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.
Notes on file formats
Tamas Kalman
Department of Mathematics, University of Southern California
Los Angeles, CA 90089, USA
Email: tkalman@usc.edu
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