Studies On The Invariants Of The Links Based On DNA Polyhedron And Protein

Knots and links are new forms of the molecular structures, which are not only recognized in naturally occurring molecules but also realized in recently synthesized compounds. These novel structures play an important role in chemical and physical properties of molecules. Link invariants will be served as the essential tools for un-derstanding and describing these structures. This paper are dedicated to establishing the links having some important significant in biological chemistry and studying their some invariants, via the relationship of graph theory and knot theory. The main content of this thesis is divided into four parts.I. Background KnowledgeIn the first part, we introduce our research background and some basic knowledge used in this thesis. In the part of our research background, we focus on describing DNA polyhedron synthesized and bacteriophage HK97virus capsid discovered in laboratory. DNA polyhedron can be divided into two types, that is I-DNA polyhedron and II-DNA polyhedron, according to their each edge composing of one or two parallel DNA chains. These facts provide our work with the research background and goal. The basic knowledge proposed in Chapter two contain some basic concepts and notations in knot theory and graph theory, which will be served as the theoretical basis.II. The HOMFLY polynomial of polyhedral links and their invariantsIn structural nanotechnology, DNA polyhedron using DNA as a building material has been synthesized in laboratories. It has became a new challenge for scientists to understand and characterize the topological structure of these molecules. Polyhedral links are proposed as a mathematical model. Our research work has mainly focused on how to obtain their link invariants for a variety of the polyhedral links. First, Polyhedral links are divided into two categories, I-polyhedral links and II-polyhedral links, according to the structure of DNA polyhedron. Based on the complexity of II-polyhedral links, they are further divided into odd and even polyhedral links according the parity of the building block, and each of both classes is discussed individually. All polyhedral links can be obtained from polyhedron via using the operation "tangle covering". Hence, we have established some relationships between their HOMFLY polynomials and polynomial invariants of the original graphs. For I-polyhedral links, this relation not only simplified the computation of HOMFLY polynomial, but also has been used to obtain HOMFLY polynomial of a typical link family called rational links. For II-polyhedral links, we further calculated the span of HOMFLY polynomial, and also discussed their genera and braid indices. As a result, the genus of even polyhedral links is zero, and their braid indices are determined by the building blocks and the polyhedral graph. In contrast, odd polyhedral links can’t be embedded on the sphere, their genus are determined completely by the face number of polyhedral graph. Moreover, the latter have more complicated structure, we only obtain the upper and lower bound of their braid index. Our research will provides a theoretical basis on the synthesis and control of the molecules with remarkably complex topology.III. Genera of the links derived from2-connected plane graphsGenus is an important invariant in knot theory, and used to classify molecular catenanes. However, it is very difficult to calculate in general. Hence for an n-component link, we often need to calculate the genera of its2n-1different oriented links. It will be a great deal of work, particularly for a link with a large number of components. To address the problem, we apply the idea of studying links via graphs to obtain their genera. First, a large family of links has been generated base on the medial graph of2-connected plane graphs. Then, we here associated the Seifert circles of an oriented link with the connected components of one of its states. By using medial graph to analyze the change of the states of links, we shown that these oriented links have the smallest number of seifert circles over all orientations of these links. At last, the genera of these links are given as a formula in terms of the link component number and the degree sum of graph. Our work not only proposed a new method to resolve the original mathematical problem, but also provided a useful tool to characterize the HK97viral capsid.IV. The polynomial invariants of rigid-vertex graphsAn even-degree and rigid-vertex graph, can be viewed as some edged-knotted or edged-linked graphs. It is a graph embedded in three dimensional Euclidean space R3, whose each vertex has an even degree. Such a graph is a mixture of the topological flexibility on its edges and the mechanical rigidity on its vertices, hence they have extensive applications in chemical and biological networks. For4-regular and6-regular rigid-vertex graph, their polynomial invariants has been established. However, for any rigid-vertex graph G, directly using the original method will lead to some complicated situations occurring, making it difficult to obtain the polynomial invariants of G. We here associate an even-degree and rigid-vertex graph with a simplified set of links via combination method. Hence we establish HOMFLY polynomial and Kauffman polynomial of these links. Our work generalizes the origin results on4-regular graph, and simplified the method used on6-regular graphs. Moreover, the even-degree and rigid-vertex graphs are determined completely by the graphical calculus of4-degree planar graphs.