There have been recent advances in the analysis and visualization of 3D symmetric tensor fields, with a focus on the robust extraction of tensor field topology. However, topological features such as degenerate curves and neutral surfaces do not live in isolation. Instead, they intriguingly interact with each other. In this paper, we introduce the notion of {\em topological graph} for 3D symmetric tensor fields to facilitate global topological analysis of such fields. The nodes of the graph include degenerate curves and regions bounded by neutral surfaces in the domain. The edges in the graph denote the adjacency information between the regions and degenerate curves. In addition, we observe that a degenerate curve can be a loop and even a knot and that two degenerate curves (whether in the same region or not) can form a link. We provide a definition and theoretical analysis of individual degenerate curves in order to help understand why knots and links may occur. Moreover, we differentiate between wedges and trisectors, thus making the analysis more detailed about degenerate curves. We incorporate this information into the topological graph. Such a graph can not only reveal the global structure in a 3D symmetric tensor field but also allow two symmetric tensor fields to be compared. We demonstrate our approach by applying it to solid mechanics and material science data sets.

翻译：近年来，三维对称张量场的分析与可视化取得进展，重点在于稳健提取张量场拓扑。然而，退化曲线和中性曲面等拓扑特征并非孤立存在，而是相互之间存在着有趣的交互作用。本文引入三维对称张量场的**拓扑图**概念，以促进此类场的全局拓扑分析。该图的节点包括退化曲线以及由域内中性曲面界定的区域，图中的边表示区域与退化曲线之间的邻接信息。此外，我们观察到退化曲线可形成环状甚至纽结，且两条退化曲线（无论是否处于同一区域）可构成链环。我们针对单个退化曲线给出定义与理论分析，以帮助理解纽结与链环产生的原因。进一步，我们区分了楔形区和三分区，从而对退化曲线进行更细致的分析，并将这一信息纳入拓扑图。此类图不仅能揭示三维对称张量场中的全局结构，还可实现两个对称张量场的比较。我们通过将其应用于固体力学与材料科学数据集来展示该方法。