Open problems on k-orbit polytopes

@article{Cunningham2018OpenPO,
  title={Open problems on k-orbit polytopes},
  author={Gabe Cunningham and Daniel Pellicer},
  journal={Discret. Math.},
  year={2018},
  volume={341},
  pages={1645-1661}
}
We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before. 

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References

SHOWING 1-10 OF 95 REFERENCES
Polytopes with Preassigned Automorphism Groups
We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also showExpand
Constructions of k-orbit abstract polytopes
OF DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the College of Science of Northeastern University April 5, 2013
Problems on polytopes, their groups, and realizations
TLDR
A collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding workshop at the Banff International Research Station in May~2005 are given. Expand
Finite polytopes have finite regular covers
We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.
A combinatorial theory of Grünbaum's new regular polyhedra, part I: Grünbaum's new regular polyhedra and their automorphism group
The new regular polyhedra, defined and investigated by Branko Grünbaum in [4], and theirn-dimensional generalizations are classified in terms of their symmetry group.
Constructions of Chiral Polytopes of Small Rank
Abstract An 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 paperExpand
A construction of higher rank chiral polytopes
TLDR
This paper shows that this construction implies the existence of chiral d-polytopes, for every rank d>=3, and describes a construction for chiral polytopes with preassigned regular facets. Expand
Regular projective polyhedra with planar faces I
Summary. This is the first of two papers in which we classify the regular projective polyhedra in $ \Bbb P^3 $ with planar faces. Here, we develop the basic notions; we introduce a new diophantineExpand
Realizations of Regular Toroidal Maps
We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types{3,6} and{6,3}.
The monodromy group of the n-pyramid
TLDR
The structure of the monodromy groups of the infinite family of pyramids over n-gons, as well as the structure of their minimal regular covers are investigated. Expand
...
1
2
3
4
5
...