Hamiltonian dynamics generated by Vassiliev invariants

This paper employs higher-order winding numbers to generate Hamiltonian motion of particles in two dimensions. The ordinary winding number counts how many times two particles rotate about each other. Higher-order winding numbers measure braiding motions of three or more particles. These winding numbers relate to various invariants known in topology and knot theory, for example Massey and Milnor numbers, and can be derived from Vassiliev-Kontsevich integrals. The invariants can be regarded as complex-valued functions of the paths of the particles. The real part gives the winding number, whereas the imaginary part seems uninteresting. In this paper, we set the imaginary part to be a Hamiltonian for particle motions. For just two particles, this gives the familiar motion of two point vortices. However, for three or more particles, the Hamiltonian generates more complicated intertwining patterns. We examine the dynamics for the case of three particles, and show that the motion is completely integrable. The intertwining patterns correspond to periodic braids; closure of these braids gives links such as the Borromean rings. The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links.