  We consider S^1-families of Legendrian knots in the standard contact R^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(S^1,R^3) of
Legendrian knots, although it is contractible in the space Emb(S^1,R^3) 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.
