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- W4300517610 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." @default.
- W4300517610 created "2022-10-03" @default.
- W4300517610 creator A5058233318 @default.
- W4300517610 creator A5066368701 @default.
- W4300517610 date "2014-05-22" @default.
- W4300517610 modified "2023-09-29" @default.
- W4300517610 title "Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds" @default.
- W4300517610 doi "https://doi.org/10.48550/arxiv.1405.5725" @default.
- W4300517610 hasPublicationYear "2014" @default.
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