The Twist Operator on Maniplexes

Douglas, Ian; Hubard, Isabel; Pellicer, Daniel; Wilson, Steve

doi:10.1007/978-3-319-78434-2_7

Cited by 8 publications

(10 citation statements)

References 11 publications

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“…Similar arguments to the ones in this section can be used to prove Corollary 12 using the trivial maniplex over M defined in [7] instead of2 M .…”

Section: Maniplex2 Mmentioning

confidence: 76%

“…We choose the base flag of2 M to be (Φ 0 , 0), where 0 ∈ Z S 2 is the 0-constant function. The following proposition summarises Section 6 of [7]. Item (6) was proven in [14].…”

Section: Maniplex2 Mmentioning

confidence: 85%

“…It was initially introduced by Danzer in [6] in the context of incidence complexes, which are combinatorial structures more general than abstract polytopes. This construction has appeared several times when studying regular polytopes and was generalised to maniplexes in [7].…”

Section: Maniplex2 Mmentioning

confidence: 99%

“…It generalises a map on a surface to higher ranks, and at the same time is a relaxation of the notion of abstract polytope. Maniplexes were first introduced in [17], and later used for example in [5] and [7].…”

Section: Introductionmentioning

confidence: 99%

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An existence result on two-orbit maniplexes

Pellicer

Potočnik

Toledo

2019

Journal of Combinatorial Theory, Series A

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A maniplex of rank n is a connected, n-valent, edge-coloured graph that generalises abstract polytopes and maps. If the automorphism group of a maniplex M partitions the vertex-set of M into k distinct orbits, we say that M is a k-orbit n-maniplex. The symmetry type graph of M is the quotient pregraph obtained by contracting every orbit into a single vertex. Symmetry type graphs of maniplexes satisfy a series of very specific properties. The question arises whether any pregraph of order k satisfying these properties is the symmetry type graph of some k-orbit maniplex. We answer the question when k = 2. 1 arXiv:1812.04148v1 [math.CO]

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“…Similar arguments to the ones in this section can be used to prove Corollary 12 using the trivial maniplex over M defined in [7] instead of2 M .…”

Section: Maniplex2 Mmentioning

confidence: 76%

“…We choose the base flag of2 M to be (Φ 0 , 0), where 0 ∈ Z S 2 is the 0-constant function. The following proposition summarises Section 6 of [7]. Item (6) was proven in [14].…”

Section: Maniplex2 Mmentioning

confidence: 85%

Section: Maniplex2 Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An existence result on two-orbit maniplexes

Pellicer

Potočnik

Toledo

2019

Journal of Combinatorial Theory, Series A

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“…The task of determining if a maniplex is the flag graph of a polytope or not, can arise when dealing with operations on polytopes, such as the Petrie operation ( [5]), the mix or parallel product ( [8]) or the Twist operation ( [4]). The problem is also often encounter when dealing with covers and quotients of polytopes ([?, ?, 9]), in particular in determining the polytopality of the so-called minimal regular cover of a polytope ( [8,7]).…”

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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Voltage Operations on Maniplexes, Polytopes and Maps

2023

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Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to unify the study of such operations and generalize them to other geometric and combinatorial structures such as abstract polytopes, hypermaps, maniplexes or hypertopes. This can be done since our technique provides a way to study classical operations in a graph theoretic setting, and thus to apply a voltage operation one only needs that the combinatorial structure in hand can be understood as an n-valent properly n-edge colored graph. For example, in the case of abstract polytopes, the partial order can be encoded into the so-called flag graph of the polytope and the voltage operation is therefore applied to such flag graph to then be recovered as a partial order. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that these operations can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.

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The Twist Operator on Maniplexes

Cited by 8 publications

References 11 publications

An existence result on two-orbit maniplexes

An existence result on two-orbit maniplexes

Polytopality of maniplexes

Voltage Operations on Maniplexes, Polytopes and Maps

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