Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes

We describe the combinatorics of three families of simple 3-dimensional polytopes which play an important role in various problems of algebraic topology, hyperbolic geometry, graph theory, and their applications. The first family $\mathcal{P}_{\leqslant 6}$ consists of simple polytopes with at most hexagonal faces. The second family $\mathcal{P}_\mathrm{pog}$ consists of Pogorelov polytopes. The third family $\mathcal{F}$ consists of fullerenes and is the intersection of the first two. We show that in the case of fullerenes there are stronger results than for the first two. Our main tools are $k$-belts of faces, simple partitions of a disc and the operations of transformation and connected sum.