Chiral 4-polytopes in space

In this talk we adopt the notion of skeletal polytope introduce by Grunbaum in the second half of the 20th Century. A polygon is then a connected 2-valent graph embedded in Euclidean space (no assumption on convexity or planarity), and in general an n-polytope is a collection of (n-1)-polytopes satisfying certain axioms. Such a structure is called chiral if it admits all possible symmetry by abstract rotations, but none by abstract reflections.

Chiral polyhedra (polytopes of rank 3) were found and classified by Schulte only in 2005. Chiral 4-polytopes were claimed not to exist in 2004. This last claim was mistaken. In this talk we present the three chiral 4-polytopes in space.