Ravels: knot-free but not free. Novel entanglements of graphs in 3-space

Abstract

Molecular and extended framework materials, from proteins to catenanes and metal-organic frameworks, can assume knotted configurations in their bonding networks (the chemical graph). Indeed, knot theory and structural chemistry have remained closely allied, due to those connections. Here we introduce a new class of graph entanglement: "ravels". These ravels-often chiral-tangle a graph without the presence of knots. Just as knots lie within cycles in the graph, ravels lie in the vicinity of a vertex. We introduce various species of ravels, including fragile ravels, composite ravels and shelled ravels. The role of ravels is examined in the context of finite and infinite graphs-analogous to molecular and extended framework nets-related to the diamond net.