Contact homology and one parameter families of Legendrian knots

Abstract

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.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper46.abs.html

References in corpus (3)

• Floer homology of families I
• Augmentations and Rulings of Legendrian Knots
• Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology

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• Spaces of tight contact structures in dimension 3
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