A Study Of Duality In The Polyhedra And Their Links

In geometry, convex polyhedra are old and important researchful objects, which contacts peoples' daily life. Polyhedral links are new theoretical results, some interlocked structures on the basis of old convex polyhedra and knot theory, which derived from the theoretical research for viral capsid structure and supermolecule structure. Polyhedral links are a class of topological links with polyhedral shape which are linked with a collection of finitely separate closed curves..On the basis of theoretical result of convex polyhedra, the thesis includes two parts of research which aim at different problems.一,Duality is an important conception and manipulation in geometry. As new linked structures, polyhedral links are constructed on the frame of polyhedra. Whether polyhedral links possess of duality, how to define, and how to manipulate are important basis researches of polyhedral links. Furthermore, polyhedral links just a particular family of knots and links, which are attentive in the field of mathematics. Facing to the huge and complicated knot table, how to class, and what kind of relationships there are between them? When duality is applied to knots theory, it 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.1. 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.2. 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: G_e and G_o, 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.二,A virus is a unit of infectious genetic material smaller than any bacteria and embodying properties placing it on the borderline between life and non-life. The vast majority of the virus capsid protein assembled into icosahedral symmetry of the structure. Caspar-Klug Theory has become a fundamental concept for the classification of icosahedral viral capsids based on the principle of quasi-equivalence. However, recent experiments have shown that there are viruses that do not follow the organisation predicted by this theory. We take these novel viruses as objects of investigation, and constructing the geometry models to explain these novel architectures, which enrich the world of polehedra and give theoretic guidance of viral investigation.The outer shells of papilloma virions and polyoma virus contain 72 pentamers, the architectures of which do not follow the Caspar-Klug (CK) "quasi-equivalence" theory. Moveover, the spherical pentagon packing problem for 72 pentagons is a mathematical problem. On the basis of the frame of dodecahedron, we apply the method of "spherical stretching" to the frame, we can obtain a novel polyhedron with I_h symmetry, which contains 72 pentagons. The novel polyhedral structure improves new theory for the simulating of viral capsid with 72 pentamers, and gives additional insight into the mathematical problem of the spherical pentagon packing problem for 72 pentagons, and enrichs the world of polehedra.