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We consider S^1-families of Legendrian knots in the standard contact R^3. Wedefine the monodromy of such a loop, which is an automorphism of theChekanov-Eliashberg contact homology of the starting (and ending) point. Weprove this monodromy is a homotopy invariant of the loop. We also establishtechniques to address the issue of Reidemeister moves of Lagrangian projectionsof Legendrian links. As an application, we exhibit a loop of right-handedLegendrian torus knots which is non-contractible in the space Leg(S^1,R^3) ofLegendrian knots, although it is contractible in the space Emb(S^1,R^3) ofsmooth knots. For this result, we also compute the contact homology of what wecall the Legendrian closure of a positive braid and construct an augmentationfor each such link diagram.

Contact homology and one parameter families of Legendrian knots