Dual complementary polynomials of graphs and combinatorial–geometric interpretation on the values of Tutte polynomial at positive integers

We introduce modular (integral) complementary polynomial κ ( κ Z ) of two variables on a graph G by counting the number of modular (integral) complementary tension–flows. We further introduce cut-Eulerian equivalence relation on orientations and geometric structures: complementary open lattice polyhedron Δ ctf , 0–1 polytope Δ ctf + , and lattice polytopes Δ ctf ρ with respect to orientations ρ . The polynomial κ ( κ Z ) is a common generalization of the modular (integral) tension polynomial τ ( τ Z ) and the modular (integral) flow polynomial φ ( φ Z ) of one variable, and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0–1 polytopes. There are dual complementary polynomials κ ̄ and κ ̄ Z , dual to κ and κ Z respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes Δ ̄ ctf + . It turns out remarkably that κ ̄ is Whitney’s rank generating polynomial R G , which gives rise to a nontrivial combinatorial–geometric interpretation on the values of the Tutte polynomial T G at all positive integers. In particular, some special values of κ Z and κ ̄ Z ( κ and κ ̄ ) count the number of certain special kinds (of equivalence classes) of orientations, including the recovery of a few well-known values of T G .