Knots as dynamical systems

Abstract

A knot is considered as an n-gon in R 3. Two potential energies for these PL knot conformations are found, mapping R 3 n → R (the configuration space to energy). These functions have the property that they “blow up” if the edges approach crossing, that is, as the knot changes type. Therefore the configuration space is divided into manifolds for each knot type by infinitely high potential walls. Therefore we have energy surfaces associated with each knot type. Descriptions of these surfaces are invariants of the knot type. For example the locations of and connections between the critical points are invariants of the knot type. In particular, the global minimum energy position for the knot could be said to be a canonical conformation of the knot. The existance and achievement of the global minimum is proved. It is shown that the energy surface for any knot of N vertices has a finite number of components. Also several topological properties of the configuration space of polygonal knots in R 3 are established. These invariant are in some sense computable. In a sequel paper, G. Buck and J. Orloff discuss computer simulations of the gradient flow of one of the energy functions.