Collective Dynamics and Topological Locking in Knotted Ring Polymers: A Novel Phenomenological Theory

Abstract: We present a novel phenomenological theory describing how topological constraints in prime-knot ring polymers induce collective (cooperative) modes of motion. In low-complexity knots, chain segments can move quasi-independently. However, as the crossing number increases, the ring's degrees of freedom become collectively coupled: distinct arc segments must move in coordinated, out-of-phase patterns to preserve the knot. We formulate this using an arc-based model in which each crossing imposes constraints that generate a coupling matrix among subchain displacements. We show how strong couplings emerge at higher knot complexity, eventually leading to topologically driven dynamical arrest. We demonstrate that torus and twist knots belong to distinct universality classes of topologically driven dynamical arrest: torus knots exhibit a gradual, stretched-exponential slowdown, while twist knots undergo a sharp, jamming-like transition. These findings establish a topological control parameter for relaxation dynamics, independent of steric effects or bending rigidity. Our results offer a unified framework connecting number of conformations, cooperative motion, and final arrested states, thus extending our fundamental understanding of entanglement in soft matter systems.