This talk concerns Hamiltonians for the motion of particles in two dimensions. A space time diagram of such motions gives a braided pattern. 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. Suppose, however, 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 3 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.