The annular Khovanov homology is an invariant for links in a thickened annulus, which generalizes the original Khovanov homology defined for links in a three-sphere. It is a special case of the theory developed by Asaeda, Przytycki and Sikora which works for links in any thickened surface. In this talk, I will introduce an analogue of the annular Khovanov homology using singular instanton Floer theory, called the annular instanton Floer homology.  It is related to the annular Khovanov homology by a spectral sequence. As an application of this spectral sequence, I will prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous).

Another application is a new proof of Grigsby and Ni’s result that tangle Khovanov homology distinguishes braids from other tangles.