Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes

We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, a causally simple spacetime $(X, g)$ can be conformally embedded as an open set into some globally hyperbolic $(\widetilde X, \widetilde g)$ and the space of light rays in $(X, g)$ is an open submanifold of the space of light rays in $(\widetilde X, \widetilde g)$. If this is always the case, this provides an approach to solving the conjectures relating causality and linking in causally simples spacetimes.