On bordered theories for Khovanov homology

Abstract

We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and Type A structures in Khovanov homology, and his algebra $\mathcal{B}\Gamma_n$, in terms of Khovanov's theory of modules over the ring $H^n$. We reprove invariance and pairing properties of Roberts' bordered modules in this language. Along the way, we obtain an explicit generators-and-relations description of $H^n$ which may be of independent interest.Comment: 87 pages; 2 figure

Planar algebras and the decategorification of bordered Khovanov homology

“…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. 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, [7]. Furthermore, using Khovanov's conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in Σ × [−1, 1] where Σ is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points.…”

“…In [5], M. Khovanov describes an approach to invariants for traditional (2n, 2m)-tangles with diagrams in a square and the decategorification of the modules used in his theory. The relationship between the tangle homology in [5] and that in [8] and [9] is the subject of recent work by Andy Manion, [7]. He shows that the relationship is novel and quite complicated.…”

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].…”

Immersed curves in Khovanov homology

“…Roberts defined candidates for type D structures, type A structures, and a box tensor product between them, which recovers the Khovanov homology of a knot from its decomposition into two tangles. Manion has reinterpreted Khovanov's algebraic framework for two-sided tangles in the bordered language and related it to Robert's bordered theories [Man17]. Our algebra B can be defined using the same bordered techniques and we expect that the corresponding bordered type D structure agrees with our invariant Д(T ); see Remarks 4.22 and 4.29.…”