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March, 2003 The N-copy of a topologically trivial Legendrian knot
K. Mishachev
J. Symplectic Geom. 1(4): 659-828 (March, 2003).


We consider Legendrian knots and links in the standard 3-dimensional contact space. In 1997 Chekanov [Ch] introduced a new invariant for these knots. At the same time, a similar construction was suggested by Eliashberg [E1] within the framework of his joing work with Hofer and Givernthal on Symplectic Field Theory ([E2],[EGH]). To a knot diagram, they associated a differential algebra A. Its stable isomorphism type is invariant under Legendrian isotopy of the knot.

In this paper, we introduce an additional structure on this algebra in the case of a Legendrian link. For a link of N components, we show that its algebra splits A =g ∈ G Ag Here G is a free group on (N - 1) variables. The splitting is determined by the order of the knots and is preserved by the differential. It gives a tool to show that some permutations of link components are impossible to produce by Legendrian isotopy.


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K. Mishachev . "The N-copy of a topologically trivial Legendrian knot." J. Symplectic Geom. 1 (4) 659 - 828, March, 2003.


Published: March, 2003
First available in Project Euclid: 17 August 2004

zbMATH: 1075.57007
MathSciNet: MR2039159

Rights: Copyright © 2002 International Press of Boston


Vol.1 • No. 4 • March, 2003
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