Tangle structure trees

TL;DR

This paper introduces tangle structure trees, a versatile data structure that visualizes all tangles and certificates in an abstract separation system, extending classical theorems and providing a constructive, software-implementable approach.

Contribution

It presents a new, comprehensive data structure for tangles that generalizes existing theorems and is applicable to broader obstruction sets, with a constructive proof and software implementation.

Findings

Unified structure for all tangles and certificates

Extends classical tangle-tree duality theorems

Constructive proof with open-source software implementation

Abstract

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.