International Congress of Mathematicians

ICM 2002 • Vol. II • 385-394Invariants of Legendrian KnotsYu. V. Chekanov*AbstractWe present two different constructions of invariants for Legendrian knots inthe standard contact space R 3 . These invariants are defined conibinatorially,in terms of certain planar projections, and are useful in distinguishing Legendrianknots that have the same classical invariants but are not Legendrianisotopie.2000 Mathematics Subject Classification: 57R17.Keywords and Phrases: Legendrian submanifold, Legendrian knot.1. Introduction1.1. Legendrian knotsA smooth knot L in the standard contact space (R 3 ,a) = ({(q,p,«)}, du—pdq)is called Legendrian if it is everywhere tangent to the 2-plane distribution ker(a)(or, in other words, if the restriction of a to L vanishes). Two Legendrian knotsare Legendrian isotopie if the they can be connected by a smooth path in thespace of Legendrian knots (or, equivalently, if one can be sent to another by adiffeomorphism g of R 3 such that g*a = (pa, where (p > 0). In order to visualizea knot in R 3 , it is convenient to project it to a plane. In the Legendrian case, thecharacter of the resulting picture will depend on the choice of the projection. Theuseful two are: the Lagrangian projection n: R 3 —¥ R 2 , (q,p,u) >-¥ (q,p), and thefront projection a: R 3 —¥ R 2 , (p,q,u) >-¥ (q,u). In Figure 1, two projections of thesimplest Legendrian knot (unknot) are shown.We say that a Legendrian knot L C R 3 is 7r-generic if all self-intersectionsof the immersed curve n(L) are transverse double points. We can represent a n-generic Legendrian knot L by its (Lagrangian) diagram: the curve n(L) C R 2 ,at every crossing of which the overpassing branch (the one with the greater valueof u) is marked. Of course, not every abstract knot diagram in R 2 is a diagram of* Moscow Center for Continuous Mathematical Education, B. Vlasievsky per. 11, Moscow119002, Russia. E-mail: chekanov@mccme.ru