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G&T Monographs |
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Contact homology and one parameter families of Legendrian
knots
Tamas Kalman
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Geometry & Topology 9 (2005)
2013–2078
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Abstract
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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.
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Keywords
Legendrian contact homology, monodromy,
Reidemeister moves, braid positive knots, torus knots
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Mathematical Subject Classification
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
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