【作者】 路丹；

【导师】 邱文元；

【作者基本信息】 兰州大学 ， 物理化学， 2009， 博士

【摘要】 凸多面体是几何学的一个古老而重要的研究对象,它与人们的生活密切相关。多面体链环是将古老的凸多面体结构和重要的纽结理论相结合,在解决近年来实验室中发现的病毒结构和合成的超分子结构的理论研究中,得到的理论成果。多面体链环是由多个环相互嵌套成的具有多面体形状的一种拓扑几何结构。本论文基于凸多面体的理论成果,从不同问题入手进行了两大方面的研究:一:对偶是几何学的一个重要概念和操作,多面体链环作为在多面体结构上构造的新型结构,是否存在对偶性,如何定义,如何操作,是关于多面体链环的一个重要的基础性研究。更进一步,多面体链环是链环结构的特例,纽结与链环一直是数学研究的热点,面对庞大而复杂的纽结表,如何分类,它们之间存在怎样的联系?我们将对偶理论引入纽结理论,给出对偶链环的定义和构造方法,为纽结的分类提供依据,同时这一构造方法的提出,为纽结超分子的合成提供新的思路,对实验的指导也具有重要的意义。1.在图论中间图理论和拓扑理论的基础上,提出对偶多面体链环的定义。以五个柏拉图多面体为例,运用“三交叉——双线覆盖”的方法构造四面体链环,六面体链环和十二面体链环,利用“球面游走”的拓扑学方法,得到相应的对偶多面体链环。结果显示,四面体链环是自对偶结构,六面体链环和八面体链环,十二面体和二十面体链环互为对偶。从手性角度考虑,对偶变换具有手性保持的特征,十个多面体链环分为六组对偶多面体链环。从构型角度考虑,四面体链环的自对偶是“平凡”的,六面体链环和八面体链环,十二面体和二十面体链环的对偶是“非平凡”的。这一研究说明多面体链环具有对偶性,其对偶变换是拓扑的。2.基于图论中反转中间图的方法和纽结理论中的缠绕理论,我们提出了构建对偶链环的新方法。这一方法定义了两种有向4-度平面图:G_e和G_o分别用E-tangles和O-tangles覆盖两种有向4-度平面图的顶点。结果得到两种对偶链环:E-dual links和O-dual links,它们具有许多不同的拓扑特征,特别是它们的手性规则。研究表明,通过有向4-度平面图和缠绕可以构建得到对偶链环。这一研究提出了对偶链环的定义及其构建的方法。对偶链环为链环的研究提供了新的思路,这一构建方法可以被用于指导手性分子的合成。二:病毒是比任何细菌都小的感染性遗传物质,介于生命与非生命之间的一类无细胞结构的生物,绝大多数病毒的衣壳蛋白装配成二十面体对称的结构,“准等价”原理是解决严格遵从二十面体对称性的病毒衣壳几何特征的有效而成熟的方法。但是随着研究手段的不断发展,更多的病毒结构被发现,它们的衣壳结构违背准等价原理。我们以这些奇特结构为研究对象,找到它们的几何模型,补充多面体结构,为病毒研究提供理论指导。乳多空病毒和多瘤病毒的72个五聚体的衣壳结构违背Caspar-Klug(CK)“准等价”原理。而且,72个五边形的球面包裹问题是一个有待解决的数学难题。我们在十二面体框架的基础上,利用“球面拉伸”的方法,得到新的具有二十面体对称性,72个五边形构造的的多面体结构。新型多面体结构为72个五聚体的病毒衣壳结构的模拟提出新的理论,为五边形排列问题提供了新的思路,丰富了多面体世界。更多还原

【Abstract】 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.更多还原