Characteristics of shape and knotting in ideal rings (Q2895715)

The properties of ideal rings (imbedded equilateral polygons in \({\mathbb R}^3\)) and ideal chains are studied. It is required that the polygons are based at the origin and are oriented. In this model, the squared end to end distance in ideal chains, the squared end to end distance of subsegments of length \(k\) in a chain of length \(n>k\), average edge products in ideal rings and the average squared radius of gyration of subsegments of length \(k\) are defined and calculated. The comparison of characteristics of ideal rings and ideal chains shows that the structure of ideal rings is similar to that of ideal chains only for very short lengths. Numerical simulations prove that the behavior of subsegments of lengths \(k\) in chains is not influenced by the ambient length of the chain. However, for the ideal rings, subsegments of length \(k\) in a ring of length \(n\) do not behave in the same way as a subsegment of length \(k\) in a ring of length \(m \not= n\). In such a case, the behavior of subsegments depends on both \(k\) and \(n\). It is also shown that the squared end to end distance and squared radius of gyration of subsegments could be used to predict a knot length which is computationally difficult to calculate.