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Summary: Shapes of Knotted Cyclic Polymers
Eric J. Rawdon, University of St. Thomas, Saint Paul, MN, USA
John C. Kern, Duquesne University, Pittsburgh, PA, USA
Michael Piatek, University of Washington, Seattle, WA, USA
Patrick Plunkett, University of California, Santa Barbara, CA, USA
Andrzej Stasiak1
, University of Lausanne, Lausanne, Switzerland
Kennth C. Millett, University of California, Santa Barbara, CA, USA
Momentary configurations of long polymers at thermal equilibrium usually deviate from
spherical symmetry and can be better described, on average, by a prolate ellipsoid. The as-
phericity and nature of asphericity (or prolateness) that describe these momentary ellipsoidal
shapes of a polymer are determined by specific expressions involving the three principal mo-
ments of inertia calculated for configurations of the polymer. Earlier theoretical studies and
numerical simulations have established that as the length of the polymer increases, the average
shape for the statistical ensemble of random configurations asymptotically approaches a char-
acteristic universal shape that depends on the solvent quality. It has been established, however,
that these universal shapes differ for linear, circular, and branched chains. We investigate here
the effect of knotting on the shape of cyclic polymers modeled as random isosegmental polygons.
We observe that random polygons forming different knot types reach asymptotic shapes that
are distinct from the ensemble average shape. For the same chain length, more complex knots
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