Low's causality conjecture for skies in (2+1)-dimensional spacetimes

Let $(M,g)$ be an orientable, time-oriented, globally hyperbolic $(2+1)$-dimensional spacetime with a Cauchy surface $\Sigma\subset M$ diffeomorphic to a subset of $\mathbb{R}^2$. Let $N$ be the manifold of null geodesics, and let $X,Y\subset N$ be the skies of events $x,y\in M$. Low's conjecture. The events $x$ and $y$ are causally related if and only if $X$ and $Y$ either intersect or are linked. The conjecture relates causal geometry of spacetime to the topology of its manifold of light rays; the paper gives counterexamples to detecting causal relatedness by nonzero linking number alone and proves the conjecture in particular cases.