Volume 9, issue 4 (2005)

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Contact homology and one parameter families of Legendrian knots

Tamas Kalman

Geometry & Topology 9 (2005) 2013–2078
 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 $Leg\left({S}^{1},{ℝ}^{3}\right)$ of Legendrian knots, although it is contractible in the space $Emb\left({S}^{1},{ℝ}^{3}\right)$ 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
Primary: 53D40
Secondary: 57M25
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