We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.

中文翻译：

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缠结树二元性：在图形、拟阵及其他领域

我们将最近的对偶定理应用于抽象分离系统中的缠结，以推导出图和拟阵中宽度参数的缠结型对偶定理。我们进一步推导出大数据集中存在簇的对偶定理。我们对图的应用包括新的缠结型对偶定理，用于树宽、路径宽度和小附着力的树分解。相反，我们表明雕刻宽度对边缘缠结是双重的。对于拟阵，我们获得了树宽的对偶定理。我们的结果可用于推导出图次要理论中宽度参数的所有经典对偶定理的简短证明，例如路径宽度、树宽度、分支宽度和秩宽度。