In Search of a Good Polygonal Knot Energy

An energy function on knots is a scale-invariant function from knot conformations into non-negative real numbers. The infimum of an energy function is an invariant which defines "canonical conformation(s)" of a knot in three space. These are not necessarily unique, and, in some cases, may even be singular. Many hierarchies of energy functions for knots in the mathematical and physical science literature have been studied, each energy function with its own characteristic set of properties. In this paper we focus on the energy functions of equilateral polygonal knots. These energy functions are important in computer studies of knot energies, and are often defined as discrete versions of energy functions defined on smooth knots. Energy functions on equilateral polygonal knots turn out to be ill-behaved in many cases. To characterize a "good" polygonal knot energy we introduce the concepts of asymptotically finite and asymptotically smooth energy functions of equilateral polygonal knots. Energy functions which are both asymptotically finite and smooth tend to have food limiting behavior (as the number of edges goes to infinity). We introduce a new energy function of equilateral polygonal knots, and show that it is both asymptotically finite and smooth. In addition, we compute this energy for several knots using simulated annealing.