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.

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and double-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodecahedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.

A new method for understanding the relations between molecular design and topological features has been developed on the basis of the Seifert construction in knot theory. Our result shows that the T2-molecular doubled knots possess the point symmetry group C2 and that the T2k-molecular doubled knots possess the point symmetry group C1 Hence both sets are topological chiral. When the rungs are cut down, topological symmetries of the molecular knots are unchanged, Our results led us to infer that the point symmetry group S1 is necessary and sufficient for the molecular topological rubber glove.