Geometry & Topology, Vol. 9 (2005) Paper no. 46, pages 2013--2078.

Contact homology and one parameter families of Legendrian knots

Tamas Kalman

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.

Keywords. Legendrian contact homology, monodromy, Reidemeister moves, braid positive knots, torus knots

AMS subject classification. Primary: 53D40. Secondary: 57M25.

E-print: arXiv:math.GT/0407347

DOI: 10.2140/gt.2005.9.2013

Submitted to GT on 3 October 2004. (Revised 24 July 2005.) Paper accepted 17 September 2005. Paper published 26 October 2005.

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Tamas Kalman
Department of Mathematics, University of Southern California
Los Angeles, CA 90089, USA

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