  Two types of average structures of a single knotted ring polymer are studied
by Brownian dynamics simulations. For a ring polymer with N segments, its
structure is represented by a 3N -dimensional conformation vector consisting of
the Cartesian coordinates of the segment positions relative to the center of
mass of the ring polymer. The average structure is given by the average
conformation vector, which is self-consistently defined as the average of the
conformation vectors obtained from a simulation each of which is rotated to
minimize its distance from the average conformation vector. From each
conformation vector sampled in a simulation, 2N conformation vectors are
generated by changing the numbering of the segments. Among the 2N conformation
vectors, the one closest to the average conformation vector is used for one
type of the average structure. The other type of the averages structure uses
all the conformation vectors generated from those sampled in a simulation. In
thecase of the former average structure, the knotted part of the average
structure is delocalized for small N and becomes localized as N is increased.
In the case of the latter average structure, the average structure changes from
a double loop structure for small N to a single loop structure for large N,
which indicates the localization-delocalization transition of the knotted part.
