Contact homology and one parameter families of Legendrian knots

Abstract

We consider S1 -families of Legendrian knots in the standard contact R3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy inhvariant of the loop (Theorem 1.1). We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S1,R3) of Legendrian knots, although it is contractible in the space Emb(S1,R3) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid (Definition 6.1) and construct an augmentation for each such link diagram. © Geometry & Topology Publications.