Recently Kronheimer and Mrowka showed that Khovanov homology detects the unknot. It's still not known if the Jones polynomial detects the unknot. 


Does annular Khovanov homology detect the unknot in an an annulus? 
More generally, does annular Jones polynomial detect the unknot in an annulus (perhaps there's a counterexample?)



Does annular Khovanov homology detect the unknot (in annulus)?
The full answer is obtained in our recent arxiv preprint [here](https://arxiv.org/abs/1907.00547). See Theorem 1.3. It shows that the annular Khovanov homology detects the unlink in the thickened annulus.


The proof relies on a spectral sequence relating the annular Khovanov homology to the annular instanton homology, sutured instanton Floer homology for sutured manifolds with tangles developed in this preprint and Batson-Seed's unlink detection result for Khovanov homology. 


