Knotted configurations with arbitrary Hopf index from the eikonal equation

The complex eikonal equation in (3 + 1) dimensions is investigated. It is shown that this equation generates many multi-knot configurations with an arbitrary value of the Hopf index. In general, these eikonal knots do not have the toroidal symmetry. For example, a solution with the topology of the trefoil knot is found. Moreover, we show that the eikonal knots provide an analytical framework in which qualitative (shape, topology) as well as quantitative (energy) features of the Faddeev-Niemi hopfions can be captured. This might suggest that the eikonal knots can be helpful in the construction of approximated (but analytical) knotted solutions of the Faddeev-Skyrme-Niemi model.Received: 5 April 2005, Revised: 16 May 2005, Published online: 19 July 2005