Fig. 2: Vortex reconnections in a helical twisted background of the chiral LC.

a, Connected sum of two trefoil knots. b, Schematic of a dischiralation vortex line with the core in the form of a region in which the helical axis field \({\bf{\chi }}\left({\bf{r}}\right)\) is singular (undefined) within a chiral LC. The top inset shows the local \({\bf{\chi }}\left({\bf{r}}\right)\) and the molecular twist at a point corresponding to the single black double arrow in a neighbourhood of the vortex line. c, Schematic of the vortex reconnections between the vortex knots of two heliknotons, where the grey and black segments indicate +1/2 and –1/2 vortex line fragments, respectively. The dashed red lines represent the locations of the corresponding cross-sections of \({\bf{\chi }}\left({\bf{r}}\right)\) depicted in planes perpendicular to the local vortex cores. The red circles highlight regions of reconnection progressing from left to middle; additional intra-heliknoton reconnections transform the dischiralation vortex knots depicted in the middle-to-right schematics. d, Two heliknotons undergoing a paired reconnection event, transforming from two trefoils (frame one) to multicomponent links (frames 3–5) coloured according to the director orientation shown in e. The red circles highlight regions in which reconnections progress through the intermediate formation of vertices of a four-valent graph, in the process indicated in c. e, Colour-mapping scheme of the vectorized director orientation n(r) based on the \({{\mathbb{S}}}^{2}\) sphere (top left), where all the possible orientations of the unit vector are uniquely represented by the colours, as illustrated for the helical structure (top right). In the flattened version of the coloured unit sphere (bottom), the arrows show the directions of increasing azimuthal and polar angles describing the orientations of n(r), where the white centre corresponds to the north pole and black periphery denotes the south pole of \({{\mathbb{S}}}^{2}\). f, Reconnections (shown in d) visualized with ribbons of splay and bend, where the dual-band and tri-band ribbons distinguish between the +1/2 and –1/2 winding numbers of dischiralation lines, respectively. Positive splay and bend regions of deformed χ(r) are shown in blue and the negative counterparts, in yellow, as depicted in g. Cross-sectional slices show the local χ(r) orientation and regions of strong splay and bend, both within their actual locations in the sample and as separate enlarged subpanels in the insets below and above. g, Schematic of the χ(r) orientation and the corresponding splay–bend geometry and the colour scheme for each dischiralation local structure type. Reconnections in d and f were initiated by reducing the voltage from 2.8 V (first three frames) to 2 V (last two frames) in a 20-μm-thick cell with an LC pitch of 5 μm. Supplementary Video 1 shows the corresponding dynamics.