Causality Detection via Symplectic Quandles

Abstract

We study whether symplectic quandle colorings can reveal causal structure encoded by "sky links" - i.e. links consisting of spheres of all light rays through two points in the space of all light rays of a spacetime. Building on the known limitations of the Alexander-Conway polynomial, we compare the connected sum of two Hopf links (which represents all causally unrelated situations) with the first two Allen-Swenberg links (that are the only known examples when this polynomial does not work). For each diagram we report both the quandle counting invariant (total number of colorings) and an enhanced version that records how many distinct colors appear in each coloring. In our tests over small finite fields, plain counts often agree across examples, but the enhanced invariant consistently separates the Hopf case from the Allen-Swenberg family, and becomes more discriminating as the field grows. A simple transfer step suggests that this effect persists along the whole sequence. These results point to enhanced symplectic quandle colorings as a practical, computable indication of causality that classical polynomials alone may miss; the first examples of this kind were discovered by Jain.