  In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences,
95(10) 111-113], the authors introduce the concept of the Tutte polynomials of
genus $g$ and announce that each matroid $M$ can be reconstructed from its
Tutte polynomial of genus $|\mathcal{B}(M)|$, where $\mathcal{B}(M)$ denotes
the family of bases of $M$. In that paper, we also announced that, for all $g$,
there exist inequivalent matroids that have the same Tutte polynomial of genus
$g$. In this paper, we prove these theorems.
