The Toulouse–Kleman homotopic classification of topological defects in ordered systems illustrated by experiments

Classification of defects in ordered systems, based on the homotopy theory, conceived by Gérard Toulouse and Maurice Kléman has a very wide range of applications. We illustrate its modus operandi with three experimental examples. We deal first with dislocations in a dissipative periodic pattern of convection rolls in a shear flow instability in nematics. As the second example we chose the captive disclination loops threaded on polymer fibers immersed in nematics. Third, we focus on objects with double topological character defined for the first time in the generic article coauthored by Gérard Toulouse: the “double et tripple anneau” Hopf links made of interlaced dislocation loops in cholesterics. Finally, we report on the recent discovery of their generalised, beads necklace version made of many minimal dislocation loops threaded, like pearls, on a string-like dislocation loops.