 
Summary: 1
MONTE CARLO EXPLORATIONS OF POLYGONAL KNOT SPACES
KENNETH C. MILLETT
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106, USA
Email: millett@math.ucsb.edu
ABSTRACT
Polygonal knots are embeddings of polygons in three space. For each n, the collection of
embedded ngons determines a subset of Euclidean space whose structure is the subject
of this paper. Which knots can be constructed with a specified number of edges? What
is the likelihood that a randomly chosen polygon of nedges will be a knot of a specific
topological type? At what point is a given topological type most likely as a function of
the number of edges? Are the various orderings of knot types by means of ``physical
properties'' comparable? These and related questions are discussed and supporting
evidence, in many cases derived from Monte Carlo explorations, is provided.
Keywords: Monte Carlo, polygonal knots, energy, thickness, HOMFLY, geometric
knots, physical knot theory
1. INTRODUCTION
The topological and geometric knotting of circles occurs in many contexts in the
