Disclinations entanglement

Again, remarkable predictions have been made on the basis of the homotopy theory regarding the properties of defects in the phase. The usual law of coalescence of defects (see 3.5.1) breaks down the combination rule is now non-Abelian. Moreover, there can be an entanglement of disclination lines, which, in turn, can give rise to what... 

One of the most spectacular consequences of noncommutativity of the group Q is the possibility of a topological entanglement of the disclinations. Originally, the problem was considered by Toulouse for biaxial nematics [42], but it apphes to any medium with a non-Abelian fundamental group [50]-[52. ... 

Figure 5.13(a) shows two entangled disclinations. The question is whether they can be transformed by continuous variations of the directors into an... 

Figure 5.13. (a) Entanglement of disclinations in a medium with a trihedron of vectors as the order parameter (b) topologically trivial and (c) nontrivial. 

The uniaxial nematic (Ny) phase usually exhibits a schlieren texture with two and four point singularities (two and four brushes). However, the texture exhibited by complex 2, shown in Fig. 4, often consists of [S] = l/2 (two-brush) disclinations similar to that exhibited by a smectic C phase. Also, this complex showed zig-zag disclinations [24] occasionally, and since this type of disclination has been observed in N phases [25] it cannot be regarded as conclusive evidence for biaxiality. It is also possible that an entanglement of disclination lines may lead to topological rigidity, as discussed by Toulouse [26] and others [27-29]. These possibilities have not been examined experimentally. Hence optical textures that... 

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