Mathematics Colloquia and Seminars

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I consider Legendrian knots in the standard contact space, that is in R^3 with zero restriction of the form y dx - dz. Such a knot is fully determined by its projection onto the xz-plane, which makes the problem of Legendrian isotopy classification of Legendrian knots an elementary looking problem concerning planar curves. It is known since the early 80-ies that topologically isotopic Legendrian knots need not to be Legendrian isotopic: they can be distinguished by two easy-to-calculate integer-valued invariants: Thurston-Bennequin number and Maslov number. Until recently, a conjecture existed that topologically isotopic Legendrian knots with equal TB and M must be Legendrian isotopic. This conjecture was destroyed in '96 by Chekanov and Eliashberg, who worked out (independently) a new invariant, which is able to distinguish between Legendrian knots, undistinguishable by the invariants known before. However, these invariants are difficult to handle, and there exists, basically, only one example to show that they work. I will show in my talk that for Legendrian knots whose diagrams satisfy an enigmatic (for me) but fairly simple condition, it may be very easy to see that they have different Ch.-El. invariants. Although my work (I hope) is still in progress, I can display a lot of examples of knots distinguishable only by Ch.-El. invariants. No special knowledge is required, and the pictures I will show seem to me to be beautiful.