It is a long-standing problem whether the Jones polynomial detects
the unknot, and it has been known that the Jones polynomial does not
detect unlinks. In the knot homology world, Kronheimer and Mrowka
proved that Khovanov homology, the categorification of Jones
polynomial, detects the unknot. On the other hand, the question
whether Khovanov homology detects unlinks remains open. In this talk,
we will show that Khovanov homology with an additional natural module
structure detects unlinks. This is joint work with Matt Hedden.