Self-Entanglement of a Tumbled Circular Chain

The spontaneous knotting of linear chains has been well studied, but little attention has been given to the self-entanglement of chains with more complex topologies. In this work, we perform experiments with granular chains that undergo tumbling motion to investigate the self-entanglement of circular chains, which lack the chain ends essential for forming knots. We study the entanglement probability and types of self-entanglements formed on linear and circular chains, using the well-studied self-entanglements on a linear chain to frame our understanding of self-entanglements on a circular chain. We describe a characterization method that views a self-entangled circular chain as a link of two components and use it to characterize the self-entanglements on circular chains with known topological descriptors from knot theory. By examining the formation pathway of several self-entanglements, we infer a general mechanism for the self-entanglement of circular chains.