Tight chiral polytopes

Abstract

A chiral polytope with Schlafli symbol $$\{p_1, \ldots , p_{n-1}\}$$ has at least $$2p_1 \cdots p_{n-1}$$ flags, and it is called tight if the number of flags meets this lower bound. The Schlafli symbols of tight chiral polyhedra were classified in an earlier paper, and another paper proved that there are no tight chiral n-polytopes with $$n \ge 6$$ . Here we prove that there are no tight chiral 5-polytopes, describe 11 families of tight chiral 4-polytopes, and show that every tight chiral 4-polytope covers a polytope from one of those families.