Ring polymers are represented on a computer with the bead-stick model. If an infinitely thin spike is inserted at some (minimum) distance r from the center of mass of a Gaussian ring of contour length N, the probability P(r,N,w) of the ring winding a nonzero number of times w about the spike is nearly given by A exp[-(r/R(g))(2)/2 sigma(2)], where A and a depend on N and w, and R(g) is the radius of gyration of the polymer. We also find that the asymptotic results of related theoretical calculations cannot be realized in systems of practical size.