Khovanov homology via 1-tangle diagrams in the annulus

Abstract: We show that the reduced Khovanov homology of an oriented link $L$ in $S^3$ can be expressed as the homology of a chain complex constructed from a description of $L$ as the closure of a 1-tangle diagram $T$ in the annulus. Our chain complex is constructed using a cube of resolutions of $T$ in a manner similar to ordinary Khovanov homology, but it is typically smaller than the ordinary Khovanov chain complex and has several unusual features, such as long differentials corresponding to pairs of successive saddles in the cube of resolutions. Our chain complex carries a natural filtration, which we use to construct a spectral sequence that converges to reduced Khovanov homology. Our results are part of a larger program to construct an analog of Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of Khovanov homology due to Hedden, Herald, Hogancamp, and Kirk, and our chain complex was predicted by this program for the case when the lens space is $S^3$.