Torus Knots and Links from Eikonal Equations and Knot Invariants for Classification of Atoms

The history of knot theory and physics has a deep roots. It started by Lord Kelvin, in 1867, when he conjectured that atoms were knotted vortex tubes of ether. In 1997, Faddeev and Niemi suggested that knots might exist as stable soliton solution in a simple three dimensional classical field theory. That opening up a wide range of possible applications in physics. In this work we consider the Eikonal equation, which is a partial differential equation describing the traveltime propagation, which is an important part of seismic imaging algorithms. We will follow the work of Wereszczynski of solving the Eikonal equation in cylindrical coordinates. We show that only torus knots and links do occur, so figure eight knot does not occur. We show that these solutions are not unique, which means the possible occurrence of the same knot type for different configurations. Using the idea of framed knots, it is shown that two Eikonal knots are equivalent if and only if they are ambient isotopic as a framed knots, i.e. if and only if they are of the same knot type and of the same twisting number.