Knot Jones Polynomials And Integral Coefficient Polynomials

This thesis mainly studies the relationship between the knot Jones polynomial and the integral coefficient polynomial.On the basis of the previous studies,using the properties of the Jones polynomials and the special values at some points,we continue to study the integral coefficient polynomials with different degrees and widths as the establishment conditions of the Jones polynomials in knots.The first part gives the preparatory knowledge used in this thesis.Some basic concepts of knots,including the definition of knots and links,R transformation,knot equivalence,etc.Then the basic knotted polynomials are given,including Alexander polynomials,Kauffman polynomials and Jones polynomials.Finally,the properties of Jones polynomials are given.In the second part,we mainly study integral coefficient polynomials are sufficient and necessary conditions for certain knot Jones polynomials under certain width.Firstly,some polynomials of width less than six are given as sufficient and necessary conditions for Jones polynomials of a knot.Secondly,it is discussed that polynomials of width six are sufficient and necessary conditions for Jones polynomials of a knot.In the third part,we mainly study the cases of eleventh-degree integer coefficient polynomial of width nine is Jones polynomial.Examples are given in each case and Arf invariants for these knots are derived.On this basis,the integer coefficient polynomial of degree eleven and width eight is obtained,which is the discrimination method of Jones polynomial.In the fourth part,we mainly study the relationship between some integer coefficients polynomials with different degrees and widths and Jones polynomials.The research focuses on the following conditions: under which conditions,the ninth-degree,thirteenth-degree,fourteenth-degree integral coefficient polynomial of a certain width is a certain knot-Jones polynomial.The examples satisfying the conditions are given,and then some coefficients are set as special values,and it is deduced that polynomials with the same degree and different widths are the discriminant methods for certain knot Jones polynomials,and their coefficients satisfy specific quantitative laws.