Chiral extensions of chiral polytopes

Abstract

Given a chiral d-polytope K with regular facets, we describe a construction for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is finite whenever K is finite. We provide explicit examples of chiral 4-polytopes constructed in this way from chiral toroidal maps.

Regular Incidence Complexes, Polytopes, and C-Groups

“…For these results, both K and L are regular polytopes. There are also good extension results for chiral polytopes (see [11,40,41]) and for hypertopes (see [15]).…”

Proper locally spherical hypertopes of hyperbolic type

“…GPR-graphs were also used in [29] to show the existence of chiral polytopes of arbitrary rank. In [8] Cunningham and Pellicer used GPR-graphs to construct chiral polytopes with prescribed chiral facets. Recently Pellicer, Toledo and Potočnik used GPR-graphs to build 2-orbit maniplexes for every rank and every symmetry type.…”

Open problems on k-orbit polytopes

“…Let us return now to the problem of extending a polytope P without prescribing the vertex-figures. Some work has been done on extensions of regular and chiral polytopes, including regular extensions of regular polytopes (see [63]), chiral extensions of regular polytopes (see [64]), and chiral extensions of chiral polytopes (see [17,73]). When P is regular, then there is a universal extension of P that covers every other regular extension of P [54, Thm.…”

“…Regular polytopes have been extensively studied; see [54] for the standard reference, and see [8,10,52,60] for a broad cross-section of current advances. Two-orbit polytopes (including chiral polytopes) have also received a lot of attention; see [36,72] for the basic notions and [5,11,17,34,43,65] for recent work. Very little is yet known about k-orbit polytopes for k ≥ 3.…”