  An Ingletonian polymatroid satisfies, in addition to the polymatroid axioms,
the inequalities of Ingleton (Combin. Math. Appln., 1971). These inequalities
are required for a polymatroid to be representable. It is has been an open
question as to whether these inequalities are also sufficient. Representable
polymatroids are of interest in their own right. They also have a strong
connection to network coding. In particular, the problem of finding the linear
network coding capacity region is equivalent to the characterization of all
representable, entropic polymatroids. In this paper, we describe a new approach
to adhere two polymatroids together to produce a new polymatroid. Using this
approach, we can construct a polymatroid that is not inside the minimal closed
and convex cone containing all representable polymatroids. This polymatroid is
proved to satisfy not only the Ingleton inequalities, but also the recently
reported inequalities of Dougherty, Freiling and Zeger. A direct consequence is
that these inequalities are not sufficient to characterize representable
polymatroids.
