Knots in Macromolecules in Constraint Space

We find a power law for the number of knot-monomers with an exponent \(0.39 \pm0.13\) in agreement with previous simulations. For the average size of a knot we also obtain a power law \(N_m=2.56\cdot N^{0.20\pm0.04}\). We further present data on the average number of knots given a certain chain length and confirm a power law behaviour for the number of knot-monomers. Furthermore we study the average crossing number for random and self-avoiding walks as well as for a model polymer with and without geometric constraints. The data confirms the \(aN\log N + bN\) law in the case of without excluded volume and determines the constants \(a\) and \(b\) for various cases. For chains with excluded volume the data for chains up to N=1500 is consistent with \(aN\log N + bN\) rather than the proposed \(N^{4/3}\) law. Nevertheless our fits show that the \(N^{4/3}\) law is a suitable approximation.