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 $\le n$ 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.

34 pages, 8 figures. We added many new examples of Legendrian homotopic Legendrian knots that realize isotopic framed knots but are distinguishable by finite order invariants of Legendrian knots. In some of these examples the complements of both Legendrian knots are overtwisted. We also extended Theorem 3.0.6 to the case of $S^1$-fibrations over nonorientable surfaces and corrected typos