Fig. 3: Orientation-dependent fusion of vortex knots.
From: Fusion and fission of particle-like chiral nematic vortex knots
a–d, POM time series showing an initial array of six vortex knots after they are perturbed from the initial equilibrium configuration by changing voltage (a); three heliknotons fuse while three others remain individualized (b); the fused trimer hybridizes with one more heliknoton to form a tetramer, alongside two individual heliknotons (c); the remaining two individual heliknotons fuse to form the final configuration of a dimer (right) next to a tetramer (left) (d). Scale bars, 10 μm. The black double arrows show the orientation of crossed polarizers. The bottom-right insets are the corresponding numerically simulated POMs. Angle \(\psi\) defines the relative in-plane angle between the long axes of two interacting heliknotons. Sample thickness d = 16 μm and pitch p = 6.9 μm; the applied voltage U = 1.7 V in a and U = 2.1 V in b–d. Temporal progression is shown by arrows between the frames, with the elapsed time marked on them. Supplementary Video 1 shows the corresponding dynamics. e, Relative heliknoton–heliknoton positions and the visualization depicting the orientation parameters (\({\theta}_{\mathrm{el}},{\phi}\)) defined relative to the inter-heliknoton separation vector \({{\bf{r}}}_{\mathrm{s}}\) and the uniform far-field helical background of the sample. f,g, Experimental (f) and numerically simulated (g) trajectories of the separation vector for the two far-right trefoils in a–d. The knot insets in f (visualized with the help of KnotPlot freeware52) show the simplified vortex knot topology at the beginning, middle and end of the interaction process. The experimental and simulated dependencies are coloured based on the elapsed time according to the colour scheme shown in f. The schematic of the insets in f have each closed-loop components represented by different colours. The colours of the knot insets in g represent the director orientations in the non-singular cores of vortices, according to the scheme shown in Fig. 2e. The corresponding dynamics are shown in Supplementary Video 3.
