Annular Khovanov homology and augmented links

TL;DR

This paper constructs a spectral sequence linking annular Khovanov homology of a link to the reduced Khovanov homology of an augmented link, providing new insights into link classification and unlink detection.

Contribution

It introduces a spectral sequence connecting annular and reduced Khovanov homologies and proves that annular Khovanov homology detects unlinks.

Findings

Classified links with minimal annular Khovanov homology rank.

Proved annular Khovanov homology detects unlinks.

Constructed a spectral sequence relating two types of Khovanov homology.

Abstract

Given an annular link $L$, there is a corresponding augmented link $\widetilde{L}$ in $S^3$ obtained by adding a meridian unknot component to $L$. In this paper, we construct a spectral sequence with the second page isomorphic to the annular Khovanov homology of $L$ and it converges to the reduced Khovanov homology of $\widetilde{L}$. As an application, we classify all the links with the minimal rank of annular Khovanov homology. We also give a proof that annular Khovanov homology detects unlinks.