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Twisted toroidal vortex-solitons in inhomogeneous media with repulsive nonlinearity

Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist, s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis m, appear in many fields, including the field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multi-component systems, or as trapped quasi-linear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0,1,2 are produced. They are stable for m<=1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in BEC, by means of the Feshbach resonance controlled by inhomogene-ous magnetic fields.