Topology-Driven Dynamics and Randomness Control in Prime Knots

Knots are entangled structures that cannot be untangled without a cut. Knots appear ubiquitously across length scales and play a key role in understanding and controlling the behaviors of complex systems. It has been known that entanglement structure alters polymer material's properties such as relaxation time, fragility, and viscosity. However, the precise link between structure and dynamics has not been well established. To help elucidate this link, we present a detailed analysis of the relation between the dynamics, topology, and complexity (crossing number) of polymer knots, focusing on a subset of closed knots called prime knots. We first identify three main motions of knots—orthogonal, aligned, and mixed motions—whose different compositions create unique dynamics for each knot. As knot complexity increases, we observe a gradual fading of dynamics, showing for the first time that connectivity alone can lead to topology-driven dynamical arrest in knots of high complexity. As knot size shrinks, knots undergo a transition from nearly stochastic motions to either non-random or "quasiperiodic" dynamics before culminating in dynamical arrest. Together, these findings demonstrate a clear link between structure and dynamics and present applications to biomedicine and nanotechnology.