The n-Category Café

A (plane diagram of a) link $L$ is something you obtain by drawing a couple of closed lines on a piece of paper, such that each time there would be a crossing you make a decision if you want to pass under or over.

Khovanov homology is the name for an assignment of a chain complex to any such link, such that this assignment is invariant under deformations of the link - known as Reidemeister moves – up to chain homotopy.

In other words, the homology of this chain complex is a strict invariant of the link. This invariant generalizes the Jones polynomial invariant of links in that the latter is reobtained as the (graded) Euler characteristic of the Khovanov chain complex.

As Lauda and Pfeiffer write:

This construction can be seen as a categorification of the unnormalized Jones polynomial, replacing a polynomial in one indeterminate $q$ by a chain complex of graded vector spaces. The coefficients of the polynomial arise as the dimensions of the homogeneous components of the graded homology groups of the chain complex in such a way that the degree corresponds to the power of $q$.

The nice thing about the precise procedure for computing the Khovanov homology of a given link – as far as currently understood – is that it can be understood as coming from a couple of “correlators” that a certain closed string 2-dimensional topological field theory assigns to a certain collection of worldsheets that are determined by the given link.

But it turns out that not every 2d TFT will do. We need one that satisfies a certain normalization condition (determining the partition function of the sphere and of the torus) and a certain funny extra invariance condition known as the four tubes law.

That four tubes law relates four different ways to propagate two closed strings to two closed strings. It is not all that involved, but for exactly which reason it is this law that ensures that we get a link invariant - and not some other law - remains conceptually a little unclear, as far as I understand.

The way one obtains a worldsheet from a link is rather nice: we form the cylinder over the entire link, except that in each neighbourhood of a crossing we “smooth out” that crossing by letting the corresponding surface be of saddle shape.

Since there are two different ways to insert a saddle for each crossing, there will be a total of $2 ^n$ different surfaces associated to a link this way.

An edge in the Khovanov chain complex associated to the link is obtained from taking certain subsets of these surfaces, computing the correlator of the given TFT on them and adding up the resulting linear maps in a certain way.

Unfortunately, the precise prescription for doing this seems to have little a priori motivation so far, except for the empirical fact that it happens to yield a link invariant in the end.

Anyway. If Khovanov homology is so closly related to closed topological strings, one is lead to wonder if there is a generalization to open/closed topological strings. This is exactly what Aaron Lauda and Hendryk Pfeiffer investigated.

A link in wich not only closed curves but also open arcs are admitted is known as a tangle. There is a more or less obvious generalization of the precription for obtaining surfaces from links to tangles which naturally leads to worldsheets with physical boundaries: “D-branes”.

Aaron and Hendryk carefully generalize every step in the construction of Khovanov homology to this more general case of tangles. They do indeed obtain an assignment of chain complexes to tangles that is invariant (up to homotopy) under deformations of tangles.