Arnold's seminar
September 7, 1999

V.Chernov
Finite-order invariants of Legendrian, of transverse, and of framed
knots in contact $3$-manifolds

ABSTRACT:
We show that for a big class of contact manifolds the groups of order
$\leq n$ invariants (with values in an arbitrary Abelian group) of
Legendrian, of transverse and of framed knots are canonically isomorphic.
On the other hand for an arbitrary cooriented contact structure on

$S^1\times S^2$ with the nonzero Euler class of the contact bundle we
construct examples of Legendrian homotopic Legendrian knots $K_1$ and $K_2$
such that they realize isotopic framed knots but can be distinguished by
finite order invariants of Legendrian knots in $S^1\times S^2$. We construct
similar examples for a big class of contact manifolds $M$ such that $M$ is a
total space of a locally trivial $S^1$-fibration over a nonorientable surface.
We show that in some of these examples the complements of $K_1$ and of $K_2$
are overtwisted.