6,579 research outputs found
Chiral extensions of chiral polytopes
Given a chiral d-polytope K with regular facets, we describe a construction
for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is
finite whenever K is finite. We provide explicit examples of chiral 4-polytopes
constructed in this way from chiral toroidal maps.Comment: 21 pages. [v2] includes several minor revisions for clarit
Tropical cycles and Chow polytopes
The Chow polytope of an algebraic cycle in a torus depends only on its
tropicalisation. Generalising this, we associate a Chow polytope to any
abstract tropical variety in a tropicalised toric variety. Several significant
polyhedra associated to tropical varieties are special cases of our Chow
polytope. The Chow polytope of a tropical variety is given by a simple
combinatorial construction: its normal subdivision is the Minkowski sum of
and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties;
several improvements suggested by Eric Katz; some rearrangemen
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Mixing Convex Polytopes
The mixing operation for abstract polytopes gives a natural way to construct
the minimal common cover of two polytopes. In this paper, we apply this
construction to the regular convex polytopes, determining when the mix is again
a polytope, and completely determining the structure of the mix in each case
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