Khovanov homology detects split links

TL;DR

This paper demonstrates that the module structure on Khovanov homology can uniquely identify split links, extending previous ideas and providing new insights into link detection methods in knot theory.

Contribution

It proves that the module structure on Khovanov homology detects split links and establishes an analogous detection result for untwisted Heegaard Floer homology.

Findings

Khovanov homology module structure detects split links

Analogous detection result for untwisted Heegaard Floer homology

Well-defined module structure on reduced Khovanov complex

Abstract

Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is well-defined up to quasi-isomorphism.