Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds · arXivDesk
1405.5725May 22, 201431 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 2
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 3-manifold M that are transverse to a nowhere-zero vector field V up to the corresponding isotopy relation. Such knots are called V-transverse. A framed isotopy class is simple if any two V-transverse knots in that class which are homotopic through V
Nearby in the stack
-transverse immersions are
V
-transverse isotopic. We show that all knot types in
M
are simple if any one of the following three conditions hold:
1.
M
is closed, irreducible and atoroidal; or
2.
the Euler class of the
2
-bundle
V⊥
orthogonal to
V
is a torsion class, or
3.
if
V
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
h
-principle for Legendrian immersions, we also construct knot types which are not Legendrian simple.