A Hypergraph Tutte Polynomial

We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion--contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.