This paper extends the methodology of the construction of polyhedral links by tangles in knot theory. Building blocks consist of odd tangles which are regions in the projection plane with 2n+1 half-twist, where n is an integer. Fixing odd tangles at the all vertices of Extend Goldberg polyhedra, and then connect them together will result in many interlocked networks. The solution to the component algorithm of 4-regular polyhedral links has been proposed. Our result shows, by counting the length of central circuits of a polyhedron, the component number of the relating polyhedral link will be presented. Using such topological models, some potential significance in biological and chemical areas is tested to be explored.

A new methodology for understanding the construction of polyhedral links has been developed on the basis of 4-regular polyhedra and knot theory. In the method, we utilize uniform 2n-tangles (n is an integer) to cover all vertexes of Extended Goldberg polyhedra, and many infinite series of interlinked and interlocked architectures have been assembled. The growth rule of links with tangle of |n|=1 and a class of topological transformation depending on the number of n are systematic enumerated. Our study reveals that these novel structures all have I symmetry and each belongs to a given topological configuration, D or L. Moreover, they provide some potential models for protein and DNA cages which have chirality.

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.

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.