Finite Order Invariants of Legendrian, Transverse, and 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.