Energy, ropelength, and other physical aspects of equilateral knots

Closed macromolecular chains may form physically knotted conformations whose relative occurrence and spatial measurements provide insight into their properties and the mechanisms acting upon them. Under the assumption of a degree of structural homogeneity, equilateral spatial polygons are a productive context within which to create mathematical models of these knots and to study their mathematical and physical properties. The ensembles, or spaces, of these knots are models of the settings within which the knots evolve in ways determined by a physical model. In this paper we describe the mathematical foundation of such models as well as such spatial, geometric, statistical, and physical properties of the configurations as mathematical energies, thickness and ropelength, average crossing number, average writhe, and volumes and surfaces areas of standard bodies enclosing the knots. We present methods with which the energy and ropelength are optimized within the families of spatially equivalent equilateral configurations. Numerical results from our implementation of these methods are shown to illustrate connections between the physical measurements and spatial characteristics of the optimized knot configurations. In addition, these data suggest potentially new connections involving their spatial properties.