Topological Mechanics of Knots and Tangles

Knotted structures play a fundamental role in the dynamics of biological and physical systems, from DNA and polymers to liquid crystals and turbulent plasmas, as well as in climbing, weaving and sailing. Despite having been empirically studied for centuries, the subtle interplay between topology and mechanics in knotted elastic materials remains poorly understood. Here, we combine optomechanical experiments with theory and simulations to analyze the behavior of knots in flexible fibers that change their color in response to mechanical deformations. Exploiting a previously unrecognized analogy with long-range ferromagnetic spin systems, we identify simple counting rules to predict the relative mechanical stability of knots and tangles, in agreement with numerical simulations and experimental measurements for commonly used climbing and sailing knots. The underlying topological principles provide a conceptual foundation for understanding the roles of twist and writhe in untangling processes, and are expected to find broad applications in the description and control of systems with complex entanglements.