Legendrian knots in overtwisted contact structures

Details Legendrian knots in overtwisted contact structures Legendrian knots in overtwisted contact structures

2004-10-05

arxiv.org/abs/math/0410122v2

Mathematics

Geometric Topology

34 pages, 29 figures; uses eplain and epsf. Minor corrections, figures fixed, references updated

Scientific paper

We prove that two Legendrian knots in a contact structure which is trivializable as a plane bundle are Legendrian isotopic provided that (1) they are isotopic as framed knots, (2) they have the same rotation number with respect to some parallelization of the contact structure, and (3) there is an overtwisted disk disjoint with both knots. (For zero-homologous knots the condition (1) reads as: (1a) they are isotopic as topological knots, and (1b) they have the same Thurston-Bennequin invariant.) Then we discuss the situation when condition (3) is not fulfilled, in particular that of non-loose Legendrian knots.

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