Contact homology and one parameter families of Legendrian knots.

摘要

I consider S1-families of Legendrian knots in the standard contact 3-space R3. I define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. I prove this monodromy is a homotopy invariant of the loop. I also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, I exhibit a loop of positive Legendrian (p, q) torus knots which is non-contractible in the space Leg( S1, R3) of Legendrian knots, although it is contractible in the space Emb(S 1, R3) of smooth knots. Important technical tools, such as the computation of contact homology and augmentations, are generalized to Legendrian closures of positive braids.