有序介质中结状缺陷的同伦分类
Homotopy classification of knotted defects in ordered media
AI总结：
本文提出了一种基于同伦理论对有序介质中结状缺陷进行分类的方法，通过四元数群Q的子群结构和共轭类分析，实现了对缺陷的系统分类。
摘要：
我们给出了有序介质中全局缺陷的同伦分类，并通过双轴向列液晶（即顺序参数空间为3球面S³对四元数群Q的商空间）的例子进行解释。作为我们的数学模型，我们考虑了从空间图的补集到空间S³/Q的连续映射，并在某种等价关系下进行分类，发现等价类由Q的六个子群枚举。通过围绕经度环的蒙德罗变换，空间图的边被标记为Q的共轭类；一旦转为平面图，这些标签可以细化为与每条弧相关联的Q元素。相同的分类方案不仅适用于Q，也适用于任意群。
英文摘要：
We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to planar diagrams, these labels can be refined to elements of $Q$ associated to each arc. The same classification scheme applies not only in the case of $Q$ but also to arbitrary groups.
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