Narrow Results

In [A type D structure in Khovanov homology, preprint (2013), arXiv:1304.0463; A type A structure in Khovanov homology, preprint (2013), arXiv:1304.0465] the author constructed a package which describes how to decompose the Khovanov homology of a link 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 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 types A and 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 I. Petkova's decategorification of bordered Floer homology, [The decategorification of bordered Heegaard-Floer homology, preprint (2012), arXiv:1212.4529v1], and shows how they recover the Jones polynomial. We also use the decategorifications to compare this approach to tangle decompositions with M. Khovanov's from [A functor-valued invariant of tangles, Algebr. Geom. Topol.2 (2002) 665–741].

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in $\mathrm{\Sigma }\times [-1,1]$ where $\mathrm{\Sigma }$ is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.