The Knots Invariants

This thesis is written on the basis of the basic definitions and properties of knots and links and the Alexander polynomial is used as a research object. Through the use of the relation, we can conduct an in-depth discussion about a kind of links with typical characteristics and further explore the nature of the knots and links.In the thesis, we mainly study a kind of link with m branches and the kind of link features that each branch only produces intersection with the other two branches at most. For such a kind of links, we start to speculate the properties of the links with three branches on the basis of links with two branches in previously published papers. Further we can explore the Alexander polynomials and properties of the links which have m branches and whose cross number of any two branches is. Next, we remove some of the restrictions and discuss the properties of the links which have m branches and whose cross numbers of any two branches are not exactly the same.When we discuss in detail each type of the links, we use the same relation to get the general Alexander polynomials of the links. It should be noted that in the process of using the formula, we always take the links with two branches as the basis for calculating. Then we are concerned about the nature of the first-order differential and the second-order differential of the Alexander polynomials in the case of t =1.we discovered that the second-differential of Alexander polynomial equals to 0 if we let the number of the branches is 3 and .By discussing the nature of the first derivative of the polynomial and second-order differential properties of Alexander polynomial with m branch, we can conclude that when the number of the branches for the links is m and the differential order p meets the condition p ￡m -2 and t =1, the p th-order differential of the Alexander polynomial for the links which has m branches equals to 0.