The Architecture of Extended Platonic Polyhedral Links

摘要

Polyhedral links proved to be effective mathematical models for new types of polyhedral molecules, especially DNA polyhedra. In this paper, we construct four types of polyhedral links based on extended Platonic polyhedra. By applying a new Euler formula and polyhedral growth law to these polyhedral links, their topological characteristics, including crossing number, component number and Seifert circle number, are computed, thus promoting the understanding of the topological structure and synthesis of extended Platonic polyhedral links. Our study indicates that, the new Euler formula and its three important parameters explain the architectures of most polyhedral links including their Euler characteristics and genus, which facilitates rational design and synthesis of new DNA molecules and intrinsically reveals the basic principles of novel structures.