Contact homology and one parameter families of Legendrian knots

Kalman, Tamas

doi:10.2140/gt.2005.9.2013

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Contact homology and one parameter families of Legendrian knots

Tamas Kalman

Geometry & Topology 9 (2005) 2013–2078

DOI: 10.2140/gt.2005.9.2013

arXiv: math.GT/0407347

Abstract

We consider $S^{1}$ –families of Legendrian knots in the standard contact $ℝ R^{3}$ . 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

Mathematical Subject Classification 2000

Primary: 53D40

Secondary: 57M25

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Publication

Received: 3 October 2004

Revised: 24 July 2005

Accepted: 17 September 2005

Published: 26 October 2005

Proposed: Yasha Eliashberg

Seconded: Peter Ozsváth, Tomasz Mrowka

Authors

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

Volume 9, issue 4 (2005)