Fig. 1: Schematics of skyrmion filaments of achiral banana-shaped particles.

a Hard curved spherocylinder with a length-to-diameter ratio L/σ = 16 modeled by tangent hard spherical beads of diameter σ and opening angle ψ = L/R = 1.6 with R the radius of curvature. The long \(\hat{{{{\boldsymbol{u}}}}}\) and polar \(\hat{{{{\boldsymbol{p}}}}}\) axes of the particle are represented by the gray and red arrows. The third particle axis is simply \(\hat{{{{\boldsymbol{m}}}}}=\hat{{{{\boldsymbol{u}}}}}\times \hat{{{{\boldsymbol{p}}}}}\). b Color legend for banana-shaped particles. The particles in the displayed configurations are colored according to the orientation of their long \(\hat{{{{\boldsymbol{u}}}}}\) particle axis using the RGB color coding as shown in the legend. c, d Nematic director field lines \(\hat{{{{\boldsymbol{n}}}}}({{{\boldsymbol{r}}}})\) of a skyrmion filament or double twist cylinder seen in perspective and from above, respectively. The orientation of the long axis \(\hat{{{{\boldsymbol{u}}}}}\) of individual particles is expected to align with the field lines, vertical and continuous at the core and tilted for higher radii. In the case of polar particles, the skyrmion filament also exhibits a polarization field P(r) (green arrows), which is not defined at the core of the skyrmion filament, identifying a λ-line in the center. This polarization field consistently aligns with the direction of the bend vector of the field lines. Therefore, we can identify the core as a discontinuity in the bend field, known as β-line³¹. e, f Typical configuration of a skyrmion filament of banana-shaped particles obtained from our simulations, displaying both cut-through and top perspectives. The simulations remarkably align with the theoretical predictions.