We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and DingGeiges. 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 -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.
Geometric Topology, Symplectic Geometry
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Cahn, Patricia and Chernov, Vladimir, "Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds" (2019). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.