 
Summary: Total Curvature and Total Torsion of Knotted Polymers
Patrick Plunkett, Michael Piatek, Akos Dobay,§ John C. Kern, Kenneth C. Millett,
Andrzej Stasiak, and Eric J. Rawdon*,#
Department of Mathematics and Computer Science, Duquesne UniVersity, Pittsburgh, PennsylVania
15282; Department of Computer Science and Engineering, UniVersity of Washington, Seattle,
Washington 98195; LudwigMaximiliansUniVersitašt, Biozentrum, Grosshadernerstrasse 2,
82152 Munich, Germany; Department of Mathematics, UniVersity of California, Santa Barbara, Santa
Barbara, California 93106; Faculty of Biology and Medicine, Laboratory of Ultrastructural Analysis,
UniVersity of Lausanne, Lausanne CH 1015, Switzerland; and Department of Mathematics, UniVersity
of St. Thomas, St. Paul, Minnesota 55105
ReceiVed December 1, 2006; ReVised Manuscript ReceiVed February 7, 2007
ABSTRACT: Previous work on radius of gyration and average crossing number has demonstrated that polymers
with fixed topology show a different scaling behavior with respect to these characteristics than polymers with
unrestricted topology. Using numerical simulations, we show here that the difference in the scaling behavior
between polymers with restricted and unrestricted topology also applies to the total curvature and total torsion.
For each knot type, the equilibrium length with respect to a given spatial characteristic is the number of edges
at which the value of the characteristic is the same as the average for all polygons. This number appears to be
correlated to physical properties of macromolecules, for example gel mobility as measured by the separation
between distinct knot types. We also find that, on average, closed polymers require slightly more total curvature
and slightly less total torsion than open polymers with the corresponding number of monomers.
