Refining a tree-decomposition which distinguishes tangles

Roberston and Seymour introduced tangles of order κ as objects representing highly connected parts of a graph and showed that every graph admits a tree-decomposition of adhesion κ in which each tangle of order κ is contained in a dierent part. Recently, Carmesin, Diestel, Hamann, and Hundertmark showed that such a tree-decomposition can be constructed in a canonical way, which makes it invariant under automorphisms of the graph. These canonical tree-decompositions necessarily have parts which contain no tangle of order κ, which we call inessential. Diestel asked what could be said about the structure of the inessential parts. In this paper we show that the torsos of the inessential parts in these tree-decompositions have branch-width κ, allowing us to further refine the canonical tree-decompositions and also show that a similar result holds for κ-blocks. We also use our methods to further rene the essential parts in such a tree-decomposition in a similar fashion.