Volume 9, issue 4 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Contact homology and one parameter families of Legendrian knots

Tamas Kalman

Geometry & Topology 9 (2005) 2013–2078

arXiv: math.GT/0407347

Abstract

We consider S1–families of Legendrian knots in the standard contact R3. 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(S1, 3) of Legendrian knots, although it is contractible in the space Emb(S1, 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