Khovanov module and the detection of unlinks

TL;DR

This paper introduces a module structure on Khovanov homology that is compatible with spectral sequences to Floer homology and demonstrates its effectiveness in detecting trivial links, advancing link detection methods.

Contribution

It establishes a natural module structure on Khovanov homology and proves its capability to detect trivial links, linking it with Floer homology and 3-manifold topology.

Findings

Module structure on Khovanov homology is natural under spectral sequences.

The module structure detects trivial links.

H_1/Torsion module structure on Heegaard Floer homology detects S^1xS^2 summands.

Abstract

We study a module structure on Khovanov homology, which we show is natural under the Ozsvath-Szabo spectral sequence to the Floer homology of the branched double cover. As an application, we show that this module structure detects trivial links. A key ingredient of our proof is that the H_1/Torsion module structure on Heegaard Floer homology detects S^1xS^2 connected summands.