Isotopy problems for saddle surfaces

Abstract

Four mutually dependent facts are proven. • A smooth saddle sphere in S 3 has at least four inflection arches. • Each hyperbolic hérisson H generates an arrangement of disjoint oriented great semicircles on the unit sphere S 2 . On the one hand, the semicircles correspond to the horns of the hérisson. On the other hand, they correspond to the inflection arches of the graph of the support function h H . The arrangement contains at least one of the two basic arrangements. • A new type of a hyperbolic polytope with 4 horns is constructed. • There exist two non-isotopic smooth hérissons with 4 horns. This is important because of the obvious relationship with extrinsic geometry problems of saddle surfaces, and because of the non-obvious relationship with Alexandrov’s uniqueness conjecture.