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

TL;DR

This paper examines the ability of medial quandles to detect causality in certain links and knots, revealing limitations under specific conditions and providing a generalized theorem on their coloring properties.

Contribution

It demonstrates that medial quandles cannot distinguish certain links for causality detection when a specific equivalence relation holds, and introduces a generalized coloring theorem.

Findings

Medial quandles fail to detect causality in certain links under specific conditions.

Alexander quandles exemplify the limitations of medial quandles in this context.

A generalized theorem on medial quandle coloring extends understanding of their distinguishing capabilities.

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.