Developments and open problems on chiral polytopes

Pellicer, Daniel

doi:10.26493/1855-3974.183.8a2

Cited by 22 publications

(14 citation statements)

References 54 publications

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“…A natural question was posed by the first author in , and also by Daniel Pellicer in [, Question 35]: What are the smallest chiral polytopes of rank

d

, for each

d ⩾ 3 ?

…”

Section: Introductionmentioning

confidence: 99%

The smallest chiral 6-polytopes

Conder

Zhang

2017

Bull. London Math. Soc.

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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 scriptP 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.

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“…A natural question was posed by the first author in , and also by Daniel Pellicer in [, Question 35]: What are the smallest chiral polytopes of rank

d

, for each

d ⩾ 3 ?

…”

Section: Introductionmentioning

confidence: 99%

The smallest chiral 6-polytopes

Conder

Zhang

2017

Bull. London Math. Soc.

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“…Many of the important unsolved problems of chiral polytopes are summarized in [13]. Problems 24-30 all concern the extension problem for chiral polytopes, signifying both the importance of that general problem and how little is known.…”

Section: Introductionmentioning

confidence: 99%

“…Problems 24-30 all concern the extension problem for chiral polytopes, signifying both the importance of that general problem and how little is known. An important partial result was given in [12], where it is shown how to build a finite chiral polytope of rank d + 1 with facets isomorphic to a finite regular polytope K of rank d. There are very restrictive conditions on the polytope K, however, so more work remains to be done even on this piece of the extension problem (Problem 27 of [13]). …”

Section: Introductionmentioning

confidence: 99%

“…In this paper we use GPR graphs (as defined in [14]) to build chiral polytopes of rank d + 1 with facets isomorphic to a given chiral polytope of rank d. In particular, Theorem 10 implies the following: This gives a partial answer to Problem 26 in [13]. We note that the assumption that the chiral d-polytope has regular facets is necessary (see [16,Proposition 9]).…”

Section: Introductionmentioning

confidence: 99%

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Chiral extensions of chiral polytopes

Cunningham¹,

Pellicer²

2014

Discrete Mathematics

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“…These are partially ordered sets which inherit their structure from derived constraints on convex polytopes. Of particular interest are the regular abstract polytopes -those in which any two maximal chains may be mapped to each other by an automorphism of the poset -whose structure is completely determined by their automorphism groups [MS02], and their close cousins the chiral polytopes [Pe12,SW91]. Both of these classes of abstract polytope have very restricted classes of groups from which their automorphism groups may be selected.…”

Section: Introductionmentioning

confidence: 99%