Minimal covers of the prisms and antiprisms

Hartley, Michael I.; Pellicer, Daniel; Williams, Gordon

doi:10.1016/j.disc.2012.06.025

Cited by 11 publications

(15 citation statements)

References 12 publications

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“…This is denoted by N M, and we say that N is a cover of M. We can use the notion of covering to create a partial order ≤ on a set S of toroidal maps, where M ≤ N if and only if N is a cover of M. A minimal cover of a map M is minimal with respect to this partial order in a set of all maps that cover M. We note here that this notion of covering can also be generalized to maps on different surfaces, and to abstract polytopes of higher ranks. If S is the set of all regular maps that cover a given map M, then the minimal elements of the partial order are the minimal regular covers of M, as studied in [9,17] for example.…”

Section: Preliminaresmentioning

confidence: 99%

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Archimedean toroidal maps and their minimal almost regular covers

et al. 2019

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The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type (4.8 2 ) is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equivelar maps on the torus, we construct these minimal almost regular covers explicitly. Case 1, and is thus omitted; this completes the proof of Theorem 4.4.

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Section: Preliminaresmentioning

confidence: 99%

“…There is also much interest in finding minimal regular covers of different families of maps and polytopes (see for example [9,16,19]). In a previous paper [7], two of the authors constructed the minimal rotary cover of any equivelar toroidal map.…”

Section: Introductionmentioning

confidence: 99%

Archimedean toroidal maps and their minimal almost regular covers

et al. 2019

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“…4.12]. Describing the monodromy group of several families of polytopes is an active area of research; see [2,3,33,56,68].…”

Section: The Flag Actionmentioning

confidence: 99%

Open problems on k-orbit polytopes

Cunningham

Pellicer

2018

Discrete Mathematics

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“…However, little is known about monodromy groups of non-regular polytopes. In the last decade, there has been an effort to understand these groups (see for example [2], [6], [12]). In particular in [6] Hartely et al study the monodromy group of the prism oven an n-gon and compute it, in terms of generators and relations.…”

Section: B Bmentioning

confidence: 99%

Products of abstract polytopes

Gleason

Hubard

2018

Journal of Combinatorial Theory, Series A

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Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group of a product, show that such a group is always an extension of a symmetric group, and give some examples in which this extension splits. arXiv:1603.03585v1 [math.CO]• If P is the prism (or the bipyramid) over a 3-polytope having the property that all its vertex figures (faces) are isomorphic to an n-gon, with n not congruent to 0 modulo 9;• If = and each Q i has rank 2.By [2], we already know that the monodromy group of a pyramid over an n-gon is an extension of S 4 by (C m ) 4 , where m = p gcd (3,p) , and that such extension splits whenever n is not congruent to 0 modulo 9. Our techniques to show Theorem B can also be used to show this result. Abstract polytopesAbstract polytopes generalise the (face-lattice) of classical polytopes as combinatorial structures. In this chapter we review the basic theory of abstract polytopes and refer the reader to [11] for a detail exposition of the subject.An (abstract) n-polytope (or (abstract) polytope of rank n) P is a partially ordered set whose elements are called faces that satisfies the following properties. It contains a minimum face F −1 and a maximum

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Minimal covers of the prisms and antiprisms

Cited by 11 publications

References 12 publications

Archimedean toroidal maps and their minimal almost regular covers

Archimedean toroidal maps and their minimal almost regular covers

Open problems on k-orbit polytopes

Products of abstract polytopes

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