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Hey Everyone! So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain homology theory. My question is, why do this? I mean this in the sense that why is it important/interesting to have this "quantum invariants" $\rightarrow$ "homology theories" transformation? If this question is perhaps to basic and /or unsuited for the site I would appreciate some bibliography on the matter as Khovanov's "A categorification of the Jones polynomial" article doesn't seem to answer this question for me. Thanks!

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Sam, why not make that an answer? If it takes three comments to get your point across, it's not a comment! :) – MTS Jan 9 '12 at 5:43
that's a good idea! – Sam Lewallen Jan 9 '12 at 6:32
up vote 14 down vote accepted

A good reason is that categorified invariants are usually more subtle than the uncategorified ones, and their additional structure gives more information about the knot/link. For example there are rather simple knots with the same Jones polynomial that are distinguished by their Khovanov homology. Furthermore, Kronheimer and Mrowka have recently proved that the Khovanov homology detects the unknot, while it is an open problem to determine if the Jones polynomial does or not.

Another very interesting categorification is the Knot Floer homology (defined by Ozsvath and Szabo, and independently by Rasmussen), whose graded Euler characteristic is the Alexander polynomial. This categorified invariant is also much subtle, for example (as proved by OS) it detects the genus of a knot, while the Alexander polynomial gives only bounds on it, and KFH detects fibred knots (Ni) while the Alexander polynomial gives obstructions about it.

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Thanks for your answer Francesco! Do you know if this was the original motivation for the categorification or a side product? Can you please give me a reference regarding knots with same Jones poly different Khovanov homologies, please? – Juan OS Jan 8 '12 at 22:13
Well I don't know what actually Khovanov was looking for when he discovered Khovanov homology, but surely when you try to "complicate" an invariant you expect to get something stronger. By the way a good introduction to the topic is , you can also find the example I was talking about – Francesco Lin Jan 8 '12 at 22:28

Physicists view categorification as adding a time dimension in gauge theory. For a description of this in the context of Khovanov homology, see Witten's notes, or you can read his lecture notes and watch his video from the Freedman conference last summer. There's a dimension shift here though that I don't understand, where the elliptic equations are in 4D, whose count gives the Jones polynomial, whereas Khovanov homology is counting solutions to a related equation in 5 dimensions. Witten also gives as an example how to categorify the Casson invariant to give Floer homology.

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As suggested by Budney and Agol's answers, one short answer to the question "why categorify [3-manifold invariants]" is "to get 4-manifold invariants."

This is directly related to the standard answer to the question "why is homology better than the Euler characteristic" which is: "functoriality." Functoriality in the case of Khovanov homology leads to maps coming from 4-dimensional cobordisms between knots, which one hopes will give interesting information about knot cobordisms (this has been realized in the work of Jake Rasmussen).

More specifically, the work of Donaldson/Floer/Witten et al in the 80's and early 90's led to an invariant for 3 and 4 manifolds which you might call "Donaldson-Floer theory" and which (more or less) fit together into what came to be called a "3+1 dimensional topological quantum field theory." D-F theory gave many exciting results about the smooth structure of 4-manifolds, but it was defined using complicated non-linear analysis and was very hard to compute.

Around the same time, there appeared another TQFT, the Witten-Reshetikhin-Turaev (WRT) theory. This was a 2+1 dimensional TQFT, and the work of Reshetikhin-Turaev showed that it had a purely combinatorial definition. Moreover, it gave invariants of knots which were directly related to the Jones polynomial.

According to the introduction of Khovanov's paper, his original motivation goes back to ideas of Crane and Frenkel on lifting the WRT theory from a 2+1 to a 3+1 theory, in the hopes of getting a combinatorially-defined invariant of smooth 4-manifolds with the same power as D-F theory. To understand, from their point of view, why this is related to "categorification" in the algebraic sense (a term which they may have helped define? I'm not sure), see their paper "Four dimensional topological quantum field theory, Hopf categories, and the canonical bases." There, they say

"the second idea motivating our construction is that replacing an algebraic structure with a similar categorical algebraic structure lifts the dimension of the corresponding TQFT by 1"

Currently, Khovanov homology is not defined for all knots and knot cobordisms in all 3 and 4 manifolds, and it gives limited (albeit very interesting!) 4 dimensional information. It remains a fundamental problem to get a complete combinatorial 4-manifold invariant generalizing Khovanov homology and fit it into a TQFT, and to understand its relationship with (categorified) representation theory and smooth 4-manifold topology.

I should add that there is a different point of view on this question, which takes the "categorification of quantum groups" as a fundamental goal and interesting problem in its own right. But this should probably be addressed by someone who knows more about it than I do!

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Let me begin with the grand daddy of all invariants, the Euler characteristic of a triangulated space. This is an easily defined and computable invariant, but the price we pay for this ease in computation is that it is a rather rough tool. It cannot distinguish too many spaces. E.g., compact oriented $3$-manifolds have Euler characteristic $0$.

The construction of homology by Poincare can be viewed as a first instance of categorification. It takes a bit more work to define this new invariant but it is more powerful. The Euler characteristic of a space is the Euler characteristic of its homology. One advantage is obvious: the Euler characteristic could be trivial without the homology being so. Put it differently, the "sum" of all parts could be zero, but the parts themselves may not be so.

The Alexander and Jones polynomials are themselves Euler characteristics of more refined objects (Ozsvath-Szabo and resp. Khovanov homology). These polynomials could be trivial, but the corresponding homologies may not be so.

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A slightly different answer than Francesco's.

Why would one want to categorify? I think it comes down to how you want to think of knots. By that I mean, the kind of invariants you study very much depend on what kind of object you think you're studying. Khovanov homology came about as a search for a more natural way to understand the Jones polynomial. The Jones polynomial, one way or another was defined in a more or less diagramatic way, which in the world of geometric topology is most naturally connected to concordance / pseudo-isotopy / cobordism categories. In my mind I'm thinking of examples coming from Morse and Cerf theory here, also knot concordance. So roughly speaking, the most natural geometric notion connected to the definition of the Jones polynomial pulls you towards things that are of a categorical nature. So in hindsight I don't think the development is surprising at all.

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