4,1

See Table 8.1. Essential polytopes of dimension at least 4, p.36, of Felikson. In dimensions 2 and 3 compact hyperbolic Coxeter polytopes are completely classified by Poincare and Andreev. Abstract: We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, and realize it in four-dimensional case.

H. Poincare, Theorie des groupes fuchsiens, Acta Math. 1 (1882), 1-62.

E. M. Andreev, On convex polyhedra in Lobachevskii spaces, Math. USSR Sb. 10 (1970), 413-440.

Anna Felikson, Pavel Tumarkin, Essential hyperbolic Coxeter polytopes, June 22, 2009.

a(4) = 20 because the essential hyperbolic Coxeter polytopes in 4 dimensions are proved to be 2 simplices, 2 Esselmann polytopes, 5 simplicial prisms, 8 4-polytopes with 7 facets, and 3 three times truncated simplices. a(5) = 6 because the essential hyperbolic Coxeter polytopes in 4 dimensions are proved to be 2 simplicial prisms, 3 5-polytopes with 8 facets, and 1 three times truncated simplex.

Sequence in context: A068612 A040387 A040385 this_sequence A076594 A102409 A128445

Adjacent sequences: A161994 A161995 A161996 this_sequence A161998 A161999 A162000

nonn,uned

Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 24 2009