Torus knots and polynomial invariants for a class of soliton equations

In this paper is shown how to interpret the nonlinear dynamics of a class of one-dimensional physical systems exhibiting soliton behavior in terms of Killing fields for the associated dynamical laws acting as generators of torus knots. Soliton equations are related to dynamical laws associated with the intrinsic kinematics of space curves and torus knots are obtained as traveling wave solutions to the soliton equations. For the sake of illustration a full calculation is carried out by considering the Killing field that is associated with the nonlinear Schrödinger equation. Torus knot solutions are obtained explicitly in cylindrical polar coordinates via perturbation techniques from the circular solution. Using the Hasimoto map, the soliton conserved quantities are interpreted in terms of global geometric quantities and it is shown how to express these quantities as polynomial invariants for torus knots. The techniques here employed are of general interest and lead us to make some conjectures on natural links between the nonlinear dynamics of one-dimensional extended objects and the topological classification of knots.