Polytopality of Maniplexes

Abstract

Given an abstract polytope $\mathcal{P}$, its flag graph is the edge-coloured graph whose vertices are the flags of $\mathcal{P}$ and the $i$-edges correspond to $i$-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex $\mathcal{M}$, on can define a poset ${\mathcal{P}}_{\mathcal{M}}$ 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 $\mathcal{M}$ in order for ${\mathcal{P}}_{\mathcal{M}}$ to be an abstract polytope. Moreover, in such case, we show that $\mathcal{M}$ is isomorphic to the flag graph of ${\mathcal{P}}_{\mathcal{M}}$. This in turn gives necessary and sufficient conditions for a maniplex to be (isomorphic to) the flag graph of a polytope.