Unknots, knots, links and necklaces made of dislocations in cholesterics

We deal with the genesis and decay of knots and links made of dislocations in a cholesteric layer confined in the gap between orthogonal cylindrical mica sheets.The genesis starts by nucleation of folded unknots driven by a compressive strain.The nucleation process of the trivial and folded unknots is studied and discussed in detail.We show that the expansion of the folded unknots leads to their lateral collisions followed by the coalescence into the torus knots, such as the trefoil knot, or into the links, such as the Hopf and Solomon links.The viscoelastic decays of knots and links follow well defined pathes involving topological and geometrical transformations.In particular, the symmetric configuration of the Hopf link is unstable with respect to the shrinking of one of its interlaced loops.The final asymmetric configuration, made of a minimal loop tethered on a large cargo loop, is called the Hopf necklace.Similarly, the double Hopf necklace is obtained by the elastic decay of the Solomon link.We emphasize that Hopf necklaces of various orders are quite ubiquitous.