AI 中文总结
本文研究中间quandle着色链环不变量在(2+1)维全局双曲时空中检测因果性的能力，证明在特定等价关系下无法区分关键链环，并给出充分条件及广义定理。
AI 中文摘要
我们研究了中间quandle在(2+1)维全局双曲时空中检测因果性的能力，通过判断其着色链环不变量能否区分两个Hopf链环的连通和与Allen和Swenberg在2020年构造的无限系列相关三组分链环。Allen和Swenberg曾提出，任何链环不变量必须能够区分这些链环，才能在该设定下检测因果性。我们证明，只要$a\sim b\Leftrightarrow a*b=a$定义了一个等价关系，这些quandle就无法做到这一点。Alexander quandle是此结果适用的一个例子。我们还给出了一个充分条件，使得中间quandle能够区分两个Hopf链环的连通和与Allen-Swenberg系列中的所有链环，并给出了这样一个quandle的例子。受此结果启发，我们还推导了一个关于特定类型tangle上中间quandle着色的广义定理，该定理有助于确定这些quandle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 distinguish 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 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 are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.
CommentsError on page 5, example 2.13. Coloring space was computed incorrectly. Several figure errors with orientations of links are forgot to be added
