Connectivity gaps among matroids with the same enumerative invariants

Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $G$-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer n, there are pairs of matroids that have the same configuration (and so the same $G$-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds n, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.