Relating tangle invariants for Khovanov homology and knot Floer homology

Abstract

Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the same algebras. For a closed diagram, this invariant computes the Khovanov homology of the knot or link. We show that forgetting the filtration, our DA bimodules are homotopy equivalent to a suitable version of the Ozsvath- Szabo bimodules. In addition to giving a relationship between tangle invariants for Khovanov homology and knot Floer homology, this gives an oriented skein exact triangle for the Ozsvath-Szabo bimodules which can be iterated to give an oriented cube of resolutions for the global construction.