  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.
