Obtaining trees of tangles from tangle-tree duality

TL;DR

This paper leverages the tangle-tree duality theorem to enhance tree-of-tangles theorems, providing bounds on node degrees and extending applicability to tangles of varying orders with simplified proofs.

Contribution

It introduces a strengthened and simplified duality theorem and applies it to improve tree-of-tangles theorems, including bounds on node degrees and handling different tangle orders.

Findings

Strengthened the duality theorem with a simplified proof.

Extended tree-of-tangles theorems to tangles of different orders.

Bounded node degrees in the improved theorems.

Abstract

We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.