Polytopality of Maniplexes

TL;DR

This paper characterizes when a maniplex's associated poset is an abstract polytope and establishes conditions under which the maniplex is isomorphic to a polytope's flag graph.

Contribution

It provides necessary and sufficient graph-theoretic conditions for a maniplex to correspond to an abstract polytope and shows the isomorphism between such maniplexes and polytope flag graphs.

Findings

Characterization of when a maniplex's associated poset is an abstract polytope.

Conditions under which a maniplex is isomorphic to a polytope's flag graph.

Establishment of a correspondence between maniplexes and abstract polytopes.

Abstract

Given an abstract polytope $\cal P$, its flag graph is the edge-coloured graph whose vertices are the flags of $\cal P$ and the $i$-edges correspond to $i$-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex $\cal M$, on can define a poset $\cal P_M$ by means of the non empty intersection of its faces. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex $\cal M$ in order for $\cal P_M$ to be an abstract polytope. Moreover, in such case, we show that $\cal M$ is isomorphic to the flag graph of $\cal P_M$. This in turn gives necessary and sufficient conditions for a maniplex to be (isomorphic to) the flag graph of a polytope.