A general approach is proposed to elucidate the topological characteristics ofmolecules with the shape of polyhedral links. For an arbitrary polyhedral graph, four classes of polyhedral links can be obtained by applying the operation of 'X-tangle covering' to the related reduced sets. The relationships between theW-polynomial of a polyhedral graph and the HOMFLY polynomial of each kind of polyhedral links are established. These relationships not only simplify the computation but also provide a method of constructing a general formula for the HOMFLY polynomial of polyhedral links.

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 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.

DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components m, of crossings c, and of Seifert circles s are related by a simple and elegant formula: szm~cz2. This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler’s formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus.

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.