Document Type

Article

Publication Date

7-2019

Abstract

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.

Keywords

Geometric Topology, Symplectic Geometry

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Rights

Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.

Comments

Author’s submitted manuscript.

Included in

Mathematics Commons

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