The smallest chiral 6-polytopes

作者：Conder, Marston D. E.^*; Zhang, Wei-Juan

来源：Bulletin of the London Mathematical Society, 2017, 49(4): 549-560.

DOI：10.1112/blms.12046

• POLYTOPES
• RANK

The 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). Every automorphism (symmetry) 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. In contrast to the situation for regular polytopes, relatively little is known about chiral polytopes indeed finite examples of rank 5 were constructed only about 10 years ago. Since then, many examples and families have been constructed, but it is a major open problem to find the smallest examples of each rank. In this paper, we describe the construction of two examples of small' chiral 6-polytopes, one self-dual of type {3,3,8,3,3} with 589824 flags and the other non-self-dual of type {3,3,4,6,3} with 18432 flags, and then use knowledge about all small regular and chiral polytopes of rank up to 4 in order to show that the non-self-dual example and its mirror image and their duals are the smallest chiral polytopes of rank 6.

• 出版日期2017-8