Complexity measures for random knots

Abstract In order to model volume exclusion effects, a linear macromolecule can be modeled as a self-avoiding walk (SAW) on 3, the simple cubic lattice in 3-space. A circular macromolecule can be modeled as a closed SAW, or a self-avoiding polygon (SAP) embedded in 3. We assume that all walks are rooted (they begin at the origin), and let n denote the number of edges (steps) in the walk. Almost all long random SAPs are knotted; it is known that if P (n) denotes the probability that a SAP of length n is knotted, then P (n) approaches one exponentially rapidly as n goes to infinity. This paper will discuss the problem of determining average properties of a long randomly chosen SAP. It will be argued that most measures of knot complexity (crossing number, span of various knot polynomials, unknotting number, genus etc.) go to infinity at least linearly with the length n of the SAP. Elsewhere it is shown that these results hold in general for any nonedge-separable graph which embeds in 3.