  Toroidal nematics are droplets of nematic liquid crystals in the form of a
circular torus. When the nematic director is subject to planar degenerate
boundary conditions, the bend-only director field with vector lines along the
parallels of all internal torodial shells is an equilibrium solution for all
values of the elastic constants. Local stability analyses have shown that an
instability is expected to occur for sufficiently small values of the twist
elastic constant. It is natural to conjecture that in this regime the global
equilibrium would be characterized by a maximum twist deflection on the
boundary of the torus, with a twist distribution over the torus' cross-section
represented by a fennel-like surface. We prove that surprisingly the stable
twist distribution is instead represented by a lily-like surface. Thus the
overall maximum twist deflection falls well within the torus. To cope with the
complexity of the elastic free-energy functional in the fully non-linear
setting, we developed an ad hoc deep-learning optimization method, which here
is also duly validated and documented for it promises to be applicable to other
similar problems, equally intractable analytically.
