Counterexamples to Two Conjectures on Tangles in Matroid Minors

Source abstract

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.