Computations and applications of Khovanov's functor valued invariant of tangles

Question

Soon after his famous paper "A categorification of the Jones polynomial", Khovanov introduced a "bordered" version. His theory assingns to every oriented even tangle a complex of (H_n,H_m)-bimodules, where H_n is a certain family of rings and n,m\geq 0 depend on the number of points on the top (resp. bottom) of the tangle.

This paper can be found here: http://arxiv.org/pdf/math/0103190.pdf

It is natural to ask if this relative version can be used as a tool for computations. So my first question is:

Q1 Are there any examples in the literature of explicit computation regarding the chain complex or the homology associated to some families of tangles?

My second question is about possible applications of this invariant.

Q2 Are there any examples in the literature where this theory has been used?

Regarding the second question I should mention that the work of Smith and Abouzaid which leads to a proof of the equivalence between Khovanov homology and Symplectic Khovanov homology actually makes use of this relative theory.

Answer 1

I recall discussing this with you in Budapest last summer, and I was curious about the same thing. Here are my thoughts a year later:

I haven't seen any computational examples in the literature (not that they don't exist), for Q1, but in my experience any computation using Bar-Natan's formal-pictures theory (dotted cobordism version) can be translated to Khovanov's $\mathcal{H}^n$-theory. The module associated to a tangle is the projective $\mathcal{H}^n$-module whose generators are the generators of the Bar-Natan complex; in other words, you just view a Bar-Natan generator as representing all possible outside closures (and choices of signs on the resulting circles). The differential is given by applying the usual 2d TQFT to the (closed-off in all possible ways) cobordisms in Bar-Natan's complex. When you make a simplification in Bar-Natan's complex, e.g. a Gaussian elimination of two generators $x$ and $y$ connected by an identity-cobordism component of the differential, there's a corresponding homotopy equivalence of the associated $\mathcal{H}^n$-modules.

There are tons of computations using Bar-Natan's theory, so if you want to see how an $\mathcal{H}^n$ computation works, you could probably take one of those and just follow through in $\mathcal{H}^n$-language. (There are probably equally good examples, but I can't resist linking to my own computations for pretzel knots that I was working on last summer, arXiv:1303.3303- you could probably take any part of that computation and apply the $\mathcal{H^n}$ functor to get a computation in that language.)

For Q2, there's a whole lot on the representation theory side of the picture. Stroppel's paper arXiv:math/0608234 (and several other related papers) present $\mathcal{H}^n$ as an idempotent truncation of a larger, quasi-hereditary algebra $A_{n,n}$, which is (if I'm saying it correctly) the endomorphism algebra of a certain block of a parabolic category $\mathcal{O}$ for $\mathfrak{gl}_{2n}$. There's a corresponding tangle theory that works over $A_{n,n}$ instead of $\mathcal{H^n}$ and still recovers Khovanov homology. The category in question is equivalent to a category of perverse sheaves on the Grassmannian $G(n,n)$, but now I'm way out of my depth! This material might be an interesting read, though, if you're looking for how $\mathcal{H^n}$ fits into the broader scheme of things.