Properties of knotted ring polymers. I. Equilibrium dimensions.

We report calculations on three classes of knotted ring polymers: (1) simple-cubic lattice self-avoiding rings (SARs), (2) "true" theta-state rings, i.e., SARs generated on the simple-cubic lattice with an attractive nearest-neighbor contact potential (theta-SARs), and (3) ideal, Gaussian rings. Extrapolations to large polymerization index N imply knot localization in all three classes of chains. Extrapolations of our data are also consistent with conjectures found in the literature which state that (1) R(g)-->AN(nu) asymptotically for ensembles of random knots restricted to any particular knot state, including the unknot; (2) A is universal across knot types for any given class of flexible chains; and (3) nu is equal to the standard self-avoiding walk (SAW) exponent (congruent with 0.588) for all three classes of chains (SARs, theta-SARs, and ideal rings). However, current computer technology is inadequate to directly sample the asymptotic domain, so that we remain in a crossover scaling regime for all accessible values of N. We also observe that R(g) approximately p(-0.27), where p is the "rope length" of the maximally inflated knot. This scaling relation holds in the crossover regime, but we argue that it is unlikely to extend into the asymptotic scaling regime where knots become localized.