Heegard Floer invariants of Legendrian knots in contact three-manifolds

Abstract

We define invariants of null-homologous Legendrian and transverse knots in contact 3manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non-loose knots in overtwisted 3-spheres. Moreover, we apply the invariants to find transversely non-simple knot types in many overtwisted contact 3-manifolds.