中文摘要：

本文利用纽结Jones多项式的性质研究了整系数多项式的性质,主要研究了某些宽度是5的7次和8次整系数多项式和纽结多项式的关系,给出整系数多项式是纽结多项式的充分必要条件,进而给出整系数多项式是交错纽结的Jones多项式的充分必要条件.同时根据这些性质给出了某些交错纽结的Arf不变量.

英文摘要：

In this paper,we studied the properties of polynomials with integer coefficients by using the properties of the Jones polynomial. We discussed mainly the relations between polynomials of knots and polynomials with in-teger coefficients,which the breath of these polynomials is 5 and the degree of the polynomials is 7 or 8. We gave sufficient and necessary conditions that the polynomials are the Jones polynomials of some knots and gave neces-sary and sufficient conditions that the polynomials are the Jones polynomials of some alternating knots. We ob-tained the properties of the Arf invariant of some knots.