Zeros Of The Jones Polynomial For Knots And Links And Their Distributions

Researching knot theory is aimed to find the isotopy invariant which can distinguish the different knots and be convenient to calculation.The discovery of Alexander polynomial is a major breakthrough in the knot theory,but Alexander polynomial is unable to distinguish knot and its mirror image.After the discovery of the Jones polynomial makes knot theory become a focus of the world of maths.At present scholars have made many achievements about Jones polynomial of knots,but they have made achievements which are just the tip of the iceberg for the distribution of zeros of the Jones polynomial.This thesis studies knots and links of the Jones polynomial and their zero points distributions.It concludes that unit roots are not the zeros of the Jones polynomial for torus knotsp,T_p,q?while?p,q?=1?by using some properties of the Jones polynomial for knots and the related knowledge of trigonometric function.It gives the Jones polynomial of pretzel links by using the relationship between Tutte polynomial and the Jones polynomial;It also discusses the zeros distribution of the Jones polynomial for the pretzel links p?k,k,l?based on mathematics knowledge limit.The first part,this chapter mainly introduces the preparatory knowledge for reading the paper.First the chapter introduces the concept of knots and links,secondly it introduces the polynomial of knots,including : Alexander polynomial,Laurent polynomial and Jones polynomial;It introduces the concept of the torus knots and the basic properties of the torus knots;Finally it introduces the concept of the pretzel links and the basic properties of the pretzel links.The second part,this chapter mainly studies the unit points which is not zero of the Jones polynomial.Firstly this chapter recommends the torus knot of the Jones polynomial and some conclusions about the torus knots;it secondly gives points that e^2p+1/n?i is not the zeros of the Jones polynomial for the torus knotp,T_p,q?while?p,q?=1? while n?8.The third part,this chapter firstly calculates the Jones polynomial of the pretzel links p?k,k,l?based on the Jones polynomial of the pretzel links p?c₁,c₂,c₃?,then it gets the zeros distribution of the pretzel links p?k,k,l? while it makes k is fixed and l tend to be infinite.