Colloidal entanglement in highly twisted chiral nematic colloids: twisted loops, Hopf links, and trefoil knots.

The topology and geometry of closed defect loops is studied in chiral nematic colloids with variable chirality. The colloidal particles with perpendicular surface anchoring of liquid crystalline molecules are inserted in a twisted nematic cell with the thickness that is only slightly larger than the diameter of the colloidal particle. The total twist of the chiral nematic structure in cells with parallel boundary conditions is set to 0, π, 2π, and 3π, respectively. We use the laser tweezers to discern the number and the topology of the -1/2 defect loops entangling colloidal particles. For a single colloidal particle, we observe that a single defect loop is winding around the particle, with the winding pattern being more complex in cells with higher total twist. We observe that colloidal dimers and colloidal clusters are always entangled by one or several -1/2 defect loops. For colloidal pairs in π-twisted cells, we identify at least 17 different entangled structures, some of them exhibiting linked defect loops-Hopf link. Colloidal entanglement is even richer with a higher number of colloidal particles, where we observe not only linked, but also colloidal clusters knotted into the trefoil knot. The experiments are in good agreement with numerical modeling using Landau-de Gennes theory coupled with geometrical and topological considerations using the method of tetrahedral rotation.