Liquid crystals confined on a curved surface involve a complex and intriguing interplay between the elastic free energy of the orientational order parameter and the geometric properties of the surface, including boundary conditions. Although conical surfaces have zero Gaussian curvature locally, it was recently reported that the ground states of liquid crystals on cones nevertheless show nontrivial behaviors in the presence of tangential boundary conditions,[1] because the Gaussian curvature at the apex not only induces geometric frustration to the orientational order field by parallel transport but also interacts with the topological defects on the cone flanks. In this work, we study ground states of liquid crystals with p-fold rotational symmetry on cones with anti-twist boundary conditions.  On the base of the cones, the anti-twist boundary conditions require that the orientational order parameter rotates by -2pi as moving a full circle counterclockwise around the base, including a contribution from the deficit angle. The orientational order field on the boundary is thus topologically equivalent to a net -1 topological defect in a flat 2D plane. We use both simulations and theory to determine the ground states of p-atic liquid crystals with anti-twist boundary conditions as a function of cone angle, and find varying numbers of defects with negative topological charge in the ground state.

[1] Vafa, F., Zhang, G. H., & Nelson, D. R. (2022). Defect absorption and emission for p-atic liquid crystals on cones. Physical Review E, 106(2), 024704.