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.

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.

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.

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.

In this paper, the novel topology of DNA knots is discussed on the basis of the Seifert construction in knot theory. Our result shows that T4k-DNA knots possess the point symmetry group S1 and T2k+1-DNA knots possess the point symmetry group C1-Hence, C, represents the most chiral and S1 the least achiral We say that a structure with C1 symmetry is topologically chiral, wile a structure with S1 Symmetry is shown to have the properties of a topological rubber glove, and that the orientability of the Seifert construction is primary tool for the left-right classification of molecular knots.