 
Summary: Characteristics of Shape and Knotting in Ideal Rings
Laura Zirbel
Department of Mathematics,
University of California, Santa Barbara,
Santa Barbara, CA 93016, USA
Email: lzirbel@math.ucsb.edu
Homepage: http://www.math.ucsb.edu/ lzirbel
Kenneth C. Millett
Department of Mathematics,
University of California, Santa Barbara,
Santa Barbara, CA 93016, USA
Email: millett@math.ucsb.edu
Homepage: http://www.math.ucsb.edu/ millett
Abstract
We present two descriptions of the the local scaling and shape of ideal rings, primarily featuring
subsegments. Our focus will be the squared radius of gyration of subsegments and the squared
internal end to end distance, defined to be the average distance between vertices k edges apart.
We calculate the exact averages of these values over the space of all such ideal rings, not just a
calculation of the order of these averages, and compare these to the equivalent values in open
chains. This comparison will show that the structure of ideal rings is similar to that of ideal
