Research Of The Knot Polynomials

In this paper, we study mainly the properties of the Jones polynomial and polynomial with integer coefficients which has important value to study, focusing on discussing the relationship between them. At the same time, by use the properties of the knot polynomial and the problem distribution of zeros of entire of the polynomial to study that what kind of coefficients of the polynomial is the Jones polynomial, and gives necessary condition when the twelfth and thirteenth polynomial with integer coefficients is a knot polynomials. We also give Arf-invariants of some knot.This article has given the main content of the article describes the basics knowledge for reading this article, including basic concepts about the knot, chain link, equivalence of the knot, isotopy, transformation of Reidemeite and so on; It introduces several common Knot polynomials, Alexander polynomial, Conway polynomial, Jones polynomial, etc. At the same time, also gives properties of them. The main study of this article is Jones polynomial, which is including directional link, isotopy invariants and properties of the Jones polynomial of the knot, and also introduced the method about calculation of Jones polynomial is knot split and also gives examples of Jones polynomial about the trefoil.This article first studied the necessary condition between the polynomial of a knot is Jones polynomial. This section discuss the a necessary conditions between the polynomial of a knot is Jones polynomial when the polynomial of a knot width is 9, followed by the introduction necessary condition of a knot width equal to 9 polynomial is the Jones polynomial of, and gives relevant examples. We also give three deductions about Arf invariant, and there exist any relationships between its polynomial coefficients.Secondly, the article focuses on the relationship between the integer coefficients polynomial with the Jones polynomial, that is, if a Laurent polynomial with integer coefficients, and it’s power thirteen, the constant term is zero, then we can find a necessary condition for it’s a Jones polynomial; And also discussed the necessary condition Laurent polynomial with integer coefficients which power is thirteen and constant term is not zero, is the Jones polynomial, in the other way, if a Laurent polynomial with integer coefficients, and it’s power thirteen, the constant term is not zero, then we can find a necessary condition for it’s a Jones polynomial; Secondly, if the discussion is a Laurent polynomial with integer coefficients, and it’s power is thirteen, the width is 5, then we can get the necessary condition for it is a Jones polynomial. If a Laurent polynomial is a polynomial with integer coefficients and its power is thirteen, the width is 6, then we can turn out it is a necessary condition of Jones polynomial.