Stable configurations of entangled graphs and weaves with repulsive interactions

Abstract: Entangled objects such as entangled graphs and weaves are often seen in nature. In the present article, two identical graphs entangled, and weaves with two different color threads are studied. A method to identify stable configurations in the three dimensional space of a given topological entangled structure, a planar graph with crossing information, is proposed by analyzing the steepest descent flow of the energy functional with repulsive interactions. The existence and uniqueness of a solution in the entangled case. In the untangled case, a weave has a unique tangle decomposition with height order, whose components are moving away each other in the order $t^{1/3}$ as time $t$ goes to the infinity.