Stable configurations of entangled systems with repulsive interactions

Entangled systems are prevalent in both biological and synthetic materials. This study examines the stable configurations of weaves consisting of two families of intertwined threads, such as warp and weft threads. By analyzing the steepest descent flow of an energy functional featuring repulsive interactions, we develop a framework for identifying stable states in ${\mathbb R}^3$. Although a weave consists of one-dimensional threads that do not intersect each other, it behaves collectively like a two-dimensional object. To describe this phenomenon, we define a non-separable component of a weave as a ``layer'' and establish the existence and uniqueness of its stable configuration. Furthermore, we show that two distinct layers drift apart with an asymptotic growth rate of $t^{1/3}$ as $t \to \infty$.