Virtual Legendrian Isotopy

TL;DR

The paper introduces virtual Legendrian isotopy, establishing a unique irreducible representative for each class and proving that virtual isotopy implies Legendrian isotopy in certain cases, thus solving a conjecture.

Contribution

It defines virtual Legendrian isotopy, proves the existence of unique irreducible representatives, and confirms a conjecture relating virtual and actual Legendrian isotopies.

Findings

Every virtual Legendrian isotopy class has a unique irreducible representative.

Virtual Legendrian knots correspond bijectively with Gauss diagram classes.

Two Legendrian knots in $ST^*S^2$ that are virtually isotopic are also Legendrian isotopic.

Abstract

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes of Legendrian links and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and the first author: two Legendrian knots in $ST^*S^2$ that are isotopic as virtual Legendrian knots must be Legendrian isotopic in $ST^*S^2.$