Finite Order Invariants of Legendrian, Transverse, and Framed Knots in
Contact 3-manifolds
(http://xxx.lanl.gov/abs/math.SG/9907118)

Author: Vladimir Tchernov
July 20, 1999, updated Aug 9, 1999.

Comments: 34 pages, 8 figures. 
Subj-class: Symplectic Geometry; Geometric Topology 
MSC-class: 53C15, 57M99 (Primary), 53C42 (Secondary)

Recently D. Fuchs and S. Tabachnikov proved that the quotients of the groups
of $C$-valued order $\leq n$ invariants by the groups of order $\leq (n-1)$
invariants of Legendrian, of transverse, and of framed knots in the standard
contact $ R^3$ are canonically isomorphic. An analogous result was obtained
by J. W. Hill in the case of Legendrian and of framed knots in the spherical
cotangent bundle of $ R^2$ (with the standard contact structure). The proofs
of these facts were based on the fact of the existence of the universal
Vassiliev invariant of framed knots in these spaces.

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.

Update (Aug 9, 1999):
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.