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 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.

A new method for understanding the relations between the molecular design and topological chirality has been developed on the basis of the Seifert construction of knot theory. Tis interesting problem of topological chirality has been solved by applying the topological symmetry concept to the molecular mobius ladders with citber 2n+1 or 2n half-twists (n≥1). Our results show that the T2n+1-Mobius ladders possess the point symmetry group C2n+1 when Lγ=(2n+1)m. and that the Tπ-Mobius ladders possess the point symmetry group Cπ when Lγ=2nm or Lγ=2nm+n(m=1,2,…). Hence both sets are topological chiral.

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.