A new method for understanding the construction of polyhedral links has been developed on the basis of the novel structure of HK97 capsid and the classification of Goldberg polyhedra. Polyhedral links, the interlinked and interlocked architectures, have been solved by analyzing and characterizing the topological structure of the two types of polyhedral links, which are RnC1ZRnC10(2nC1) and RnC1ZRnC20n. Where, nZ1, 2,., R1Z12 and R represents the number of the interlocked rings. Our results show that these two types of polyhedral links both have I symmetry and therefore are said to possess chirality.

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.

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.

A new methodology for understanding the construction of Extended Goldberg polyhedra has been developed on the basis of Goldberg polyhedra by using our methods of the ‘spherical rotating’ and the ‘spherical stretching’. The spherical rotating describes the deformation of rotating polygons on a sphere; the spherical stretching depicts the deformation of stretching spaces between polygons on a sphere. Our results show that these Extended Goldberg polyhedra are a kind of novel geometrical objects of icosahedral symmetry and are considered to explain some viral capsids.