Cooperative Motions and Topology-Driven Dynamical Arrest in Prime Knots

TL;DR

This paper explores how the topology of prime knots influences their dynamic motions, revealing a transition from stochastic to arrested states and identifying cooperative motions that modulate conformations, with implications for nanoscale materials.

Contribution

It introduces a classification of knot motions, demonstrates topology-driven dynamical arrest, and uncovers cooperative motions in complex knots, advancing understanding of knot dynamics and topology-dynamics links.

Findings

Identification of three main motion groups: orthogonal, aligned, mixed.

High complexity knots exhibit dynamical arrest driven by topology.

Some knots show cooperative motions modulating conformations.

Abstract

Knots are entangled structures that cannot be untangled without a cut. Topological stability of knots is one of the many examples of their important properties that can be used in information storage and transfer. Knot dynamics is important for understanding general principles of entanglement as knots provide an isolated system where tangles are highly controlled and easily manipulated. To unravel the dynamics of these entangled topological objects, the first step is to identify the dominant motions that are uniquely guided by knot structure and its complexity. We identify and classify motions into three main groups -- orthogonal, aligned, and mixed motions, which often act in unison, orchestrating the complex dynamics of knots. The balance between these motions is what creates an identifiable signature for every knot. As knot complexity increases, the carefully orchestrated dynamics is gradually silenced, eventually reaching a state of topologically driven dynamical arrest. Depending on their complexity, knots undergo a transition from nearly stochastic motions to either non-random or even quasiperiodic dynamics before culminating in dynamical arrest. Here, we show for the first time that connectivity alone can lead to a topology-driven dynamical arrest in knots of high complexity. Unexpectedly, we noticed that some knots undergo cooperative motions as they reach higher complexity, uniquely modulating conformational patterns of a given knot. Together, these findings demonstrate a link between topology and dynamics, presenting applications to nanoscale materials.