On the structure of the $6 \times 6$ copositive cone

In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those components which are essential, i.e., which are not embedded in the boundary of other components. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the $6 \times 6$ copositive cone. As an application, we construct an example of a copositive $6 \times 6$ matrix with unit diagonal which does not belong to the Parrilo inner sum of squares relaxation ${\cal K}^{(1)}_6$.