Tangle structure trees
Hanno von Bergen, Reinhard Diestel

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 -tangles of an abstract separation system for very general obstruction sets . It simultaneously also displays certificates 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 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…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Digital Image Processing Techniques · Advanced Combinatorial Mathematics
