We show how the length scale hierarchy, resulting from different interaction strengths in an optically-trapped spin-1 23 Na Bose-Einstein condensate, can lead to intriguing core deformations in singular topological defects. In particular, a point defect can be unstable with respect to the formation of a stable half-quantum vortex ring (an 'Alice ring'), providing a realistic scheme to use dissipation as a sophisticated state engineering tool. We compute the threshold for stability of the point monopole, which is beyond the current experimental regime.PACS numbers: 03.75.Lm,03.75.MnThe rich order parameter space of multi-component Bose-Einstein condensates (BECs) can admit truly 3D topological excitations [1,2,3,4,5], beyond the simple quantized vortices of single-component BECs. Such structures are of interest in a wide range of physical contexts, but dilute atomic BECs offer the unusual advantage that we can fully explore, e.g., the short-range physics in topological defect cores, where the order parameter may explore a larger space than the usual ground state manifold. In this Letter we show how this can result in rich and surprising core structures, by demonstrating a spontaneous deformation of a singular point defect to an energetically stable half-quantum vortex ring.We examine the recently presented case [5] of a defect analogous to the 't Hooft-Polyakov monopole [6], in a antiferromagnetic, or polar, spin-1 BEC [7]. We will show that it is only in the strongly antiferromagnetic regime, which is not attained in current experiments, that its stable core will be the spherically symmetric hedgehog, with a total density depression, of Ref. [5]. In the weakly antiferromagnetic regime that currently holds, the wavelength at which the antiferromagnetic constraint may be violated is much larger than that at which the total density constraint fails. The stable defect core therefore extends to this larger size, and holds non-zero average spin instead of a density zero. The singular point defect itself deforms to a circle: a half-quantum vortex ring (Figs. 1 and 2), called an 'Alice ring' by high energy physicists [8,9,10], which carries a topological charge similar to delocalized magnetic 'Cheshire' charge [11]. This forms an interesting connection between ultra-cold atom experiments and elementary particle physics. It also shows that dissipation, often an obstacle in state engineering, can sometimes perform the intricate final step in producing an exotic object.We consider the BEC of spin-1 atoms. In the absence of a magnetic trapping potential, the macroscopic BEC wave function is determined by a spinor wave function Ψ with three complex components [12,13]. The Hamiltonian density of the classical Gross-Pitaevskii (GP) mean-The stable half-quantum vortex ring (Alice ring), when the energy of an initial spherically symmetric monopole was minimized by continuously deforming the field configuration. The asymptotic distribution of the spin quantization axis d(r) (left), for r ≫ ξa, forms the radial hedgehog. For visu...