The Chirality And Some Topological Index Of Polyhedral Links

In the structural chemistry, DNA and protein folding can show the shape of a polyhedron. In recent years, in the laboratory, many DNA polyhedron and protein catenanes were synthesized and found. For examples, DNA Plato polyhedron, DNA Archimedean polyhedron and bacteriophage HK 97 capsid. As for this, we are first propose these DNA or protein folding Frameworks by the model of polyhedral links, and bring forward some methods for construction of polyhedral links. We also give their chirality, polynomial, genus and braid index by knot theory and graph theory. The construction of polyhedral links and their characterization can provide new the-ories and methods for structural chemistry, stereochemistry and constitution of virus morphology mathematics. In the paper, the central contents are following:Firstly, we generalize the model of three-crossing-polyhedral links, and construct a new type of polyhedral links by the means of branched alternating closed braids and double lines covering. We are also give some conditions to determine the chirality of the polyhedral links in terms of generalized Tutte and Kauffman polynomials. Our results show that each regular branched alternating closed polyhedral links are chiral. This means the model of bacteriophage HK97, toplogically linked protein catenane, is chiral.Secondly, we generalize the conception of three-crossing-curves and double-lines covering to a general situation, and propose the conception of n-crossing curves, and use it to construct a type of n-pyramidal links. On the base of the links, we calculate their genus, and obtain (1) m-pyramidal link is more complex than the n-pyramidal link, where m> n. (2) There are three-crossing-tetrahedral catenanes can be designed in a surface with genus one. And there exit only two pyramidal catenanes can be designed in a surface with genus more than one. This provides a new guiding principle to topology-aided molecular design in a given surface and a rational evidence for the synthesis in the laboratory.Thirdly, inspired by the model of tangled DNA polyhedra, we construct a type of cycle-crossover polyhedral links, which is also the model of DNA polyhedron. Apart from this, we also calculate their Jones polynomial by chain polynomial and Kauffman bracket polynomial.Fourthly, we give a formulas of the genus and a upper boundary of braid index of three-crossing polyhedral links, which are described to be and where n(P) denotes the number of vertices of the polyhedron P. Studies on the model of the links have some signification on the chemistry and biology, which can displayed in the three aspects. (1) They can help us to order and classify protein polyhedral catenanes; These three-crossing-polyhedral links with the same crossing number are belong to the same class. (2) Characterize the complex of protein polyhedral catenanes; The higher the genus of the polyhedral link has, the more complex it is. (3) They can propose theoretical foundation for topology-aided molecular design in a given surface; Only those supramolecular can be synthesized on a surface with the same genus.