Narrow Results

Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded n-gons 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 n-edges 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.

We present some commonly-used models for polymers in different experimental conditions, including some discussion of implementation issues. We then analyze how the size and shape of polymers are affected by length and the existence of knotting.