Characterizing knots in polymer coil and globule phases

We examine the statistics of knots with numerical simulations of a model polymer, spanning high temperature (coil) and low temperature (globule) phases. All monomers in the model interact via a Lennard-Jones potential, while adjacent beads are connected by flexible springs. Although relatively simple, this model provides a realistic description of polyethylene, thus relating the simulation results to specific polymers. Equilibrated configurations are generated by a combination of pivot, ``slithering snake'' and local Monte Carlo moves. The knot type is determined by applying Taylor's reduction method$^{1}$ and calculating the Alexander polynomial after connecting the end monomers (of an open polymer). We find that knots are common in the globule phase and under confinement, but rare in coils. We also associate a typical size with the knots, and find knots to be small (tight) in the swollen phase, and large (loose) in the dense phase. $^{1}$ W.R. Taylor, Nature \textbf{406}, 916 (2000)