In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, by constructing a spectral sequence from reduced Khovanov homology to a form of instanton knot homology and then showing that the latter has rank 1 only for the unknot. In this talk, we will discuss joint work with John Baldwin in which we apply techniques from contact geometry to prove that the rank of instanton knot homology also detects the trefoils. As a corollary, we conclude that Khovanov homology is also a trefoil detector.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/4234
[3] http://www.mpim-bonn.mpg.de/node/7451