Medial quandles's capability of detecting causality and properties of their coloring on certain links and knots

Abstract

I investigated the capability of medial quandle, quandle whose operation satisfying that $(a_1*b_1)*(a_2*b_2)=(a_1*a_2)*(b_1*b_2)$, to detect causality in (2+1)-dimensional globally hyperbolic spacetime by determining if they can distinguished the connected sum of two Hopf links from an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. I show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles is an example that this result can apply to. Inspired by this result, I also derived a generalized theorem about the coloring of medial quandles on a specific type of tangles, which help to determine whether this quandles can distinguish between a wider range of knots and links or not.