medial 拟群检测因果性的能力及其对特定链环和纽结的着色性质
On Medial Quandle and Detecting Causality
AI总结：
研究medial拟群检测（2+1）维整体双曲时空因果性的能力，证明其在特定等价关系下无法区分关键链环（含Alexander拟群实例），并推导其对特定缠结着色的广义定理以助力区分更广泛纽结链环。
摘要：
我研究了medial拟群（其运算满足$(a_1*b_1)*(a_2*b_2)=(a_1*a_2)*(b_1*b_2)$的拟群）在（2+1）维整体双曲时空中检测因果性的能力，方法是判断它们能否区分两个Hopf链环的连通和与Allen和Swenberg在2020年构造的一系列相关三分支链环——后者提出，任何链环不变量若要在此设定下检测因果性，必须能区分这些链环。我证明，只要$a\backsim b\backsim a*b=a$定义等价关系，这些拟群就无法做到这一点；Alexander拟群是该结果适用的例子。受此启发，我还推导了关于medial拟群对特定类型缠结着色的广义定理，这有助于确定此类拟群能否区分更广泛的纽结和链环。
英文摘要：
We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distin- guish between the connected sum of two Hopf links and 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. We use algebraic methods to obtain a comprehensive analysis of the ability of me- dial quandles to complete this task. In particular, we obtain firmly useful rules to determine whether a given medial quandle is capable of distinguishing the links, and find a method of constructing a large family of medial quandle that are able to distinguish the connected sum of two Hopf links and every link in the Allen-Swenberg infinite series , which therefore gives powerful candidates for detecting causality in the spacetime of our setting
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