Counterexamples to Two Conjectures on Tangles in Matroid Minors

We give counterexamples to two conjectures of Brettell, Jowett, Oxley, Semple, and Whittle concerning tangles in matroid minors. We construct a matroid $P_0$ with a tangle $\mathcal{T}$ of order seven. For each $i\in \{4,5,6,7\}$, let ${\mathcal{T}}_i$ denote its order-$i$ truncation. No proper minor $N$ of $P_0$ admits, for every such $i$, a tangle of order $i$ whose breadth equals that of ${\mathcal{T}}_i$, and $P_0$ is not weakly $4$-connected. We also construct a matroid $P_1$ with a tangle of order four and breadth six, but no internally $4$-connected minor of $P_1$ admits a tangle of order four.