A new method for understanding the construction of dual links has been developed on the basis of medial graph in graph theory and tangle in knot theory. The method defines two types of oriented 4-valent plane graph: Ge and Go, whose vertices are covered by E-tangles and O-tangles, respectively. The result shows that there are two types of dual links: E-dual links and O-dual links, which have many differences in topological properties, especially their chiral rule. In our paper, we show that dual links can be constructed by oriented 4-valent plant graphs and tangles. This research puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.

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.

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.

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.

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.