A two-dimensional topological representation theorem for matroid polytopes of rank 4

The Folkman–Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank r can be represented as a pseudosphere arrangement on the (r−1)-dimensional sphere Sr−1, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank 4. We introduce 2-weak configurations of points and pseudocircles (2-weak PPC configurations) on S2 and prove that every uniform matroid polytope of rank 4 can be represented by a 2-weak PPC configuration. As an application, we provide a proof of Las Vergnas conjecture on simplicial topes for the case of uniform matroid polytopes of rank 4.