The energy spectrum of knots and links

Abstract

KNOTTED and linked structures arise in such disparate fields as plasma physics, polymer physics, molecular biology and cosmic string theory. It is important to be able to characterize and classify such structures. Early attempts to do so¹ were stimulated by Kelvin's² recognition of the invariance of knotted and linked vortex tubes in fluid flow governed by the classical Euler equations of motion. The techniques of fluid mechanics are still very natural for the investigation of certain problems that are essentially topological in character. Here I use these techniques to establish the existence of a new type of topological invariant for knots and links. Any knot or link may be characterized by an 'energy spectrum'—a set of positive real numbers determined solely by its topology. The lowest energy provides a possible measure of knot or link complexity.