Static solitons with nonzero Hopf number

We investigate a generalized nonlinear O$\mathrm{}\left(3\right)\mathrm{}$ $\sigma$ model in three space dimensions where the fields are maps from $R^3\cup \left\{\infty \right\}$ to $S^2$. Such maps are classified by a homotopy invariant called the Hopf number which takes integer values. The model exhibits soliton solutions of closed vortex type which have a lower topological bound on their energies. We numerically compute the fields for topological charge 1 and 2 and discuss their shapes and binding energies. The effect of an additional potential term is considered and an approximation is given for the spectrum of slowly rotating solitons.