Topological Classification And Chiral Analysis Of DNA Triangular Prism Links

In recent decades,DNA polyhedra with DNA molecules as building materials has been synthesized in large quantities in experiments.DNA polyhedral molecules have attracted much attention from scientific researchers due to their unique geometric and topological characteristics.Polyhedral links are the interlocked and interlinked architectures formed by treating DNA as a thin strand,which are presented as mathematical models to explain the topological structures of DNA polyhedra.Link invariants are important tools for describing and classifying these structures.This dissertation aims to construct some polyhedral links with biochemical significance and study their invariants based on the relation between graph theory and knot theory.The research contents of this dissertation are as followsIn Chapter I,we mainlny introduces the research background and basic knowledge of this dissertation.In terms of research background,we mainly introduce the research progress of DNA polyhedral molecules synthesized in experiments.DNA polyhedra can be divided into three categories,including DNA Platonic polyhedra,DNA Archimedean polyhedra and other DNA polyhedra.These experimental advances provide background and goals for our research work.In terms of basic knowledge,it mainly involves some basic concepts and symbols in graph theory and knot theory,as well as chiral analysis of knots and links,which provides a theoretical basis for our researchIn Chapter ?,DNA triangular prisms with one helical turn on each edge has been realized recent.ly by one or five synthetic DNA single strands with rationally designed sequences.In this chapter,we determine all possible existing topological structures for DNA triangular prisms with such double-helical edges.Here triangular prism links are assembled as the mathematical models for DNA triangular prisms by using six oriented t.wist tangles as basic building blocks.In this process,we firstly show that only 22 orientations are allowed to exist in triangular prism links.Then there are 366 link types of triangular prism links identified from all generated link diagrams by considering the symmetry of triangular prism.We note that each type includes infinite triangular prism links by changing the number of building blocks on each edge.Our research provides a general classification for the topological structures of triangular prism links,and also can be served a.s a list of cancdidates for further synthesized DNA triangular prisms with required topological structures.In Chapter III,we performed chiral classification for 366 link types of triangular prism links by calculating their link invariants.Firstly,three link invariants including the component number,crossing number and writhe number are calculated for triangular prism links.Applying the component number,we showed that there are 178 triangula.r prism link diagrams with an even number of components,which are all chiral.The crossing number and writhe number,combined with the inequality introduced by Kauffman are used to identify the chirality of 111 link types of triangular prism links with odd component number.For the remaining 77 link types of t.riangular prism links,we must resort to a more powerful invariant of oriented links:the HOMFLY polynomials.Hence a new algorithm is to supposed for establishing a general formula for t.he lowest-degree terms of HOMFLY polynomial for each link type.In particular,the second lowest-degree term of HOMFLY polynomial of the link D(2b2n?,4a2n?,2b2n?,a2n?)64 is given separately.These polynomials are shown to be asymmetrical over the variant v,and hence the remaining 77 triangular prism links are all chiral.These results altogether show that 366 link types of triangular prism links are all chiral,which means that the chirality of DNA triangular prisms can be determined by their topological structures.Hence our work provides a new perspective for predicting and controlling the chiral structure of DNA triangular prism from the theoretical viewpoint.