Geometry & Topology

Contact homology and one parameter families of Legendrian knots

Tamas Kalman

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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.

Article information

Geom. Topol., Volume 9, Number 4 (2005), 2013-2078.

Received: 3 October 2004
Revised: 24 July 2005
Accepted: 17 September 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Legendrian contact homology monodromy Reidemeister moves braid positive knots torus knots


Kalman, Tamas. Contact homology and one parameter families of Legendrian knots. Geom. Topol. 9 (2005), no. 4, 2013--2078. doi:10.2140/gt.2005.9.2013.

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