scholarly journals The graph of an abstract polytope

1973 ◽  
Vol 4 (1) ◽  
pp. 336-346 ◽  
Author(s):  
Katta G. Murty
Keyword(s):  
Abstract Polytope

scholarly journals Constructions of Chiral Polytopes of Small Rank

2011 ◽  
Vol 63 (6) ◽  
pp. 1254-1283 ◽  
Author(s):  
Antonio Breda D’Azevedo ◽  
Gareth A. Jones ◽  
Egon Schulte
Keyword(s):  
Automorphism Group ◽  
Small Rank ◽  
Abstract Polytope ◽  
Chiral Polytopes ◽  
General Method

AbstractAn abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4, and 5.

scholarly journals Categories: Between Cubes and Globes. Sketch I

2019 ◽  
Vol 64 (12) ◽  
pp. 1125
Author(s):  
Y. Bespalov
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Geometric Interpretation ◽  
Small Category ◽  
General Context ◽  
Fixed Target ◽  
Abstract Polytope ◽  
Usual Theory ◽  
Finite Partially Ordered Set ◽  
Natural Way

For a finite partially ordered set I, we define an abstract polytope PI which is a cube or a globe in the cases of discrete or linear poset, respectively. For a poset P, we have built a small category ♦P with finite lower subsets in P as objects. This category ♦P = ♦P+♦P- is factorized into a product of two wide subcategories ♦P+ of faces and ♦P- of degenerations. One can imagine a degeneration from I to J ⊂ I as a projection of an abstract polytope PI to the subspace spanned by J. Morphisms in ♦P+ with fixed target I are identified with faces of PI . The composition in ♦P admits the natural geometric interpretation. On the category ♦I of presheaves on ♦I , we construct a monad of free category in two steps: for a terminal presheaf, the free category is obtained via a generalized nerve construction; in the general case, the cells of a nerve are colored by elements of the initial presheaf. Strict P-fold categories are defined as algebras over this monad. All constructions are functorial in P. The usual theory of globular and cubical higher categories can be translated in a natural way into our general context.

scholarly journals Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes

10.37236/7070 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Gabe Cunningham
Keyword(s):  
Regular Polytopes ◽  
Abstract Polytope ◽  
Chiral Polytopes

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

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