The two-variable hypergraph Tutte polynomial via embedding activities

We prove that the two-variable Tutte polynomial of hypergraphs can be defined via embedding activities. We also prove that embedding activities of hypergraphs yield a Crapo-style decomposition of $\mathbb{Z}^E$, thus generalizing Bernardi's results from graphs to hypergraphs. We also show that hypergraph embedding activities do not fit into the $\Delta$-activity framework of Courtiel. Based on this observation, we construct a graph with an activity notion that yields a Crapo decomposition, but cannot be obtained as a $\Delta$-activity.