In this paper, we introduce two combinatorial operations and a knot-theoretical approach forgeneration and description of fullerene architectures. The 'Spherical rotating-vertex bifurcation' operationapplied to original fullerene polyhedra can lead to leapfrog fullerenes. However, the 'Sphericalstretching-vertex bifurcation' operation applied to fullerene generates a family of related polyhedra, whichgo beyond the scope of fullerenes. These related cages, the cubic tessellations containing not only 5-gonsand 6-gons but also 3-gons and 8-gons, are potential candidates in carbon chemistry. By using a simplealgorithm based on knot theory, these two homologous series of molecule graphs can be transformed intovarious polyhedral links. For these interlocked architectures, it is now possible to quantify their propertiesby knot invariants. By means of this application, we show connections (1) between knot polynomials andfullerene isomers determination, (2) between knot genus and fullerene complexity and (3) betweenunknotting numbers and fullerene stability. Our results suggest that techniques coming from knot theoryhave potential applications and offer novel insights in predicting several structural and chemical propertiesof fullerene polyhedra.

A new method for understanding the construction of dual links has been developed on the basis of medial graph in graph theory and tangle in knot theory. The method defines two types of oriented 4-valent plane graph: Ge and Go, whose vertices are covered by E-tangles and O-tangles, respectively. The result shows that there are two types of dual links: E-dual links and O-dual links, which have many differences in topological properties, especially their chiral rule. In our paper, we show that dual links can be constructed by oriented 4-valent plant graphs and tangles. This research puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.

In this paper, the novel topology of Platonic polyhedral links is discussed on the basis of the graph theory and topological principles. This interesting problem of the dual polyhedral links has been solved by using our method of the ‘sphere-surface-movement’. There are three classes of dual polyhedral links which can be explored: the tetrahedral link is self-dual, the hexahedral and octahedral link, as well as the dodecahedral and icosahedral link are dual to each other. Our results show that the duality of self-dual tetrahedral link is ‘trivial’, and the duality of hexahedral and octahedral link as well as dodecahedral and icosahedral link are ‘nontrivial’. This study provides further insight into the molecular design and theoretical characterization of the new polyhedral links.

Polyhedral links, interlinked and interlocked architectures, have been proposed for the description and analysis of knotted configurations in DNA and proteins. Qiu et al. fabricated cubic polyhedral and carbon nanotube links by the means of three cross curves and double lines covering, and analyzed their chirality by point groups. We present, in this paper, a novel method by replacing three cross curves with branched alternating closed braids to construct a new type of polyhedral links on arbitrary convex polyhedra. We give some conditions to determine the chirality of the polyhedral links in terms of generalized Tutte and Kauffman polynomials. Accordingly, we show that each regular branched closed braid link is chiral. This result shows that the model of bacteriophage HK97 capsid, topologically linked protein catenane, is chiral.

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic solids by using our method of the 'n-branched curves and m-twisted double-lines covering'. There are five classes of platonic polyhedral links we can construct: the tetrahedral links; the hexahedral links; the octahedral links; the dodecahedral links; the icosahedral links. The tetrahedral links, hexahedral links, and dodecahedral links are, respectively, assembled by using the method of the '3-branched curves and m-twisted double-lines covering', whereas the octahedral links and dodecahedral links are, respectively, made by using the method of the '4-ranched curves' and '5-branched curves', as well as 'm-twisted double-lines covering'. Moreover, the analysis relating topological properties and link invariants is of considerable importance. Link invariants are powerful tools to classify and measure the complexity of polyhedral catenanes. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA polyhedral catenanes.