Linking, Legendrian Linking and Causality

The set N of all null geodesics of a globally hyperbolic (d+ 1)-dimensional spacetime (M, g) is naturally a smooth (2d– 1)-dimensional contact manifold. The sky of an eventx in M is the subset X of N consisting of all null geodesicsthrough x, and is an embedded Legendrian submanifold of N diffeomorphicto S^{(d – 1)}. It was conjectured by Low that for d = 2two events x and y are causally related if and only if X andY are linked (in an appropriate sense). We use the contact structureand knot polynomial calculations to prove this conjecture incertain particular cases, and suggest that for d = 3 smoothlinking should be replaced with Legendrian linking.