Conjectures on the Khovanov Homology of Legendrian and Transversely Simple Knots

Abstract: 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.