Homotopy classification of defects in liquid crystals

Now we briefly discuss the topological homotopy theory that classifies defects in liquid crystals and judges the stability of the defects according to the symmetry of liquid crystals. 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by 

For a three-dimensional nematic liquid crystal for example, the r = 0 case corresponds for example to a defect with d = 2, which means a discli-nation wall for r = 1, d = 1 corresponds to a disclination line for r = 2, d = 0 corresponds to a disclination point. It is known that the order vector space of three dimensional nematic liquid crystals is the projection plane P2 Its homotopy group of the zero rank (r = 0) is 

In the other words, there is no topologically stable disclination wall. 

This is a cycle group of two elements, one of two elements is in fact the disclination line of m = 1/2, the other element corresponds to the m = 1 disclination line, which is not topologically stable. 

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