  A theorem of Kronheimer and Mrowka states that Khovanov homology is able to
detect the unknot. That is, if a knot has the Khovanov homology of the unknot,
then it is equivalent to it. Similar results hold for the trefoils and the
figure-eight knot. These are the simplest of the Legendrian simple knots. We
conjecture that Khovanov homology is able to distinguish Legendrian and
Transversely simple knots. Using the torus and twist knots, numerical evidence
is provided for all prime knots up to 19 crossings. The conjectured Legendrian
simple knots from the Legendrian knot atlas have also been included, and no
knots with identical Khovanov polynomials were found.
