Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds

TL;DR

This paper classifies loose Legendrian knots in 3-manifolds by analyzing knots transverse to a vector field, extending previous results and identifying conditions for simplicity, with implications for Legendrian knot theory.

Contribution

It generalizes the classification of loose Legendrian knots in 3-manifolds and introduces criteria for knot simplicity based on manifold and vector field properties.

Findings

All knot types are simple under certain topological conditions.

Constructs examples of homotopic knots with differing simplicity.

Shows existence of non-Legendrian simple knot types.

Abstract

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.