  In two previous papers, the author showed how to decompose the Khovanov
homology of a link $\mathcal{L}$ into the algebraic pairing of a type D
structure and a type A structure (as defined in bordered Floer homology),
whenever a diagram for $\mathcal{L}$ is decomposed into the union of two
tangles. Since Khovanov homology is the categorification of a version of the
Jones polynomial, it is natural to ask what the type A and type D structures
categorify, and how their pairing is encoded in the decategorifications. In
this paper, the author constructs the decategorifications of these two
structures, in a manner similar to Ina Petkova's decategorification of bordered
Floer homology and shows how they recover the Jones polynomial. We also give a
new proof of the mutation invariance of the Jones polynomial which uses these
decomposition techniques.
