Tangles, Trees and Flowers

A tangle of order k in a connectivity function λ may be thought of as a "k-connected component" of λ. For a connectivity function λ and a tangle in λ of order k that satisfies a certain robustness condition, we describe a tree decomposition of λ that displays, up to a certain natural equivalence, all of the k-separations of λ that are non-trivial with respect to the tangle. In particular, for a tangle in a matroid or graph of order k that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or graph that displays, up to a certain natural equivalence, all of the k- separations of the matroid or graph that are non-trivial with respect to the tangle.