Knots in Macromolecules in Constraint Space

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We find a power law for the number of knot-monomers with an exponent 0.39 ±0.13 in agreement with previous simulations. For the average size of a knot we also obtain a power law Nₘ=2.56· N⁰.²⁰±⁰.⁰⁴. 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 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 N + bN rather than the proposed N⁴/³ law. Nevertheless our fits show that the N⁴/³ law is a suitable approximation.