In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
"A generalization of the Tutte polynomials." Proc. Japan Acad. Ser. A Math. Sci. 95 (10) 111 - 113, December 2019. https://doi.org/10.3792/pjaa.95.111