Self-entanglement in ring polymers

We represent ring polymers with the off-lattice rod-bead model. We produce perfect unbiased model instances that have between 32 and 2048 beads. These rings are produced at several different values of bead radius, to represent polymers in solvents of different quality. These instances are subject to a novel smoothing operation that facilitates the determination of their topological state, which is represented with the Alexander polynomial. We observe that the probability of observing a trivial knot (P) has a decreasing exponential dependence on the contour length (N) of the polymer, or that P=exp(−N/N0). The characteristic length (N0) varies by many orders of magnitude depending on chain flexibility and solvent quality. For Gaussian rings in a theta solvent, the characteristic length (N0) is 2.6×102. For a good solvent, N0∼8×105. We also explore the expectation value of the minimum number of crossings and suggest that it probably cannot be represented as a power law or exponential function of N. We also suggest that sufficiently large rings will always be composite, and not prime.