The two-variable hypergraph Tutte polynomial is the polymatroid Tutte polynomial of the integer base vectors determined by a connected hypergraph.
For 
, form its incidence bipartite
graph, with one graph vertex for each element
of 
and each hyperedge in 
, and a graph edge joining
to 
when 
. The bases are the nonnegative integer vectors 
for which some spanning tree 
of that bipartite graph
satisfies 
at every hyperedge graph
vertex. These vectors, called hypertrees in this setting, are counted once even
if several spanning trees realize the same vector
(Tóthmérész 2026).
Tóthmérész (2026) proves that internal and external embedding activities give this two-variable polynomial, extending the embedding-activity construction of the ordinary Tutte polynomial. The embedding data are cyclic orders at the vertices of the incidence bipartite graph and a starting node-edge pair. The resulting polynomial is independent of these choices. The paper also extends computation-tree activities to polymatroids, with both minimum and maximum variants. The conjecture about a different, alternative definition of embedding activity remains open (Tóthmérész 2026, §5).
