Fields 2009 Knot Homologies Proposal
This is a part of a proposal for a 2009 Knot Theory Program at the Fields Institute.
In the 1980s a group of people, lead by Jones, Drinfel'd, Witten, Reshetikhin, Turaev and Vassiliev revolutionized knot theory (and other parts of low dimensional topology) finding a vast array of new and (at the time) unexpected knot (and 3-manifold) invariants. Much work had gone into understanding these new invariants. A lot remains open, but yet, by the end of the 1990s it seemed that the surprise wore off and we got used to the fact that knots were related to Lie algebras and to quantum field theory; we even came to understand this relationship quite well.
Then in 1999 came Khovanov and got us all confused once again (confused is of course the best state a mathematician can be in; the struggle out of that state is the primary drive for progress). He found a chain complex, naturally associated with knot diagrams, whose homology is a knot invariant and whose Euler characteristic (interpreted in an appropriate way) is the good old Jones polynomial that started the revolution of the 1980s.
Why is that So Exciting?
Homology's Stronger than its Euler Characteristic
The first and probably least significant reason is that the newly discovered homology theory is a stronger invariant than the Jones polynomial and it is computable (though not too easily) even for pretty large knots. Thus we can expect years of study and hundreds of papers establishing this or that property of Khovanov homology for this or that class of knots. One aim of the Fields program will be to take a share of that loot.
The second reason is much better. Generally speaking, homology is "functorial". A map between spaces provides no relationship between their Euler characteristics, but always yields a map between their homologies. Without this we wouldn't be proving the Brouwer fixed point theorem in the first class of every algebraic topology course; it is the primary reason why homology is interesting.
The excellent news is that Khovanov homology is likewise "functorial", for the appropriate (4-dimensional) notion of "morphisms" between (3-dimensional) knots. Hence we can expect Khovanov homology to be qualitatively better than the Jones polynomial, leading to much more interesting topology. The early signs (a lovely theorem by Rasmussen) suggest that this is indeed the case. There ought to be further applications to the functoriality of the Khovanov homology and at the Fields program we are sure to look for them and find them.
It's the New Kid on the Block
The third reason is the most speculative, yet in our humble opinion, it is by far the most exciting.
Nobody expected Khovanov homology. The Jones polynomial has its natural place in the world of quantum algebra and topological quantum field theories. Khovanov homology yet doesn't. Could it be that Khovanov homology is an accident? Not really, for in 2004 came Khovanov and Rozansky and showed that the HOMFLY polynomial has a lift to a homology theory, much like Khovanov lifts Jones. So the reasonable expectation is that Jones and HOMFLY lift to homological theories because their context, or at least a part of their context, can be lifted.
That context is Lie algebras, quantum algebra and quantum field theory; we can now fairly expect that these great subjects are merely the “Euler” shadows of even bigger structures.
Math hardly ever gets more exciting than this. The young and smart and the old and wise are converging and they will eventually unravel these bigger structures for everybody's joy. At the Fields program we will learn what they will already have done and finish what they won't.
A Word about Khovanov-Rozansky
We should add a word about the Khovanov-Rozansky homology (KRH), whose Euler characteristic is the HOMFLY polynomial. There is something extraordinary about the KRH construction. KRH associates a complex with an ordinary differential satisfying d2 = 0 to a knot or a link. But to a tangle, a "knot part", it associates a differential satisfying d2 = ω where (these are so called "matrix factorizations").
There is a general nature to the KRH use of such non-standard differentials. It seems surprising to us that such differentials were not used previously as steps towards the construction of "honest" differentials, and it seems unlikely to us that non-standard differentials will not find future applications. Yet while the idea behind those non-standard differentials is simple, there is not yet a simple and conceptual explanation for why they must arise and the way they arise in "categorifying" the Lie algebra sl(n) which lurks behind the HOMFLY polynomial.
It may well be that a simple and conceptual explanation of the KRH construction will be the key to understanding how all other Lie algebras (and other objects as well?) may be categorified. Thus studying the Khovanov-Rozansky homology will be one of our prime goals at the Fields Institute program.