Answers to questions about medial layer graphs of self-dual regular and chiral polytopes

An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which captures important properties of a geometric polytope, for any dimension $n$. For even $n\ge 2$, the incidences between elements in the middle two layers of the Hasse diagram of $\mathcal{P}$ give rise to the medial layer graph of $\mathcal{P}$, denoted by $\mathcal{G}=\mathcal{G}(\mathcal{P})$. If $n=4$, and $\mathcal{P}$ is both highly symmetric and self-dual of type $\{p,q,p\}$, then a Cayley graph $\mathcal{C}$ covering $\mathcal{G}$ can be constructed on a group of polarities of $\mathcal{P}$. In this paper we address some open questions about the relationship between $\mathcal{G}$ and $\mathcal{C}$ that were raised in a 2008 paper by Monson and Weiss, and describe some interesting examples of these graphs. In particular, we give the first known examples of improperly self-dual chiral polytopes of type $\{3,q,3\}$, which are also among the very few known examples of highly symmetric self-dual finite polytopes that do not admit a polarity. Also we show that if $p=3$ then $\mathcal{C}$ cannot have a higher degree of $s$-arc-transitivity than $\mathcal{G}$, and we present a family of regular $4$-polytopes of type $\{6,q,6\}$ for which the vertex-stabilisers in the automorphism group of $\mathcal{C}$ are larger than those for $\mathcal{G}$.