Journal of Combinatorics

Volume 13 (2022)

Number 2

Obtaining trees of tangles from tangle-tree duality

Pages: 251 – 287



Christian Elbracht (Fachbereich Mathematik, Universität Hamburg, Germany)

Jakob Kneip (Fachbereich Mathematik, Universität Hamburg, Germany)

Maximilian Teegen (Fachbereich Mathematik, Universität Hamburg, Germany)


We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems [7] 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.


tangles, tree-decompositions, abstract separation systems, tangle-tree duality, tree of tangles

2010 Mathematics Subject Classification

05C05, 05C40, 05C83

Received 23 February 2021

Accepted 27 February 2021

Published 30 March 2022