  We give an analogue of the Tutte polynomial for hypermaps. This polynomial
can be defined as either a sum over subhypermaps, or recursively through
deletion-contraction reductions where the terminal forms consist of isolated
vertices. Our Tutte polynomial extends the classical Tutte polynomial of a
graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon
graph polynomial), and it is a specialization of the transition polynomial via
a medial map transformation. We give hypermap duality and partial duality
identities for our polynomial, as well as some evaluations, and examine
relations between our polynomial and other hypermap polynomials.
