This is a list of types of polytopes where it is based off of the relationships between elements.
Compound
A polytope where some of the edges are not connected by any element to another.
Exotic
A polytope where more than two facets are on a ridge.
Exotical
A polytope where the elements are exotic but the polytope itself isn't exotic.
Fissary
A polytope with compound vertex figures but is not compound.
Complexified
A polytope with compound facets but is not compound.
Degenerate
A polytope where every facet is d-1 hypercoplanar with another and there are finite elements.
Unbounded
Has at least one element at infinity.
Tesselation
A polytope where all of the facets are d-1 hypercoplanar and there are infinite elements.
Manifoldic
A polytope where all of the facets lie on a manifold.
Coincidic
A polytope where the facets coincide.
Cofacetal
A polytope where some of the facets are d-1 hypercoplanar.
Skew
A polytope with an infinite amount of facets that are not d-1 hypercoplanar.
Solid
A polytope which is not coincidic, n-1 hypercoplanar, on a non-orb manifold, nor a compound.