monoidal 1-category. But that’s just a braided category! And the Jones polynomial comes from a braided category. So the whole theory of Khovanov homology is just that we have some 2-monoidal 2-category KHOV whose split Grothendieck group is just the 2-monoidal 1-category JONES (or Rep(U_q(sl_2)) if you’re so inclined).

Alright, now we’re ready to ask the question. Why is it that if you start with an abelian 2-monoidal 1-category its categorification is triangulated? Restated, why are all interesting known 2-monoidal 2-categories triangulated? Why doesn’t that issue appear at the 2-monoidal 1-category level? What’s going on here?

Posted by: Noah Snyder on May 29, 2007 10:11 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Noah, thanks, but what’s does it mean for a (2-monoidal) 2-category to be triangulated? I thought we were talking about triangulated 1-categories. Perhaps if I knew anything about KHOV, it would be clear.

Posted by: James on May 29, 2007 11:20 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Good question. Let me try to unpack things a little bit. The category that Scott was talking about was a category of (complexes of formal direct sums of) cobordisms between Temperley-Lieb diagrams. However, Temperley-Lieb diagrams themselves form a category! The objects are unions of points and the Temperley-Lieb diagrams are the morphisms. So this all fits into a 2-category: objects = points, morphisms = TL diagrams, 2-morphisms = complexes of cobodisms of TL diagrams modulo the Bar-Natan relations. Note that we are only taking complexes in the 3rd (cobordism) dimension. We don’t allow complexes of unions of points (with TL diagrams between them).

So the triangulated categories that come up are categories of morphisms in a 2-category. So I guess I’m calling a 2-category triangulated if its morphism categories are triangulated. Maybe we need a bit more. I’m not entirely sure what the right definition is. Somehow the triangulated structure only comes in on the 2-morphism level.

Posted by: Noah Snyder on May 30, 2007 12:33 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Let’s see if I understand Scott’s post.

We start with the 2-category Cob(su 2)Cob(su_2) as he described. Then we form a new 2-category Kom(Cob(su 2))Kom(Cob(su_2)) whose hom-categories are the (derived categories?) of the (chain-complex version of the?) hom-categories in Cob(su 2)Cob(su_2).

And Kom(Cob(su 2))Kom(Cob(su_2)) is a 2-monoidal 2-category, in other words a braided monoidal 2-category.

Then we decategorify Kom(Cob(su 2))Kom(Cob(su_2)) by forming the category whose hom-sets are the grothendieck groups of the hom-categories in Kom(Cob(su 2))Kom(Cob(su_2)). The resultant braided monoidal category turns out to be equivalent to TLTL.

In this sense one says that “Cob(su 2)Cob(su_2) is a categorification of TLTL”. Is that right?

I’m interested in nice explicit examples of “braided monoidal 2-categories with duals”.

Posted by: Bruce Bartlett on May 30, 2007 12:35 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

I think you have this basically right Bruce. But a couple clarifications.

Cob(su_2) is a monoidal 2-category with duals (though clarifying exactly to what extent it “has duals” is an area of active research).

Given any 2-category C, we can make Kom(C) which has the old objects, but where the 1-morphisms are complexes (that is a bunch of 1-morphisms f_i with 2-morphisms d_i: f_i -> f_i+1) and 2-morphisms are chain maps up to homotopy.

In particular we have a new 2-category Kom(Cob(su_2)). This 2-category isn’t just monoidal, it’s braided monoidal! (It also still has duals to some extent, though see the above caveat.)

So a simply form of the question (without reference to triangulated stuff) is why do we seem to get braided monoidal 2-categories by looking at complexes in certain monoidal 2-categories?

Finally, n-category theory cafe readers should note that “canopolis” means roughly “monoidal 2-category where the 0- and 1-morphisms have duals.” This will help you read the relevant Khovanov homology literature.

Posted by: Noah Snyder on May 30, 2007 1:58 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

In Noah Snyder’s 10:11 post yesterday he mentioned that KhoHo gives invariants of 2-tangles. But for closed surfaces these invariants are trivial. See: Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory, by Tanaka, Khovanov’s invariant for closed surfaces, by Rasmussen, and our paper Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers with Masahico Saito and Shin Satoh. Maybe Scott Morrison can tell me if one has a non-closed cobordism if these invariants are non trivial — I am not sure what that means. In the closed case you are looking for a number.

Later in this thread, (12:35 AM) Bruce Bartlett asked for explicit examples of braided monoidal 2-categories with duals. Of course, he means something other than the catgory of 2-tangles:
Higher-Dimensional Algebra IV: 2-Tangles, JB and Laurel Langford.
So anyway, the joke around here was when any representation theorist would visit, I would ask, do you have a good example of a braided monoidal 2-category with duals. And no one did.

So I stopped asking the question. When we (CJKLS) figured out the quandle cocycle invariant Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces the question ceased having meaning to me. But I have a very narrow minded point of view of the subject: The reason for finding a braided monoidal category with duals was to find a knotted surface invariant.

Our original plan of attack was to try to use Neuchl’s cocycles (I don’t think that paper is on the ArXiv?) to construct knotted surface invariants.

So there are a couple of good lead’s for Bruce’s question. First, see if Neuchl’s example has duals. A good approach would be to finish the work that Masahico, Laurel, and I started: 1st get the knotted surface invariant, the duality structure will poop-out of JB’s work with Laurel. The other approach that I always thought would work would be to just construct the category you desire from the quandle cocycles.

There are new reasons that *I* believe that this will work. In Cohomology of the Adjoint of Hopf Algebras and Cohomology of Categorical Self-Distributivity we are seeing cocycles appearing very 2-categorically looking.

Posted by: Scott Carter on May 30, 2007 7:21 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Scott wrote:

So anyway, the joke around here was when any representation theorist would visit, I would ask, do you have a good example of a braided monoidal 2-category with duals. And no one did.

So I stopped asking the question. When we (CJKLS) figured out the quandle cocycle invariant Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces the question ceased having meaning to me. But I have a very narrow minded point of view of the subject: The reason for finding a braided monoidal category with duals was to find a knotted surface invariant.

Right — from that viewpoint, once you get your invariant of knotted surface invariant you can forget about braided monoidal 2-categories. But, if you actually like braided monoidal 2-categories — as some of us do — you can turn the process around: take any invariant of knotted surfaces and look for the braided monoidal 2-category underlying it! Of course this can only work if your invariant can be defined on 2-tangles (not just closed knotted surfaces).

Another thing: I think Aaron Lauda and Hendryk Pfeiffer believed they could use their work to squeeze a braided monoidal 2-category out of Khovanov homology. However, they felt there was not enough interest on the part of the Khovanov homology crowd to make this work worthwhile. I hope this changes someday.

Posted by: John Baez on May 31, 2007 12:55 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

I mentioned a promising candidate for an easy-ish braided monoidal 2-category with duals a while ago at the end of this post. I think I can now even take this further : every rational vertex operator algebra equipped with a finite group of automorphisms should give you an invariant of 2-tangles :-)

Posted by: Bruce Bartlett on May 31, 2007 8:49 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Bruce wrote

I think I can now even take this further […]

That’s really cool, Bruce! I wish I had more time currently to walk along such quantum paths with you – along the third edge, you know.

Unfortunately, right now I am way down a different edge. So I’ll just trust that once I get back to the Quantum-Do I’ll just pick up your thesis and be enlightened. :-)

Posted by: urs on May 31, 2007 10:26 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

The trouble is I’d need some help for this project (of linking vertex operator algebras to invariants of 2-tangles, via explicit examples of objects inside braided monoidal 2-categories with duals).

I think I could come up with the braided monoidal 2-category with duals : that’s actually the easy part. Caveat : one might need to weaken slightly the notion of a braided monoidal 2-category with duals as presented in John and Laurel’s 2-tangles paper… and you’d need to check that this weakening doesn’t do violence to the construction of 2-tangle invariants.

So I guess I need help from (a) an expert on vertex operator algebras (since I know very little about them) and (b) an expert on 2-tangles. And “higher” help (of the higher categorical nature) would always be appreciated too.

Posted by: Bruce Bartlett on May 31, 2007 1:08 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

The trouble is I’d need some help for this project

Let the Café be with you.

Posted by: urs on May 31, 2007 1:28 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Bruce Bartlett asks about 2-tangles. I think I am familiar with these. But Vertex Operator Algebras still scare me.

JB says, “But, if you actually like braided monoidal 2-categories, as some of us do, you can turn the process around: take any invariant of knotted surfaces and look for the braided monoidal 2-category underlying it! Of course this can only work if your invariant can be defined on 2-tangles (not just closed knotted surfaces).”

The cocycle invariants *should* give such invariants. This will be analogous to Masahico’s work with Kheira Ameur On classical tangles . Maybe there are problems with colorings of the boundary, but I don’t see how that can prevent the construction of the invariant. So I am pretty sure there is a braided monoidal 2-category sitting there.

Posted by: Scott Carter on June 1, 2007 10:56 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

However, they felt there was not enough interest on the part of the Khovanov homology crowd to make this work worthwhile.

I find this statement rather disturbing. How many of the very best pieces of science and mathematics might not have occurred had their authors thought like this?

Posted by: David Corfield on June 3, 2007 9:34 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

David wrote:

How many of the very best pieces of science and mathematics might not have occurred had their authors thought like this?

Don’t put it in the subjunctive like that! It’s a very common occurence, more the norm than the exception. Young scientists often ask themselves if working on a lengthy and difficult project will pay off in a tenured job or not. So, research gets focused on ‘fashionable’ areas.

You may consider this regrettable, but it’s far from clear-cut. I’m sure most people would say the projects Pfeiffer and Lauda are actually doing are more interesting than constructing braided monoidal 2-categories. Maybe even they think so. And, maybe it’s true.

Of course, I’m trying to convince the world otherwise.

Posted by: John Baez on June 3, 2007 6:55 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Young scientists often ask themselves if working on a lengthy and difficult project will pay off in a tenured job or not. So, research gets focused on ‘fashionable’ areas.

Oh, so that’s why it took me so long to get a job.

Maybe even they think so.

A totally different kettle of fish, if so.

Posted by: David Corfield on June 3, 2007 9:30 PM | Permalink | Reply to this

The Human Factor

David wrote:

A totally different kettle of fish, if so.

The interesting thing is: the tastes of others are not totall different from ones own — except for a few rebels like you.

To what extent does a young scientist develop his or her tastes based on the the current fashions? To what extent should this happen? Ones tastes don’t develop in a vacuum. Complete flouting of ones intellectual environment can lead to crackpottery. Learning by example is a wonderful thing. On the other hand, completely slavish devotion to fashion leads to mediocrity. Somewhere between lies the golden mean.

Ideally, the funding agencies that support the hard scientists would also want sociologists, historians and philosophers to study these questions. To ignore them is less than wholly rational! But alas, right now the ‘soft sciences’ suffer from neglect — perhaps because few people realize how incredibly important the human factors are in every endeavor!

But you know all this.

Posted by: John Baez on June 3, 2007 9:58 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

In subjects with a more experimental component, I think it’s quite common when one has a “vague idea” to estimate the amount of work involved in setting up experiments, data collection and analysis and estimate whether fully working things up is likely to be worth the effort based on how interesting, useful and important the idea could turn out to be. I know nothing about Lauda and Pfeiffer’s work, but can easily imagine making stuff rigorous might be an analogue of experimental work.

A more interesting question is whether Lauda and Pfeiffer thought “they wouldn’t be interested because they’re an insular community” or “they wouldn’t be interested because having a B-M 2-category doesn’t really advance the things they’re interested in”? In the first case you can be persistent, in the second case it’s a tougher sell.

Posted by: dave tweed on June 3, 2007 11:11 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Let’s not talk about my friends Aaron Lauda and Hendryk Pfeiffer in this way anymore. They might not like it. We can talk about the sociological issues we’re discussing without using them as examples.

I only brought their names up for mathematical reasons: namely, I believe that it’s possible to construct braided monoidal 2-categories using the math behind Khovanov homology, in part because they thought it might be doable, and I trust their judgement.

Posted by: John Baez on June 4, 2007 1:01 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

If I have offended anyone, my apologies. My only reason for repeating the names was to avoid tripping myself up in multiple “they”s.

Let me try and reframe the scientific enquiry point: I have definitely had ideas that seemed like they’d probably “work”, tried to estimate the amount of experimental work they’d require to validate and decided not to do proceed. That’s why I was a bit surprised David C was so perturbed by the suggestion that in science what one works on is partly based on the amount of “extraneous” effort. I have also been in the situation of deciding an idea, whilst interesting itself was very likely “the end of the line” in the methodology and wouldn’t be extendable, which thus wouldn’t lead to new papers on the idea, which then wouldn’t lead to citations.

If anyone wants to discuss the scientific philosophy issues behind this phenomenon, using me as an example if they need one, they’re welcome to.

Posted by: dave tweed on June 4, 2007 12:10 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Of course, there is a very complex mix of considerations at play when deciding what to work on, one which is dependent on the stage of one’s career. What perturbs me is when people refuse to follow what they believe to be the right thing to do, and the dominant consideration is not that they have been exposed to good reasons to adopt another course, but because they only care about winning favour.

I was talking about this to a fellow philosopher a while ago, and he commented on how many young people say to themselves that they’ll play the game until they’re established, but then find they’re too far committed to return back to what they originally wanted to do when they finally have a secure post.

Posted by: David Corfield on June 4, 2007 12:27 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

…how many young people say to themselves that they’ll play the game until they’re established, but then find they’re too far committed to return back to what they originally wanted to do when they finally have a secure post.

On the other hand, it can be supremely nerve-wracking to be a young, unestablished academic and to do what you think is interesting, despite being told (by supporters) that phrases like “co-C object” are the sorts of things that get you not-hired.

That said, I find it incredibly disappointing that popularity of ideas holds such weight in academic mathematics. If I were the type to proceed along popular lines I’d have dropped the math side of my major back in college, finished the computer science, gotten picked up in time to catch the tail end of the dot-com bubble, and worked my way into some obscenely high-paid consulting gig. I’m in this line of work because I buck the trends, and I’m not very likely to stop now.

Posted by: John Armstrong on June 4, 2007 3:49 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Oh, and another thing… This is really just a plug for my own work, but I think the difference between categorifying the Kauffman bracket and categorifying the su2 quantum group skein relations (which I’ll go ahead and confound with the Jones polynomial) is closer to being understood. See math.GT/0701339 for my and Kevin Walker’s explanation of how categorifying quantum su2 gives you a fully functorial version of Khovanov, not one that’s just functorial ‘up to sign’, a problem which plagued the theory for years.

Posted by: Scott Morrison on May 28, 2007 2:10 AM | Permalink | PGP Sig | Reply to this

Re: Link Homology and Categorification in Kyoto

David wrote:

As Aaron is not in the same position as Derek and Jeff…

Since the conference in Kyoto ended on the 25th, I’m uncertain of Aaron’s position — but his velocity is several hundred kilometers per hour, zipping back to Columbia as we speak.

I’ll try to coax him to post when he’s de-jetlagged.

Posted by: John Baez on May 27, 2007 5:37 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Since the conference web page has a schedule with notes from the lectures, it doesn’t make sense to tell you what each person talked about. So instead I will give my general impression of the conference. It was very exciting to see all the different ways that link homology theories can be understood. Everything from the triangulated categories, geometric picture worlds, derived categories of coherent sheaves, geometric representation theory, matrix factorizations, topological string theory, and gauge theory. It is quickly becoming difficult for the average mathematician to keep up with all of the sophisticated and beautiful tools that are being used to understand and develop link homology theories. But the conference did a great job of supplying many introductory talks on these topics.

I was particularly interested in learning more about the matrix factorization approach to link homology and Lev Rozansky gave a great introduction. Sergei Gukov’s lectures described how ideas from gauge theory and string theory could be used to choose potentials to plug into the matrix factorization approach and get interesting link homologies.

Fans of n-categories will be excited to see non trivial examples of braided monoidal 2-categories emerging “in the wild”. Many of the link homology theories are defined for tangle cobordisms, or 2-tangles as n-category theorists like to call them.

There was also a six part lecture series on knot Floer homology given by Ciprian Manolescu and Peter Ozsvath. There are sort of opposite stories with knot Floer homology and Khovanov homology. Khovanov homology began as a combinatorial invariant and its geometric origins are just beginning to be understood. Knot Floer homology on the other hand began as a geometric theory using symplectic topology inspired by gauge-theory. At the conference we got to hear how this theory can now be defined purely combinatorially. So far knot Floer homology has not been extended to tangle cobordisms which leaves some exciting work still left to be done.

Posted by: Aaron Lauda on June 4, 2007 1:51 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

Fans of n-categories will be excited to see non trivial examples of braided monoidal 2-categories emerging “in the wild”.

With duals? Could you give us hints as to what some of these beasts look like?

Posted by: David Corfield on June 5, 2007 8:57 AM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

The closest that I’ve seen to an argument that these BM 2-categories have duals is pages 23-24 of Morrison and Walker’s paper math.GT/0701339.

Posted by: Noah Snyder on June 5, 2007 6:10 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

The functoriality of the planar algebra operations ensure that we can build a ‘city of cans’ (hence the name canopolis) any way we like, obtaining the same result: either constructing several ‘towers of cans’ by composing morphisms, then combining them horizontally, or constructing each layer by combining the levels of all the towers using the planar operations, and then stacking the levels vertically. (p. 61)

Urs, looks like you’ve been pipped to the use of tin can imagery.

Posted by: David Corfield on June 5, 2007 6:47 PM | Permalink | Reply to this

Re: Link Homology and Categorification in Kyoto

The last part of Gukov’s notes linked to above sounds really fascinating.

Ever since I learned about Khovanov homology (not that I really know much about it beyond the mere basic notions, and even these I am about to begin forgetting) I was wondering what its natural interpretation would be. Like the Jones polynomial is a Wilson loop observable in a 3-dimensional quantum field theory, the Khovanov thing should be related to “Wilson surfaces” in 4-dimensional quantum field theory.

I have no idea what is actually known about this. But on his latter slides, Gukov seems to at least hint at lots of relations to known TQFT, ranging from Donaldson-Witten invariants, to – apparently – the Kapustin-Witten realization of geometric Langlands, including t’Hooft operators and all that.

It is not clear from Gukov’s slides whether he is just reviewing interesting known aspects of 4D gauge theory there, or if he actually claims to see a direct relation of Khovanov homology to these known aspects (beyond the mere fact that it is to be expected that there is a relation).

Does anyone know?

Posted by: urs on May 30, 2007 9:51 AM | Permalink | Reply to this

Via feedback from Not Even Wrong #: check out Dror Bar-Natan’s

Dream Map of Quantum Groups \subset Knot Theory

Posted by: urs on June 1, 2007 11:59 AM | Permalink | Reply to this

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