Defects in frustrated media: Classification by homology groups

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Abstract
Supercooled liquids and metallic glasses have been modeled recently by icosahedral and cuboctahedral nematic liquid crystals. The defects, which are forced into these media by frustration, have been classified by homotopy theory. Here, the defects are grouped into classes, which form an Abelian group isomorphic to the first homology group of the order-parameter space. This classification is more general than the homotopic because two defects are considered equivalent also, if one can be transformed into the other via catalyzation by a third defect line. In this scheme only one class exists for the icosahedral nematics, and two classes exist for the cuboctahedral nematics. Homology groups are also presented for tetrahedral nematics, for two-dimensional triangular crystals, for simple-cubic crystals, and for the polytope {3,35}.