The Links–Gould polynomial conjecture for detecting causality

Let $X$ be a $(2+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $\Sigma\neq S^2,\,\mathbb R P^2$, and let $\mathrm{LG}$ denote the Links–Gould polynomial of the sky link in the light-ray manifold.

Links–Gould polynomial conjecture. The invariant $\mathrm{LG}$ completely captures causality in such spacetimes.

The paper proves that $\mathrm{LG}$ distinguishes all known Allen–Swenberg links from the causally unrelated link $H\#H$, but leaves the assertion for all such spacetimes conjectural. The claim is motivated by the fact that the Alexander–Conway polynomial is a specialization of $\mathrm{LG}$ and has known deficiencies in this problem.