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Combinatorics and hyperbolic geometry of families of 3-dimensional polytopes: fullerenes and right-angled polytopes.

Banff International Research Station for Mathematical Innovation and Discovery

We study combinatorial properties of families of simple 3-dimensional polytopes defined by their cyclic and strongly cyclic k-edge-connectivity. Among them are flag polytopes and Pogorelov polytopes, which are polytopes realizable in the Lobachevsky (hyperbolic) space L3 as bounded polytopes of finite volume with right dihedral angles. The latter class contains fullerenes â simple 3-dimensional polytopes with only pentagonal and hexagonal faces. We focus on combinatorial constructions of families of polytopes from a small set of initial polytopes by a given set of operations. Here we will present the classical result by V.Eberhard (1891) for all simple 3-polytopes, more recent results by A.Kotzig (1969), D.Barnette (1974, 1977), J.Butler (1974), T.Inoue (2008), and V.D.Volodin (2011), and their improvements by V.M.Buchstaber and the author (2017-2019). For fullerenes we have a more strong result. We also study polytopes realizable in L3 as polytopes of finite volume with right dihedral angles. On the base of E.M. Andreev's theorem (1970) we prove that cutting off ideal vertices defines a one-to-one correspondence with strongly cyclically 4-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family is obtained by cutting off a matching of a polytope from the same family or the cube with at most two nonadjacent orthogonal edges cut producing all the quadrangles. We refine D.Barnette's construction of this family of polytopes and give its application to right-angled polytopes. We refine the construction of ideal right-angled polytopes by edge-twists described in the survey by A.Yu.Vesnin (2017) on the base of results by I.Rivin (1996) and G.Brinkmann, S.Greenberg, C.Greenhill, B.D.McKay , R.Thomas, and P.G.Wollan (2005), and analyse its connection to D.Barnette's construction via perfect matchings. We make a conjecture on behaviour of volume under operations generalizing results by T.Inoue (2008) and give arguments confirming it.

Mathematics; Convex and discrete geometry; Statistical mechanics, structure of matter; Discrete mathematics

video/mp4

Author affiliation: Moscow State University

2020-03-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73846

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/