Contact homology and one parameter families of Legendrian knots

Abstract

We consider $S^1$–families of Legendrian knots in the standard contact $ℝR^3$. 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 invariant of the loop. 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 $Leg\left(S^1,{ℝ}^3\right)$ of Legendrian knots, although it is contractible in the space $Emb\left(S^1,{ℝ}^3\right)$ of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram.