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

We introduce modular (integral) complementary polynomial κκ (κZκZ) of two variables on a graph GG 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 ΔctfΔctf, 0–1 polytope Δctf+, and lattice polytopes Δctfρ with respect to orientations ρρ. The polynomial κκ (κZκZ) is a common generalization of the modular (integral) tension polynomial ττ (τZτZ) and the modular (integral) flow polynomial φφ (φZφZ) 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 κ̄ and κ̄Z, dual to κκ and κZκZ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes Δ̄ctf+. It turns out remarkably that κ̄ is Whitney’s rank generating polynomial RGRG, which gives rise to a nontrivial combinatorial–geometric interpretation on the values of the Tutte polynomial TGTG at all positive integers. In particular, some special values of κZκZ and κ̄Z (κκ and κ̄) count the number of certain special kinds (of equivalence classes) of orientations, including the recovery of a few well-known values of TGTG.