Journal of Symplectic Geometry

Volume 1 (2001)

Number 4

The N-copy of a topologically trivial Legendrian knot

Pages: 659 – 828



K. Mishachev


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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