Knots in finite memory walks

We investigate the occurrence and size of knots in a continuum polymer model with finite memory via Monte Carlo simulations. Excluded volume interactions are local and extend only to a fixed number of successive beads along the chain, ensuring that at short length scales the excluded volume effect dominates, while at longer length scales the polymer behaves like a random walk. As such, this model may be useful for understanding the behavior of polymers in a melt or semi-dilute solution, where exactly the same crossover is believed to occur. In particular, finite memory walks allow us to investigate the role of local interactions in the transition from highly knotted ideal polymers to almost unknotted self-avoiding polymers. Even though knotting decreases substantially when a few next-nearest neighbor interactions are considered, we find that the knotting probability of a polymer chain of modest length of 500 steps only decays slowly as a function of the range of the excluded volume interaction. In this context, we also find evidence that for length scales up to the interaction length the knotting behavior of the finite memory walk resembles that of a self-avoiding walk (effectively suppressing small knots), while for larger length scales it resembles that of a random walk.