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 $V^{\perp}$ 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.

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Received 12 October 2016

Accepted 16 July 2019

Published 30 July 2020