Knotted Configurations with Arbitrary Hopf Index from the Eikonal Equation

Abstract

The complex eikonal equation in $(3+1)$ dimensions is investigated. It is shown that this equation generates many multi soliton configurations with arbitrary value of the Hopf index. In general, these eikonal hopfions do not have the toroidal symmetry. For example, a hopfion with topology of the trefoil knot is found. Moreover, we argue that such solitons might be helpful in construction of approximated analytical knotted solutions of the Faddeev-Niemi model.