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.Comment: 37 pages, 4 figure