Self-entanglement of a tumbled circular chain

Abstract

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. Our experimental results show that an increase in circular chain length leads to an increase in entanglement probability and entanglement complexity until a plateau is reached, similar to the trends observed with linear chains. By examining the formation pathway of several self-entanglements, we infer a general mechanism for the self-entanglement of circular chains.