The decategorification of bordered Khovanov homology

Abstract

Abstract. In [14], [15] the author showed 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 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, [13], 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.

Planar algebras and the decategorification of bordered Khovanov homology

“…The module I 2n comes from the idempotents of the differential bigraded algebra BΓ 2n used in bordered Khovanov homology [9], [8], and which supports the decategorification of differential bigraded modules over BΓ 2n , [10]. The maps Z P are a combinatorial generalization to planar surfaces of the combinatorics of decategorification in [10]. Our motivation, besides the simplicity of the construction, is 1) to use these maps to compare the constructrions of bordered Khovanov homology to Khovanov's tangle homology, and 2) to give a simple generalization of the Jones polynomial to tangles with good compositional properties, that is different from those the author could find in the literature.…”

“…The module I 2n comes from the idempotents of the differential bigraded algebra BΓ 2n used in bordered Khovanov homology [9], [8], and which supports the decategorification of differential bigraded modules over BΓ 2n , [10]. The maps Z P are a combinatorial generalization to planar surfaces of the combinatorics of decategorification in [10].…”

“…In particular, the construction in [9] takes a tangle diagram T in a disc D, with a marked point * ∈ ∂D: * and associates to it a bigraded differential module T ]] over the differential bigraded algebra BΓ n , where T has 2n endpoints on ∂D. In [10]…”

“…In particular, the construction in [9] takes a tangle diagram T in a disc D, with a marked point * ∈ ∂D: * and associates to it a bigraded differential module T ]] over the differential bigraded algebra BΓ n , where T has 2n endpoints on ∂D. In [10] the author describes the decategorification of T ]] as a map [ P ]]] : Z[q 1/2 , q −1/2 ] −→ I 2n P associated to an inside tangle, i.e. one with signature (n 0 ).…”

“…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 [10]. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology ([9]) and its difference from that in Khovanov's tangle homology, [5], without being encumbered by any extra homological algebra.…”