Legendrian knots in overtwisted contact structures

TL;DR

This paper establishes conditions under which Legendrian knots in overtwisted contact structures are Legendrian isotopic, emphasizing the role of framing, rotation number, and overtwisted disks, and discusses non-loose knots.

Contribution

It provides new criteria for Legendrian isotopy in overtwisted contact structures, including cases with and without disjoint overtwisted disks.

Findings

Legendrian knots are isotopic if they are framed isotopic, have same rotation number, and an overtwisted disk is disjoint.

For zero-homologous knots, isotopy requires topological knot isotopy and equal Thurston-Bennequin invariants.

Discussion of non-loose Legendrian knots when the overtwisted disk condition is not met.

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.