Abstract:
Abstract polytopes are combinatorial structures obeying certain axioms that generalise both classical convex geometric polytopes (like the Platonic solids) and maps on surfaces (such as Klein’s quartic). Of particular interest are the polytopes with maximal possible symmetry subject to certain natural constraints. Symmetry of a polytope can be measured by its automorphisms. Every automorphism of an abstract polytope P is uniquely determined by its effect on any ‘flag’, which is a maximal chain of elements of increasing rank in the corresponding poset. The most symmetric polytopes are regular, with all flags lying in a single orbit, but an equally interesting class of examples which are not quite regular are the chiral polytopes, for which the automorphism group has two orbits on flags, with any two flags that differ in just one element lying in different orbits. Intuitively, a polytope is chiral if it has maximal “rotational symmetry” but lacks symmetry by “reflection”. Chirality is a fascinating concept that does not arise in the classical theory of convex polytopes. Examples of chiral polytopes have been difficult to find and construct. In 2005, Conder, Hubard and Pisanski identified the smallest examples of chiral polytopes of ranks (dimensions) 3 and 4, and found the first known examples of finite chiral polytopes of rank 5 (which are now known to be the smallest of that rank). The existence of chiral polytopes of each rank greater than 2 was proved by Daniel Pellicer in 2010, but his examples were extremely large and somewhat difficult to realise. Until now, no “nice” family of chiral polytopes has been found for arbitrary ranks. It is still an open problem to find alternative constructions for families of chiral polytopes of relatively small order, or with easily described automorphism groups. Chiral polytopes continue to be surprisingly rare in comparison with regular polytopes, even though the latter possess a higher degree of symmetry. This thesis makes a contribution to the problem by giving some new methods of constructing chiral polytopes. One of the things we achieve in this thesis is to identify the smallest chiral polytopes of rank 6. We find that up to duality and reflection, there is a unique smallest chiral 6-polytope, and it has type {3,3,4,6,3} with 18432 flags, which is much bigger than expected. (The smallest chiral polytopes of ranks 3, 4 and 5 have only 40, 240 and 1440 flags, respectively.) This is proved by using known small regular and chiral polytopes of rank up to 4. Also we describe a small self-dual chiral 6-polytope of type {3,3,8,3,3} with 589824 flags. This is currently the smallest known self-dual chiral polytope of rank 6. Next, we introduce a new covering method that allows the construction of some infinite families of chiral polytopes. These families have the property that each member of a family has the same rank as the smallest member, but the size of the polytopes in the family grows linearly with one (or more) of the parameters making up its ‘type’ (or Schl¨afli symbol). Also the automorphism group of each polytope in this family is an extension of an abelian normal subgroup by the automorphism group of the smallest one. In particular, we announce the existence of some new infinite families of chiral polytopes of ranks 3, 4, 5 and 6, constructed using this method. We remark that with Jicheng Ma (a former Auckland PhD student from China), we have developed a similar method to construct infinite families of regular polytopes. This work was also done during this PhD project, but we do not include it into this thesis. Thirdly, we describe a new construction for chiral polytopes as abelian covers of regular polytopes. Two different methods are introduced. One is general, and suitable for constructing chiral polytopes with large covering group. In particular, we use this method to construct many chiral polytopes of type {4,4, . . . ,4} with rank up to 6. The other one is somewhat technical, and gives chiral covers of some polytopes of other types with smaller covering group and smaller rotation group.