A direct and elementary proof of the well-definedness of the interior and exterior polynomials of hypergraphs

T. K'{a}lm'{a}n (A version of Tutte’s polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1\mathrm{/}x,1)$ and $(1,1\mathrm{/}y)$ to hypergraphs. The two polynomials are defined under a fixed ordering of hyperedges, and are proved to be independent of the ordering using techniques of polytopes. In this paper, similar to the Tutte’s original proof we provide a direct and elementary proof for the well-definedness of the interior and exterior polynomials of hypergraphs.