What is Knot Homology?
So I’m back from a week in Faro, Portugal, talking about various things surrounding the ideas of “knot homology”. So what is it? Well, this will be a bit of a loose treatment of the subject, and may not be completely on the mark. I like David Corfield’s idea that a mathematician is a sort of storyteller, and I’m not about to let mere history get in the way of a good história. Besides, I’ll get to most of the details in my main line sooner or later.
First I should mention the Bracket polynomial and the Jones polynomial. Jones was studying a certain kind of algebra when he realized that the defining relations for these algebras were very much like those of the braid groups. In fact, he was quickly able to use this similarity to assign a Laurent polynomial — one which allows negative powers of the variable — to every knot diagram that didn’t change when two diagrams differed by a Reidemeister move. That is, it was a new invariant of knots.
The Jones polynomial came out of nowhere, from the perspective of the day’s knot theorists. And it set the whole field on its ear. From my perspective looking back, there’s a huge schism in knot theory between those who primarily study the geometry and the “classical topology” of the situation and those who primarily study the algebra, combinatorics, and the rising field of “quantum topology”. To be sure there are bridges between the two, some of which I’ll mention later. But the upshot was that the Jones polynomial showed a whole new way of looking at knots and invariants.
Immediately in its aftermath a huge number of interpretations and generalizations poured forth. One of the most influential was Louis Kauffman’s “state-sum” model: the Bracket. This is an invariant of regular isotopy instead of ambient isotopy, which basically means we throw out Reidemeister I moves. Meanwhile, I glossed over above that the Jones polynomial actually applies to oriented links, where there’s a little arrow saying “go this way around the loop”. This is a subtle distinction between the Bracket and the Jones polynomial that many authors steamroll over, but I find it important for my own reasons.
Anyhow, the Bracket also assigns a Laurent polynomial to every diagram in a way that’s invariant under an apropriate collection of moves. It does this by taking each crossing and “splitting” it in two ways — turning an incoming strand to the left or the right rather than connecting straight across. For a link diagram with crossings there are now “states” of the diagram. Now we assign each diagram a “weight” and just add up the weights for all the different states. Thus: “state-sum”. If we choose the rule for weighting states correctly we can make the resulting polynomial into an invariant.
So now we flash forward from the mid-’80s to the late ’90s. Mikhail Khovanov, as a student of Igor Frenkel at Yale, becomes interested in the nascent field of categorification. Particularly, he was interested in categorifying the Lie algebra . That is, he needed to find a category and functors from that category to itself that satisfied certain relations analogous to the defining relations of the Lie algebra structure. He did this using some techniques from a field called “homological algebra”. I’ll eventually talk about it, but for now ask Michi.
But as it happens, this Lie algebra has a very nice category of representations. It’s a monoidal category with duals. In fact, every object is its own dual, and we can (morally, at least) build them all from a single fundamental representation. That means that it’s deeply related to the category of so-called “unoriented Temperley-Lieb” diagrams, which is (roughly) to categories with duals as the category of braids is to braided monoidal categories.
A Temperley-Lieb diagram is just a bunch of loops and arcs on the plane. The arcs connect marked points at the top and bottom of the diagram (like braid strands) while the loops just sorta float there, and none of the strands cross each other at all. So if there are no arcs, there’s just a bunch of separate loops. And we care about this because the states of a link in the definition of the Bracket are just bunches of separate loops too!
So we can take each state and read it in terms of this homological categorification of . And we can read a combination of states — a state sum — in such homological terms as well. So the defining relations of the bracket become “chain homotopies” — natural isomorphisms — in the homological context of the categorification. Thus we have a homological categorification of the Bracket model of the Jones polynomial.
And again, it just came out of nowhere and has immediately revolutionized the field. Homology theories are hot right now. This high-level approach has been broken down by Khovanov and Rozansky into a combinatorial formulation, which knot theory groups like those at George Washington University and the University of Iowa have latched onto. The field of “Heegaard Floer homology” has been nudged closer and closer to the combinatorial Khovanov framework from its origins in analytic problems. Other knot invariants are lining up to be categorified along the same lines. And all the while the incredibly rich structure of Khovanov homology itself is being spruced up and neatened, leading to a series of clear examples to act as guideposts for those probing higher categorical structures in general.
And that’s what we just spent the last week talking about in Faro.