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Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 86–147 (Mi tm4010)  

Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions

N. Yu. Erokhovets

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Abstract: We study combinatorial properties of polytopes realizable in the Lobachevsky space $\mathbb L^3$ 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00671
16-51-55017
The work was supported by the Russian Foundation for Basic Research, project nos. 17-01-00671 and 16-51-55017-GFEN.


DOI: https://doi.org/10.4213/tm4010

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 78–134

Bibliographic databases:

UDC: 514.172.45+514.132+519.17
Received: December 30, 2018
Revised: March 11, 2019
Accepted: March 13, 2019

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, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 86–147; Proc. Steklov Inst. Math., 305 (2019), 78–134

Citation in format AMSBIB
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\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 Tr. Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 86--147
\publ Steklov Math. Inst. RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4010}
\crossref{https://doi.org/10.4213/tm4010}
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\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 305
\pages 78--134
\crossref{https://doi.org/10.1134/S0081543819030064}
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