A Note on Generic Tangle Algorithms
Christian Elbracht, Jay Lilian Kneip, Maximilian Teegen
TL;DR
This note formalizes three base algorithms for tangles within abstract separation systems: a Naïve tangle search that enumerates all $\mathcal{F}$-tangles; a Tangle-tree duality procedure that either certifies nonexistence via an $S$-tree or yields a partial orientation shared by all tangles; and a Tree-of-Tangles method that builds a nested set of separations in a larger universe to distinguish real maximal tangles through local improvements. The approaches are framed in a general ASS context with oracle access to $\mathcal{F}$, and include termination and runtime analyses under standard duality assumptions. Collectively, they provide model algorithms for exploring tangle structures, offering templates for implementations and insights into complexity, with practical performance improved by data-structuring and leveraging partial information. The work emphasizes the balance between full tangles and extendable tangles, enabling explorations in tangle theory without requiring exhaustive enumeration in all cases, and supplies foundational tools for constructing nested distinguishing sets and certificates in abstract separation frameworks.
Abstract
In this note we gather the theoretical outlines of three basic algorithms for tangles in abstract separation systems: a naive tree search for finding tangles; an algorithm which outputs a certificate for the non-existence of tangles if possible, and otherwise a way to jump-start the naive tree search; and a way to obtain a tree-of-tangles.