Isotopy and energy of physical networks

While the structural characteristics of a network are uniquely determined by its adjacency matrix in physical networks, such as the brain or the vascular system, the network's three-dimensional layout also affects the system's structure and function. We lack, however, the tools to distinguish physical networks with identical wiring but different geometrical layouts. To address this need, here we introduce the concept of network isotopy, representing different network layouts that can be transformed into one another without link crossings, and show that a single quantity, the graph linking number, captures the entangledness of a layout, defining distinct isotopy classes. We find that a network's elastic energy depends linearly on the graph linking number, indicating that each local tangle offers an independent contribution to the total energy. This finding allows us to formulate a statistical model for the formation of tangles in physical networks. We apply the developed framework to a diverse set of real physical networks, finding that the mouse connectome is more entangled than expected based on optimal wiring.