In this paper we set up a topological model which can be used to describe closed circular (cc) DNAs on the basis of Seifert construction in knot theory. We visually prove that the backbones of T2n-ccDNAs (n=1,2,…) in the crystal which are topological chiral possess the point symmetry group C and that the backbones of T2n-ccDNAs in solution which are topological achiral possess the point symmetry groups Sn2, Sn and S2n. Topological achirality leads us to believe that right handed DNA can be flipped over into a left handed hcilx. Further it also proveides a sound basis for the investigation of this surprising manifestation of DNA supereoiling. Replication. transcription and recombination.

In this paper, we introduce two combinatorial operations and a knot-theoretical approach forgeneration and description of fullerene architectures. The 'Spherical rotating-vertex bifurcation' operationapplied to original fullerene polyhedra can lead to leapfrog fullerenes. However, the 'Sphericalstretching-vertex bifurcation' operation applied to fullerene generates a family of related polyhedra, whichgo beyond the scope of fullerenes. These related cages, the cubic tessellations containing not only 5-gonsand 6-gons but also 3-gons and 8-gons, are potential candidates in carbon chemistry. By using a simplealgorithm based on knot theory, these two homologous series of molecule graphs can be transformed intovarious polyhedral links. For these interlocked architectures, it is now possible to quantify their propertiesby knot invariants. By means of this application, we show connections (1) between knot polynomials andfullerene isomers determination, (2) between knot genus and fullerene complexity and (3) betweenunknotting numbers and fullerene stability. Our results suggest that techniques coming from knot theoryhave potential applications and offer novel insights in predicting several structural and chemical propertiesof fullerene polyhedra.

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.

In this paper, the novel topology of Platonic polyhedral links is discussed on the basis of the graph theory and topological principles. This interesting problem of the dual polyhedral links has been solved by using our method of the ‘sphere-surface-movement’. There are three classes of dual polyhedral links which can be explored: the tetrahedral link is self-dual, the hexahedral and octahedral link, as well as the dodecahedral and icosahedral link are dual to each other. Our results show that the duality of self-dual tetrahedral link is ‘trivial’, and the duality of hexahedral and octahedral link as well as dodecahedral and icosahedral link are ‘nontrivial’. This study provides further insight into the molecular design and theoretical characterization of the new polyhedral links.

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.