Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

Wilson, Steve

doi:10.3390/sym4020265

Cited by 22 publications

(23 citation statements)

References 17 publications

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“…To generalize the results of this paper to higher-dimensional objects, we use the idea of a maniplex, first introduced in [13]. We summarize the definitions here: an nmaniplex is a pair (Ω, [r 0 , r 1 , r 2 , .…”

Section: Maniplexes and Higher Dimensionsmentioning

confidence: 99%

Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Koike

Pellicer

Raggi

et al. 2017

Electron. J. Combin.

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Section: Maniplexes and Higher Dimensionsmentioning

confidence: 99%

Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Koike

Pellicer

Raggi

et al. 2017

Electron. J. Combin.

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“…Maniplexes were first introduced by Wilson in [21], aiming to unify the notion of maps and polytopes. In this subsection we review their basic theory.…”

Section: Maniplexesmentioning

confidence: 99%

“…Any (n + 1)-polytope can be thought as an n-maniplex. However, some maniplexes do not induce polytopes (see [21]). …”

Section: Maniplexesmentioning

confidence: 99%

“…The combinatorial structure of (n − 1)-maniplexes and n-polytopes is completely determined by an edge-coloured n-valent graph with chromatic index n, often called the flag graph. In particular, maps correspond to 2-maniplexes [21]. The symmetry type graph of a map is the quotient of its flag graph under the action of the automorphism group.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry Type Graphs of Polytopes and Maniplexes

et al. 2015

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We extend the notion of symmetry type graphs of maps to include maniplexes and (abstract) polytopes, using them to study k-orbit maniplexes (where the automorphism group has k orbits on flags). In particular, we show that there are no fully-transitive k-orbit 3-maniplexes with k > 1 an odd number. We classify 3-orbit maniplexes and determine all face transitivities for 3-and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a maniplex, given its symmetry type graph. Finally, we extend these notions to oriented maniplexes, and we provide a classification of oriented 2-orbit maniplexes and a generating set for their orientation-preserving automorphism group.

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“…Abstract polytopes generalize convex polytopes (or more precisely, their face lattice) to more general combinatorial objects and are defined as posets with certain conditions. Maniplexes were first introduced by Steve Wilson in [12] to somehow unify the study of maps and abstract polytopes. They generalize maps on surfaces to higher dimensions and, at the same time, they generalize (the flag graphs of) abstract polytopes.…”

Section: Introductionmentioning

confidence: 99%

Polytopality of maniplexes

Garza-Vargas

Hubard

2018

Discrete Mathematics

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Given an abstract polytope P, its flag graph is the edge-coloured graph whose vertices are the flags of P and the i-edges correspond to i-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex M, on can define a poset P M by means of the non empty intersection of its faces. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex M in order for P M to be an abstract polytope. Moreover, in such case, we show that M is isomorphic to the flag graph of P M . This in turn gives necessary and sufficient conditions for a maniplex to be (isomorphic to) the flag graph of a polytope.In this section we introduce the reader to the basic notions of abstract polytopes, maps, edge-coloured graphs and maniplexes. Abstract polytopesConvex polytopes generalize the notion of polyhedra for higher dimensions. Abstract polytopes are combinatorial objects whose incidence structure resemble the incidence structure of convex polytopes. In fact, each convex polytope can be regarded as an abstract polytope. We give the basics about abstract polytopes and refer the reader to [6] for more details of their study.An (abstract) n-polytope (or a polytope of rank n) is a ranked partially ordered set,

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Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

Abstract: This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group.

Cited by 22 publications

References 17 publications

Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Symmetry Type Graphs of Polytopes and Maniplexes

Polytopality of maniplexes

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