Exact knot solutions in a generalized Skyrme-Faddeev model

We propose a generalized Skyrme-Faddeev type theory with an additional scalar field. In a special case of model parameters, one has a theory which admits exact knot solutions given by a class of exact toroidal solitons from the Aratyn-Ferreira-Zimerman (AFZ) integrable CP1 model. In a general case, the theory admits an exact knot solution for a unit Hopf charge. For higher Hopf charges, we perform numeric analysis of the solutions and obtain estimates for the knot energies using an energy minimization procedure based on ansatz with AFZ field configurations and with rational functions. We show that AFZ configurations provide better approximate solutions. The corresponding knot energies are in good agreement with a standard law for the low energy bound, E-H similar or equal to Q(H)(3/4).