Fig. 6: Formation of complex knots through the fusion of simple ones.

a,b, Two \(Q=2\) fused knot dimers shown in a, which previously formed through fusion of pairs of individual trefoil-shaped vortex knots (Extended Data Fig. 6), hybridize together to form a tetramer shown in b. The fusion was driven by switching voltage from ~3.7 V to ~4 V. Insets show simplified vortex knot schematics, where each closed loop is differently coloured. c, Graph state formed by reconnections within a vortex knot tetramer, with the circled four-valent node of vortex lines that can be resolved into different knot states depending on relinking; detailed configurations of vortex lines and the corresponding simplified knot schematics of the entire knots are illustrated in the boxed inset, where the top visualizations use the scheme shown in Fig. 2g and in the bottom ones, each closed-loop component is differently coloured. d–g, Two \(Q=3\) trimers shown in d, each obtained by fusing three elementary heliknotons, sequentially reconnect to produce a \(Q=6\) heliknoton (e), which evolves into transient (f) and then stable complex graph with several nodes (g) while conserving Q = 6; for dynamics, see Supplementary Video 6. The inset in d schematically shows distinctly coloured closed-loop vortex components prior to the reconnection depicted in e. The detailed configurations of vortex lines in the region of fusion are shown in the bottom insets of e–g according to the scheme also used in c. h,i, Eight \(Q=1\) heliknotons arranged closely together, as shown in h, fuse to form a vortex graph with \(Q=8\), in response to applying 4 V (h,i). j, A fused state of 18 elementary heliknotons relaxed from a perturbed lattice to form an interconnected graph with \(Q=18\) (Extended Data Fig. 8; Supplementary Video 7 shows the dynamics of the simulated POM); the complex knot is produced via the fusion of individual knots from an array by pulsing (three times) with voltage amplitudes between 1.5 V and 3 V. The simulated POMs of the knotted structures are shown as insets in h–j, which are obtained for crossed polarizers indicated by white double arrows. Parameters used in simulations are sample thickness d = 25 μm and pitch p = 5 μm; vortex cores are coloured by the n(r) orientation according to the scheme shown in Fig. 2e.