Two combinatorial operations and a knot theoretical approach for fullerene polyhedra

In this paper, we introduce two combinatorial operations and a knot-theoretical approach for generation and description of fullerene architectures. The ‘Spherical rotating–vertex bifurcation’ operation applied to original fullerene polyhedra can lead to leapfrog fullerenes. However, the ‘Spherical stretching–vertex bifurcation’ operation applied to fullerene generates a family of related polyhedra, which go beyond the scope of fullerenes. These related cages, the cubic tessellations containing not only 5-gons and 6-gons but also 3-gons and 8-gons, are potential candidates in carbon chemistry. By using a simple algorithm based on knot theory, these two homologous series of molecule graphs can be transformed into various polyhedral links. For these interlocked architectures, it is now possible to quantify their properties by knot invariants. By means of this application, we show connections (1) between knot polynomials and fullerene isomers determination, (2) between knot genus and fullerene complexity and (3) between unknotting numbers and fullerene stability. Our results suggest that techniques coming from knot theory have potential applications and offer novel insights in predicting several structural and chemical properties of fullerene polyhedra. MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2010) 347-362