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

We introduce a modular (integral) complementary polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $ε$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $Δ_\textsc{ctf}(G,ε)$, the complementary open 0-1 polytope $Δ^+_\textsc{ctf}(G,ε)$, and the complementary open lattice polytopes $Δ^ρ_\textsc{ctf}(G,ε)$ with respect to orientations $ρ$. The polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) is a common generalization of the modular (integral) tension polynomial $τ(G,x)$ ($τ_\mathbbm{z}(G,x)$) and the modular (integral) flow polynomial $ϕ(G,y)$ ($ϕ_\mathbbm{z}(G,y)$), and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0-1 polytopes $Δ^+_\textsc{ctf}(G,ρ)$. There are dual complementary polynomials $\barκ(G;x,y)$ and $\barκ_{\mathbbm z}(G;x,y)$, dual to $κ$ and $κ_{\mathbbm z}$ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes $\barΔ^+_\textsc{ctf}(G,ρ)$. It turns out that the polynomial $\barκ(G;x,y)$ is Whitney's rank generating polynomial $R_G(x,y)$, which gives rise to a combinatorial interpretation on the values of the Tutte polynomial $T_G(x,y)$ at positive integers. In particular, some special values of $κ_\mathbbm{z}$ and $\barκ_\mathbbm{z}$ ($κ$ and $\barκ$) count the number of certain special kinds (of equivalence classes) of orientations.

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