Detecting Causality with the Links--Gould Polynomial

Abstract

The conjectures of Low and Natario--Tod, and Penrose's question on Arnold's Problem list ask if causality in spacetimes can be formulated in terms of linking of spheres of light rays in the manifold of all light rays. For $(2+1)$-dimensional spacetimes, this link happens in the manifold coverable by a solid torus $S^1\times \mathbb R^2$. This was solved positively by Chernov and Nemirovski, which raises the question of which link invariants can be used to study causality. Chernov, Martin and Petkova proved that Heegaard--Floer and Khovanov homology completely capture causality. Allen--Swenberg conjectured that the Jones polynomial, which is obtained as an alternating Euler characteristic from Khovanov homology, is also sufficient. But they constructed complicated examples of links $\mathrm{AS}(n)_{n=1}^{\infty}$ that suggest that the Alexander--Conway polynomial -- which is the Euler characteristic of Heegaard--Floer homology -- is not enough. The Links--Gould polynomial is a quantum invariant that specializes to the classical Alexander--Conway polynomial in two different ways and somewhat surprisingly inherits some of its characteristic classical features. We show that it distinguishes all the Allen-Swenberg links from the link of causally unrelated events and hence detects causality in all known examples where the Alexander--Conway polynomial is not sufficient. This suggests that it may completely capture causality. The work on the categorification of the Links--Gould Polynomial is an ongoing and hard problem, and it is not a subject of this paper. As a corollary, we also compute the Seifert genus of all Allen--Swenberg links.