Knot and its invariants

摘要

In this paper, we will discuss knots and its invariants. Invariants include tri-coloring, Morse function, Genus, Euler characteristic, and Seifert surface. Knots are actually commonly seen in our daily life. People are making a knot when they are typing their ties or shoes. Also, the origin of knots dates back to ancient times, when people kept records by typing knots. What’s more, knots have great academic importance. They are used in the research on different fields, such as the researches about topology, chemical elements, protein, DNA and so on. Since knots are closely connected with us, we will explore the properties of knots in a more sophisticated way in this paper. First, we will discuss how to use tri-coloring to identify different kinds of knots. By using tri-coloring, you can discover that two knot diagrams may look quite different but are actually the same. Second, we will focus on Morse function. After seeing the vivid knot diagrams, we can use another way other than figures to represent different knots. Third, we will combine genus, Euler characteristic and Seifert surface together. These invariants will help us to further study the properties and connections of various types of knots.