Faithfulness in maniplexes

Abstract Maniplexes generalize the flag graphs of maps and polytopes, and they have enabled the use of new techniques for constructing and analyzing polytopes. Each maniplex gives rise to a natural ranked poset of the faces and their incidence, but it is not always possible to recreate the maniplex from the poset. The first possible problem is that the mapping from flags of the maniplex to maximal chains of the poset may not be injective; in this case we say that the maniplex is unfaithful . The second is that the resulting poset may not be thin (meaning that some sections of rank 1 may have more than two proper faces), in which case the flag graph of the poset is not a maniplex. In this paper we describe a theory of faithfulness in maniplexes, laying the groundwork for future study in this area. We also classify the unfaithful maps (3-maniplexes) with at most two flag orbits, and we prove that the mix of two polytopes is a faithful maniplex that is thin if the polytopes are orientable.