It was shown that the covering of C60 “Buckminsterfullerene” is basically sumanenic, with the empty π-electron faces being only pentagons. Four series of cages, tessellated by sumanenic patterns S[r] = [r:(5,6) r/2], were generated by sequences of map operations, and their topology described. Among these cages, which all show all_R[5] 2-factors, those belonging to the series designed on the dual pair Dodecahedron/ Icosahedron by iterating the P 5 operation and closing by Le operation, show a unique term Omega signature, thus being classified as the C60 series. C60 itself also shows the unique signature and all the members of its family show large HOMO-LUMO gap values, larger than of the cages belonging to the other three series herein built up. Coverings are given in terms of circulenes/flowers and the relation with the Omega and Ring polynomials is evidenced. Analytical formulas for the net parameters and for the used polynomials are given.