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 $V^{\perp}$ 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.

中文翻译：

---

$3$-歧管中松散的勒让德里亚结和伪勒让德里亚结

我们证明了面向 3 流形中松散 Legendrian 结的完整分类定理，概括了 Dymara 和 Ding-Geiges 的结果。我们的方法是对一个 $3$-流形 $M$ 中的结进行分类，这些结横向于一个无处零向量场 $V$ 直到相应的同位素关系。这种结称为$V$-transverse。如果该类中通过 $V$-横向浸入同伦的任何两个 $V$-横向结是 $V$-横向同位素，则框架同位素类是简单的。我们证明，如果满足以下三个条件中的任何一个，则 $M$ 中的所有结类型都是简单的： $1.$ $M$ 是封闭的、不可约的和环形的；或 $2.$ 与 $V$ 正交的 $2$-bundle $V^{\perp}$ 的欧拉类是扭转类，或者 $3.$ 如果 $V$ 是紧密接触结构的共向矢量场。最后，我们构建了成对的同伦结类型的例子，这样一个简单，一个不简单。作为 Legendrian 浸入的 $h$-principle 的结果，我们还构建了不是 Legendrian 简单的结类型。