Figure 14.

Defect loops tie links and knots in chiral nematic colloids. (a) A twisted Saturn ring is an unknot and appears spontaneously around a microsphere in a chiral nematic. The molecular arrangement in all the panels is illustrated in the top-right corner. (b–e) The defect loops of the colloidal dimer, trimer and tetramers are also unknots. (f) The first non-trivial topological binding of a colloidal cluster is the Hopf link. Two separate defect loops are interlinked. In the lower panels (a–f) the loop conformations were calculated using numerical Landau-de Gennes modeling. (g–j) A series of alternating torus knots and links on colloidal particle arrays of m × n particles are knitted by the laser. The actual defect loops are schematically redrawn to show the relaxation mapping from the experimental planar projection to the final knot diagram. The knots are labeled using the standard notation C, i, N, where C is the minimum number of crossings, i distinguishes the different knot types and N counts the number of loops. Scale bars, 5 µm. Adapted from Ref. [87].