Abstract:
We study combinatorial properties of polytopes realizable in the Lobachevsky space L3 as polytopes of finite volume with right dihedral angles. On the basis of E. M. Andreev's theorem we prove that cutting off ideal vertices of right-angled polytopes defines a one-to-one correspondence with strongly cyclically four-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family can be obtained by cutting off a matching of a polytope from the same family or of the cube with at most two nonadjacent orthogonal edges cut, in such a way that each quadrangle results from cutting off an edge. We refine D. Barnette's construction of this family of polytopes and present its application to right-angled polytopes. We refine the known construction of ideal right-angled polytopes using edge twists and describe its connection with D. Barnette's construction via perfect matchings. We make a conjecture on the behavior of the volume under operations and give arguments to support it.
Citation:
N. Yu. Erokhovets, “Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 86–147; Proc. Steklov Inst. Math., 305 (2019), 78–134
\Bibitem{Ero19}
\by N.~Yu.~Erokhovets
\paper Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions
\inbook Algebraic topology, combinatorics, and mathematical physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 86--147
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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\crossref{https://doi.org/10.4213/tm4010}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 305
\pages 78--134
\crossref{https://doi.org/10.1134/S0081543819030064}
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Linking options:
https://www.mathnet.ru/eng/tm4010
https://doi.org/10.4213/tm4010
https://www.mathnet.ru/eng/tm/v305/p86
This publication is cited in the following 4 articles:
N. Yu. Erokhovets, “Canonical geometrization of orientable $3$-manifolds defined by vector colourings of $3$-polytopes”, Sb. Math., 213:6 (2022), 752–793
Nikolai Yu. Erokhovets, “Cohomological Rigidity of Families of Manifolds Associated with Ideal Right-Angled Hyperbolic 3-Polytopes”, Proc. Steklov Inst. Math., 318 (2022), 90–125
N. Yu. Erokhovets, “Theory of families of polytopes: fullerenes and Pogorelov polytopes”, Moscow University Mathematics Bulletin, 76:2 (2021), 83–95
Nikolai Erokhovets, Contemporary Mathematics, 772, Topology, Geometry, and Dynamics, 2021, 107
Citation in format AMSBIB
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