A two-dimensional topological representation theorem for matroid polytopes of rank 4. (May 2020)

• Record Type: Journal Article
• Authors: Miyata, Hiroyuki
• Is Part Of: European journal of combinatorics. Volume 86(2020)
• Publication Date: 2020-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2019.103065
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: 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.