Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connectedmatroids that are representable over a fixed finite field and that have the same Tuttepolynomial abound. In particular, for most prime powers q, we construct infinite familiesof sets of 3-connected matroids for which the matroids in a given set are non-isomorphic,are representable over GF(q), and have the same Tutte polynomial. Furthermore, thecardinalities of the sets of matroids in a given family grow exponentially as a function ofrank, and there are many such families.In Memory of Gian-Carlo Rota