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 $S^{r-1}$, 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 $S^2$ 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 $4$.