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Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds
We prove a complete classification theorem for loose Legendrian knots in an
oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our
approach is to classify knots in a -manifold that are transverse to a
nowhere-zero vector field up to the corresponding isotopy relation. Such
knots are called -transverse. A framed isotopy class is simple if any two
-transverse knots in that class which are homotopic through -transverse
immersions are -transverse isotopic. We show that all knot types in are
simple if any one of the following three conditions hold: is closed,
irreducible and atoroidal; or the Euler class of the -bundle
orthogonal to is a torsion class, or if is a
coorienting vector field of a tight contact structure. Finally, we construct
examples of pairs of homotopic knot types such that one is simple and one is
not. As a consequence of the -principle for Legendrian immersions, we also
construct knot types which are not Legendrian simple.Comment: 31 pages, 13 figures. Version 2 contains an additional theorem on
Legendrian knots with overtwisted complements. Version 3 has a revised
introduction and new title; the results are identical to version
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