Topological Properties and Characterizations

Abstract: There are three important types of structural properties that remain unchanged under the structural transformation of condensed matter physics and chemistry. They are the properties that remain unchanged under the structural periodic transformation-periodic properties. The properties that remain unchanged under the structural multi scale transformation-fractal properties. The properties that remain unchanged under the structural continuous deformation transformation-topological properties. In this paper, we will describe topological properties and characterizations in three layers: the first layer is intuitive concept, characterizations and applications, the second layer is logical physics understanding of topological properties, characterizations and applications, and the third layer is the nature of topological properties and its power. Duality and trinity are viewed as intrinsically topological objects and are recognized as common knowledge shared among human society activity, mathematics, physics, chemistry, biology and many other kinds of nature science and technology. Some important methods used so far to characterize the topological properties, including topological index, topological order, topological invariant, topology class and the topology partition are discussed. The theories of molecular topology, topological quantum matter including topological insulators, topological metal and topological superconductors and topological quantum computing are reviewed. The development of the topological duality connection between the qubit and singularity via topological space time is briefly introduced. We will see the use of iterated function systems (IFS)to simulate the connection between singularities and their qubit control codes. The novel applications of topology in integrated circuits technology are also discussed in this paper.