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A theorem of Kronheimer and Mrowka states that Khovanov homology is able todetect 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 thefigure-eight knot. These are the simplest of the Legendrian simple knots. Weconjecture that Khovanov homology is able to distinguish Legendrian andTransversely simple knots. Using the torus and twist knots, numerical evidenceis provided for all prime knots up to 19 crossings. The conjectured Legendriansimple knots from the Legendrian knot atlas have also been included, and noknots with identical Khovanov polynomials were found.

Conjectures on the Khovanov Homology of Legendrian and Transversely Simple Knots