Geometry & Topology 9 (2005) 2013–2078

DOI: 10.2140/gt.2005.9.2013

Abstract

We consider S¹–families of Legendrian knots in the standard contact RR³. 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,R³) of Legendrian knots, although it is contractible in the space Emb(S¹,R³) 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

Primary: 53D40

Secondary: 57M25

References

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