Does annular Khovanov homology detect the unknot (in annulus)? 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?)
 A: The full answer is obtained in our recent arxiv preprint here. 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.  
A: The answer is yes if we assume all the components of the link are null-homologous; see Yi Xie's preprint here.  This is a combination of an annular version of singular instanton link homology and the Kronheimer-Mrowka spectral sequence, together with a result of Batson-Seed that Khovanov homology detects the unlink.
This is intermediate between the result in Adam Lowrance's answer and the full result, which as far as I know is still open.  I think that Yi, possibly with collaborators, has an idea for how to remove the null-homologous assumption; you should contact him if you want more details. 
