A type A structure in Khovanov homology

Abstract

Abstract. Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle which complements the type D structure previously defined by the author. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. The homotopy type of the type A structure is a tangle invariant, and homotopy equivalences of the type A structure result in chain homotopy equivalences on the Khovanov chain complex. We can use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. This approach adds to the literature even in the case of a connect sum, where the techniques here will allow an exact computation of Khovanov homology from the stuctures for two tangles coming from the summands. Several examples are included, showing in particular how we can compute the correct torsion summands for the Khovanov homology of the connect sum. A lengthy appendix is devoted to establishing the theory of these structures over a characterstic zero ring.

On bordered theories for Khovanov homology

“…We will use only a subset of the full algebraic machinery; however, we will work with coefficients in Z rather than Z/2Z. For this sign lift, we will follow the conventions of Roberts in [9] and [8].…”

“…Given a Type D structure ( D, δ) and a Type A structure ( A, m n : n ≥ 1) over B, the natural way to pair them is known as the box tensor product. It yields a differential bigraded abelian group A ⊠ D; see Lipshitz-Ozsváth-Thurston [4] for more details and algebraic properties of ⊠ over Z/2Z, and see Roberts [8] for a definition over Z with the sign conventions we will use.…”

“…The first theory is a reformulation of Khovanov's functor-valued invariant [1] in the bordered language. The second theory was introduced by Lawrence Roberts in [9] and [8].…”

“…Theorem 1.0.5 (Proposition 6.2.7 and Proposition 6.3.10). The Type A structure A([T 2 ] Kh ) over BΓ n , and the Type D structure D([T 1 ] Kh ) over BΓ n , are isomorphic to the Type A and D structures Roberts associates to T 2 and T 1 in [8] and [9]. By Proposition 6.4.3, the pairing ⊠ is the same over B⊙m(B ! )…”

Bordered theory for pillowcase homology

“…Examples of analogous bordered theories developed for other invariants are: bordered Heegaard Floer homology [24], [25]; bordered theory for knot Floer homology [33], [34], [35]; bordered theories for Khovanov homology [36], [37], [27]. Step (3) for Heegaard Floer homology was done in [40], and for knot Floer homology in [34], [35].…”