While the structural characteristics of a network are uniquely determined by its adjacency matrix^1,2,3, 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.

网络的结构特征由其邻接矩阵^1,2,3唯一确定在诸如大脑或血管系统等物理网络中，网络的三维布局也会影响系统的结构和功能。但是，我们缺少区分布线相同但几何布局不同的物理网络的工具。为了满足这一需求，在这里我们介绍了网络同位素的概念，它表示可以在没有链接交叉的情况下彼此转换的不同网络布局，并显示出一个图形链接数即一个数量，它捕获了布局的纠缠，定义了不同的同位素类。我们发现网络的弹性能量线性依赖于图的链接数，表明每个局部缠结对总能量都有独立的贡献。这一发现使我们能够为物理网络中缠结的形成建立统计模型。我们将开发的框架应用于各种实际物理网络，发现基于最佳布线的鼠标连接组比预期的更纠结。