Current synthetic biology has witnessed a revolution that natural DNA molecule steps onto a broad scientific area by assembling a large variety of threedimensional structures with the connectivity of polyhedra. A mathematical model of these biomolecules is crucial to clarify the biological self-assembly principle, and unravel a first-step understanding of biological regulation and controlling mechanisms. In this paper, mechanisms of two different enzymatic actions on DNA polyhedra are elucidated through theoretical models of polyhedral links: (1) topoisomerase that untangles DNA polyhedral links produces separated single-stranded DNA circles through the crossing change operation; (2) recombinase generates a class of polyhedral circular paths or polyhedral knots by applying the crossing smoothing operation. Furthermore, we also discuss the possibility of applying two theoretical operations in molecular design of DNA polyhedra. Thus, our research provides a new sight of how geometry and topology of DNA polyhedra can be manipulated and controlled by enzymes, as well as has implications for molecular design and structural analysis of structural genome organization.

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 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 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.