Defects topological classification

One should bear in mind that the topological classification of defects in cholesteric and other layered media such as smectics and ordinary crystals is limited by the condition of the layers equidistance. As a result, some transformations between defects that belong to the same class require very high energy barriers comparable to the energy barriers between different classes. Transformation X X within the class Cx represents such an example. 

As already mentioned, the layered structure of cholesteric materials imposes certain limitations on the topological classification of defects based on ho-motopy groups a more general theory is still lacking. In this section we discuss macroscopic defects such as focal conic domains and oily streaks whose existence depends crucially on the layered character of ordering. 

Topology of Director Fields Homotopy Groups and Classification of Defects... 

The classification of defects in nematics represents a straightforward example of the applications of homotopic group theory [14, 15], The reader is referred to reviews of the subject [19-21, 23, 52]. This topological approach confirms the absence of walls, the existence of Mobius lines, the mutual annihilation of thin threads and the existence of singular points. More importantly, it shows that defects combine and merge according to the rules of multiplication of the two-element Abelian group Z2. 

The same result can be obtained for biaxial nematics [42] from a topological point of view, the classifications of defects in cholesterics and biaxial nematics are identical. Calculation of the fundamental group for iR = SO 2i)/D2 requires knowledge beyond the scope of this chapter. We simply present the result (for details, see [2], [37], [42]) ... 

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