Relating tangle invariants for Khovanov homology and knot Floer homology

TL;DR

This paper establishes a relationship between tangle invariants for Khovanov homology and knot Floer homology by comparing DA bimodules, providing new insights into their algebraic structures and exact sequences.

Contribution

It demonstrates that the DA bimodules for Khovanov and knot Floer homologies are homotopy equivalent when filtration is ignored, linking two major knot invariants.

Findings

Homotopy equivalence of DA bimodules for both invariants

Oriented skein exact triangle for Ozsvath-Szabo bimodules

Iterated oriented cube of resolutions for the global construction

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.