Machon Thomas, Alexander Gareth P
Department of Physics and Centre for Complexity Science , University of Warwick , Coventry CV4 7AL, UK.
Proc Math Phys Eng Sci. 2016 Jul;472(2191):20160265. doi: 10.1098/rspa.2016.0265.
We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in [Formula: see text], showing that the distinct homotopy classes have a 1-1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of lines in cholesterics.
我们根据扭曲上同调给出了具有弱锚定边界条件和任意缺陷集的一般区域向列型织构的全局同伦分类，并针对三维空间中打结和链接缺陷的情况给出了显式计算，表明不同的同伦类与分支双覆盖的第一同调群存在一一对应，该分支双覆盖在旋错环上分支。我们进一步表明，该群中对应于二阶元素的那些类的子集具有平面代表元，并根据梅隆子刻画了其他类的阻碍。平面织构是全局缺陷拓扑的一个特征，在任何局部刻画中都未体现。最后，我们描述了全局分类与最近关于向列型液滴的实验的关系，以及四阶元素与胆甾相中线的存在的关系。
