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June 2009 Topologically trivial Legendrian knots
Yakov Eliashberg, Maia Fraser
J. Symplectic Geom. 7(2): 77-127 (June 2009).


The first part of this paper contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e., Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. These techniques were never published in detail when the classification result was announced over ten years ago. The final part of the present paper contains a systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e., classification up to a global contactomorphism) of topologically trivial exceptional Legendrian knots in overtwisted contact $S^3$ according to the values of the invariants tb, r. We show, moreover, that such knots only occur for one of the infinitely many overtwisted contact structures on $S^3$. We remark that our tight classification result also implies that any topologically trivial loose Legendrian knots with same value of (tb, r) in an overtwisted contact 3-manifold are in fact Legendrian isotopic if $tb < 0$.


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Yakov Eliashberg. Maia Fraser. "Topologically trivial Legendrian knots." J. Symplectic Geom. 7 (2) 77 - 127, June 2009.


Published: June 2009
First available in Project Euclid: 17 April 2009

zbMATH: 1179.57040
MathSciNet: MR2496415

Rights: Copyright © 2009 International Press of Boston


Vol.7 • No. 2 • June 2009
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