Analysis Of Carbon Nanotubes And DNA Polyhedra By Knot Theory

Polyhedral links are a class of topological links with polyhedral shape which are linked with a collection of finitely separate closed curves. Research on polyhedral links enhances the understanding of the characterization and description of DNA and protein, and provides a possible theoretical model for molecular design as well. On the basis of our research on polyhedral links, this thesis constructs a series of new toroidal fullerene links, carbon nanotube links, viral polyhedral link model, and DNA polyhedral links, and analyzes and discusses them using knot theory. Moreover, it develops novel methods for theoretical characterization of complex molecules and assembly of protein catenanes. The main results are as follows:1. A topological model of toroidal fullerene link is constructed, by the method of 'three-cross-curve and double-line covering' applied to toroidal fullerene. Two types of toroidal fullerene links are depicted. The crossing of link's reduced diagram is 3 times in numbers that of vertices of the fullerene planar graph. The numbers of components in the links are the same as the faces on the toroidal fullerene. Compared with D_2d symmetry of toroidal fullerenes, these links of toroidal fullerene possess the D₅ point group.2. Carbon nanotube links are interlinked and interlocked links with nanotube sharps. A novel constructional method is described based on the structure of carbon nanotube and the knowledge of polyhedral links. Two types of carbon nanotube links are introduced and analyzed: (5, 0) and (5, 5) carbon nanotube links, both being D₅ or C₅ symmetric. The model opens a door for molecular design and mathematical analysis on nanotube structures. By assigning an orientation to the links we analyze the mathematical properties of these oriented links. Our results show that both types of links are chiral. The recurrence formulae for the writhe, self-writhe, linking number and components of these links are also established.3. A novel method for constructing Goldberg polyhedral links and knots is described based on the structure of viral genome organization. Four types of Goldberg polyhedral links are constructed from Goldberg polyhedra by replacing all vertexes with crossed configurations and all edges with twisted double-lines. Knots are interlaced cyclic structures while links are at least two cyclic structures mutually interlaced. The paper targets on how to solve an issue which obtains unicursal knot from a multicursal link. As links are multicursal, knots can be considered as unicursal curves. Polyhedral knots can be derived by transforming polyhedral links. The transformation of polyhedral links to polyhedral knots is in fact a process transforming from multicursal to unicursal. This process is realized by two methods of crossings change in polyhedral links. Our results show that certain Goldberg polyhedral links and knots we construct serve as models for viral genome. The crossed configuration of three branches are, especially, more suitable for modeling viral genome organization.4. A variety of DNA knots and links are detected and a large number of DNA polyhedra and knots are synthesized in the laboratory. And then separating and distinguishing these molecules is a critical issue. Topological techniques have already played a role in identifying DNA knots and links. In particular, knot theory gives a very nice way to model DNA recombination. A new methodology for understanding the construction of polyhedral links has been developed on the basis of DNA circles. The topology of how DNA knots and links are formed as a result of polyhedral links are described, starting with a substrate consisting of an unknot, or T(m,2). This study provides further insight into the molecular design, as well as theoretical characterization of the DNA catenanes.