Virtual Legendrian isotopy
Volume 234 / 2016
Abstract
An elementary stabilization of a Legendrian knot $L$ 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 knot $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. A class of virtual Legendrian isotopy is called a virtual Legendrian knot.
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 knots 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 $ST^*S^2.$
