Fig. 3.

Electrically driven topological transformation of a soliton. (A–C) Computer-simulated cross-sections of the soliton at different U obtained in the plane orthogonal to n₀ (Upper) and in the vertical plane containing n₀ (Lower). The vector fields are shown with arrows colored according to the corresponding points on ${\mathbb{S}}^2$ (A, Upper, Inset), and U is indicated for each structure. (D–H) The 3D preimages in ℝ³ of points on ${\mathbb{S}}^2$ indicated as cones in the Upper Right Insets. (D and F) Correspond to the solitons shown in A and C, respectively, while E, G, and H correspond to B. The Lower Right Insets show schematics of the linking of preimages, with the signs of crossings marked in red. Arrows on preimages represent their consistently determined circulations. In D, preimages of distinct points on ${\mathbb{S}}^2$ form a pair of Hopf links with a net linking number −2. In E, G, and H, a preimage can be a single loop ($0_1$) or two separate closed-loops ($0_1^2$), depending on the location of corresponding points on ${\mathbb{S}}^2$, with different subspaces of ${\mathbb{S}}^2$ separated by a boundary at θ_c = 85°; for all combinations of distinct points on ${\mathbb{S}}^2$, the total linking number is −2. In F, a preimage is a single loop, and the linking number is −1 for all pairs of distinct points on ${\mathbb{S}}^2$. For more details on the structure analyzed in B, E, G, and H, see Movie S1. (I) θ_c and Q of the soliton vs. U, with abrupt changes of θ_c at U = 2.6 V and Q at U = 4.2 V, as marked and shown using different background colors. Schematics of preimage linking within each voltage range are shown as Insets. Computer simulations were done for elastic constants of AMLC-0010 (Table S2) and d/p = 1.5.