I want to say a little here about Ben’s recent paper, or rather, the questions I started asking when first hearing about wall-crossing morphisms from Nadya. You don’t need to have read the paper to read this, or vice versa, but it might be more fun that way! In particular, I’m not going to say anything at all about matrix factorizations, because they’re still too hard for me.
In particular, I want to explain the statement — “The cone over the wall-crossing morphism simplifies.” — and say why that matters, and what it simplifies to.
First of all, what is a wall-crossing morphism? First of all, it’s just a map from the homology of a positive crossing to the homology of a negative crossing. (In every variation of Khovanov homology around these days, tangles are kosher, so there’s no need to say silly things about crossings embedded in otherwise fixed larger links.)
Moreover, in a functorial homology theory, this morphism should just be induced by the band cobordism which changes the crossing. In terms of Carter and Saito’s `elementary string interactions’, more commonly these days called `movies’, that cobordism is:
In the su(2) case, it’s easy to write down (or look up) the chain maps associated to these elementary cobordisms (saddle, Reidemeister 1, Reidemeister 1, saddle). We get
Each column here is the Khovanov complex of the corresponding tangle in the movie. The vertical arrows are all saddle cobordisms. The horizontal arrows in the middle row are all drawn in explicitly; the top and bottom rows I’ve left out, because the composition is zero.
So what do we get? Just the chain map which is the identity on the oriented resolution, and zero elsewhere.
Now, what is Nadya and Ben’s prescription for the Khovanov homology of a singular (in the Vassiliev sense) knot? Just replace each singular crossing with the cone over this wall-crossing morphism. This is just the sort of thing we’re always meant to do in Khovanov homology — `classically’, when studying finite type invariants, you extend to singular knots by looking at the difference between the two honest crossings, and we’re replacing that difference with a little two-step complex. When you do this, you get a great big cube. On the vertices are Khovanov homologies of the various honest knots coming from our singular knot, and the edges are all wall-crossing morphisms. You can then hopefully simplify this cube, and indeed calculate its homology.
However, a repeated lesson in Khovanov homology has been that one should try to simplify earlier rather than later. Both Bar-Natan’s program for computing Khovanov homology, and Ben’s (not quite working right?) program for computing Khovanov-Rozansky homology work on this principle. They try to find smaller up-to-homotopy representatives of the homology of sub-tangles of a link, and then tensoring those together, rather than just building the full cube in one step. Applying this lesson to the construction for singular knots pays off nicely.
To begin, let’s just think about a knot with a single singular crossing. I’ll write for the two resolutions, for the tangle outside of the singular crossing, and for a single positive and negative crossing. We can replace with , and then write this as . Throughout, the tensor product symbol is really the planar algebra operation: “tensor these two complexes together, combining the objects in the appropriate planar way”. (See §A.4 of http://tqft.net/functoriality, for example, if this is unfamiliar.) This gives us another way of thinking about the construction for singular knots. Instead of having a cube of Khovanov homologies of regular knots, we can imagine it as a modifying the usual prescription for Khovanov homology: in addition to “write down this cone for a positive crossing, that cone for a negative crossing, and tensor everything together”, we add the instruction “write down this cone, (it’s an iterated cone, because is itself a cone) for every singular crossing”, and still just tensor everything together.
Hopefully this makes it clear that being able to simplify the cone over the wall-crossing morphism is interesting, and useful if you actually wanted to do computations.
We’re almost there now! I just need to show you how to simplify that cone (it’s easy, pretty much the canonical example of finding a simpler up-to-homotopy representative), and then perhaps explain why the foam model for su(n) Khovanov homology says that everything keeps working exactly the same way for arbitrary n.
But it’s time for me to ride my bike home through sunny Santa Barbara, and eat some beets I bought yesterday. So perhaps more later.