Detecting Causality with Conjugation Quandles over Dihedral Groups

Abstract

We study whether quandle colorings can detect causality of events for links realized as skies in a $(2+1)$-dimensional globally hyperbolic spacetime $X$. Building off the Allen--Swenberg paper in which their $2$-sky link was conjectured to be causally related, they showed that the Alexander--Conway polynomial does not distinguish that link from the connected sum of two Hopf links, corresponding to two causally unrelated events. We ask whether the Alexander--Conway polynomial together with different types of quandle invariants suffice. We show that the conjugation quandle of the dihedral group $D_5$, together with the Alexander--Conway polynomial, does distinguish the two links and hence likely does detect causality in $X$. The $2$-sky link shares the same Alexander--Conway polynomial but has different $D_5$ conjugation-quandle counting invariants. Moreover, for $D_5$ the counting invariant alone already separates the pair, whereas for other small dihedral groups $D_3$, $D_4$, $D_6$, $D_7$ even the enhanced counting polynomial fails to detect causality. In fact we prove more that the conjugation quandle over D5 distinguishes all the infinitely many Allen-Swenberg links from the connected sum of two Hopf links. These results present an interesting reality where only the conjugation quandle over $D_5$ coupled with the Alexander--Conway polynomial can detect causality in $(2+1)$ dimensions. This results in a simple, computable quandle that can determine causality via the counting invariant alone, rather than reaching for more complicated counting polynomials and cocycles.