Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds

Details Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds

1999-07-19

arxiv.org/abs/math/9907118v2

Mathematics

Symplectic Geometry

34 pages, 8 figures. We added many new examples of Legendrian homotopic Legendrian knots that realize isotopic framed knots bu

Scientific paper

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.

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