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.