Manturov Projection for Virtual Legendrian Knots in $ST^*F$

Kauffman virtual knots are knots in thickened surfaces $F\times R$ considered up to isotopy, stabilizations and destabilizations, and diffeomorphisms of $F\times R$ induced by orientation preserving diffeomorphisms of $F$. Similarly, virtual Legendrian knots, introduced by Cahn and Levi~\cite{CahnLevi}, are Legendrian knots in $ST^*F$ with the natural contact structure. Virtual Legendrian knots are considered up to isotopy, stabilization and destabilization of the surface away from the front projection of the Legendrian knot, as well as up to contact isomorphisms of $ST^*F$ induced by orientation preserving diffeomorphisms of $F$. We show that there is a projection operation $proj$ from the set of virtual isotopy classes of Legendrian knots to the set of isotopy classes of Legendrian knots in $ST^*S^2$. This projection is obtained by substituting some of the classical crossings of the front diagram for a virtual crossing. It restricts to the identity map on the set of virtual isotopy classes of classical Legendrian knots. In particular, the projection $proj$ extends invariants of Legendrian knots to invariants of virtual Legendrian knots. Using the projection $proj$, we show that the virtual crossing number of every classical Legendrian knot equals its crossing number. We also prove that the virtual canonical genus of a Legendrian knot is equal to the canonical genus. The construction of $proj$ is inspired by the work of Manturov.