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.
