Legendrian knots in overtwisted contact structures

Abstract

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.

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

References in corpus (4)

• On the classification of tight contact structures I
• The Legendrian satellite construction
• Vassiliev invariants of Legendrian, of transverse and framed knots in contact 3-manifolds
• Computable Legendrian invariants

Cited by in corpus (12)

• Loose Legendrian embeddings in high dimensional contact manifolds
• Handle moves in contact surgery diagrams
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Loose Legendrians and the plastikstufe
• Bridge trisections in ${\mathbb{CP}}^2$ and the Thom conjecture (with Corrigendum)
• Non-looseness of non-loose knots
• Topologically Trivial Legendrian Knots
• Legendrian Hopf links
• Invariants of Legendrian knots from open book decompositions
• The Group of Contactomorphisms of the Sphere Fixing an Overtwisted Disk
• Neighbourhoods and isotopies of knots in contact 3-manifolds
• On knots in overtwisted contact structures