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.

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.

Current synthetic biology has witnessed a revolution that natural DNA molecule steps onto a broad scientific area by assembling a large variety of threedimensional structures with the connectivity of polyhedra. A mathematical model of these biomolecules is crucial to clarify the biological self-assembly principle, and unravel a first-step understanding of biological regulation and controlling mechanisms. In this paper, mechanisms of two different enzymatic actions on DNA polyhedra are elucidated through theoretical models of polyhedral links: (1) topoisomerase that untangles DNA polyhedral links produces separated single-stranded DNA circles through the crossing change operation; (2) recombinase generates a class of polyhedral circular paths or polyhedral knots by applying the crossing smoothing operation. Furthermore, we also discuss the possibility of applying two theoretical operations in molecular design of DNA polyhedra. Thus, our research provides a new sight of how geometry and topology of DNA polyhedra can be manipulated and controlled by enzymes, as well as has implications for molecular design and structural analysis of structural genome organization.

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.

A mathematical methodology for understanding the complexity of Platonic and Archimedean polyhedral links has been developed based on some topological invariants from knot theory. Knot invariants discussed here include rossing number, unknotting number, genus and braid index, which are considered significant in viewofDNAnanotechnology. Our results demonstrate that the braid index provides the most structural information; hence, it can be used, among four knot invariants, as the most useful complexity measure. Using such an invariant, it indicates that the complexity of polyhedral links is directed by the number of their building blocks. The research introduces a simple but important concept in the theoretical characterization and analysis of DNA polyhedral catenanes.