Planar algebras and the decategorification of bordered Khovanov homology

We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-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 his 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 A. Manion. Furthermore, using Khovanov's conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in $\Sigma \times [-1,1]$ where $\Sigma$ 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.