The Properties And Roots’ Distribution Of The Knot Polynomials

The knots change continuously in space,and it is difficult for us to judge whether two knots are same through observation.It’s more difficult to find the intrinsic relationship between two knots,so mathematicians begin to study the algebra knowledge to research knots.and the polynomial has become one of the focus of studying.Although the knot is continuously changing,there are some invariant properties before and after the knots change.The knot theory mainly studies those invariant properties which can help us distinguish different knots conveniently and efficiently.The Alexander polynomial promotes the development of knot theory.Later,Jones polynomial attracts more attention of mathematicians.In this paper,we use knot theory,algebra,sine cosine function,differential knowledge to study a necessary condition of Jones polynomials.being based on previous studies,we use the three order derivative of Jones polynomial to study the relationship of polynomial coefficients.Firstly,we introduce some knowledge of knot theory,which will be conducive to a better understanding of this paper,including the basic definition,R₁,R₂,R₃transform of knots,and several classes of polynomials.Secondly,according to the Jones polynomial’s value in some points,we calculate and summarize some results about polynomials which are the conditions for Jones polynomials,in other words,it meets the need of the coefficients.Thus we obtain the Conway polynomial properties of knots.The polynomial index can be positive or negative,which are involved in the paper.Finally,we list some knots’Jones polynomials,and we can study the characteristics of its zeros.The zeros of knotp（k,l,l）are all distributed on the ｜t｜=1no matter of k→∞or l→∞.