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-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-perpendicular to 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.