Conjectures about virtual Legendrian knots and links

TL;DR

This paper formulates and proves conjectures relating virtual Legendrian knots to classical Legendrian knots, exploring their properties, non-existence results, and connections to causality in spacetimes, with proofs in specific low-dimensional cases.

Contribution

It introduces conjectures generalizing known results to virtual Legendrian knots and proves them in 2D and (2+1)D cases, including a version of Arnold's 4 cusp conjecture.

Findings

Proved conjectures for 2D manifolds and (2+1)D spacetimes.

Established non-existence of positive virtual Legendrian self isotopy.

Formulated and proved a version of Arnold's 4 cusp conjecture.

Abstract

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self isotopy for the class of the fiber of $ST^*M$ and the conjectural relation of virtual Legendrian isotopy to causality in generalized spacetimes. We prove the conjectures in the case of $2$-dimensional $M$ and $(2+1)$-dimensional spacetimes. We also formulate and prove the version of the Arnold's $4$ cusp conjecture for virtual isotopies.