Tangle structure trees

We introduce a comprehensive data structure, tangle structure trees, which simultaneously displays all the $\mathcal{F}$-tangles of an abstract separation system for very general obstruction sets $\mathcal{F}$. It simultaneously also displays certificates $\sigma\in\mathcal{F}$ for any non-existence of such tangles, or for the non-extendability of low-order tangles to higher-order ones. Our theorem can be applied to produce the structures of the classical tree-of-tangles and tangle-tree duality theorems, both for graph tangles and for their known generalizations to more general separation systems. It extends those theorems to obstruction sets $\mathcal{F}$ that need not define profiles (as they must in all known trees of tangles) or consist of stars of separations (as they must in traditional tangle-tree duality). Our existence proof for these structure trees is constructive. The construction has been implemented in open-source software available for tangle detection and further analysis.