Khovanov homology via 1-tangle diagrams in the annulus

TL;DR

This paper introduces a new, more efficient chain complex for computing reduced Khovanov homology of links in the 3-sphere, using 1-tangle diagrams in the annulus, and establishes a spectral sequence converging to the homology.

Contribution

It constructs a novel chain complex from 1-tangle diagrams in the annulus that simplifies calculations and connects to a broader program for links in lens spaces.

Findings

The chain complex is typically smaller than the ordinary Khovanov complex.

It features long differentials related to pairs of saddles in the cube of resolutions.

A spectral sequence from the chain complex converges to reduced Khovanov homology.

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 $S^3$.