Isotopy problems for saddle surfaces

Four mutually dependent facts are proven. •A smooth saddle sphere in S3S3 has at least four inflection arches.•Each hyperbolic hérisson HH generates an arrangement of disjoint oriented great semicircles on the unit sphere S2S2. 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 hHhH.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.