某些链环琼斯多项式的零点性质

Properties of zeros of the Jones polynomial of some links

主要研究某一类排叉链环的琼斯多项式零点的性质与分布.利用纽结与链环琼斯多项式的表达形式以及一些性质,结合三角函数有关知识,将前人推断的某些可能为琼斯多项式零点的单位根代入纽结或链环的琼斯多项式当中逐一进行证明,发现其并不为琼斯多项式的零点,并进行归纳总结,得到一类排叉纽结与一类排叉链环零点的性质.首先,给出本论文需要的一些理论知识.包括纽结、链环的基本概念,亚历山大多项式、琼斯多项式的定义,拧数和区域的划分等一些基本概念,给出了排叉纽结与二桥链环的性质.其次,利用某些纽结或链环琼斯多项式的表达形式,结合三角函数的相关知识证明某些单位根一定不是一类排叉纽结或一类排叉链环的琼斯多项式的零点.

This paper was intended to mainly focus on the properties and distribution of zeros of the Jones polynomial of certain pretzel knots and links.The expressions and some properties of the Jones polynomial of knots and links were used in combination with knowledge of trigonometric functions to substitute some unit roots that were inferred to be zeros of the Jones polynomial by previous scholars into the Jones polynomial of knots or links and to prove each of them.It was found that they were not zeros of the Jones polynomial.In addition,some properties of a class of pretzel knots and a class of pretzel link zeros were summarized.In the first part,some necessary theoretical knowledge was introduced,including the basic concepts of knots and links,the definitions of the Alexander and Jones polynomials,and number of twists and division of regions.Moreover,the pretzel knots and two-bridge links were profiled.In the second part,some expressions of the Jones polynomial of knots or links were used in combination with the know of trigonometric functions to prove that some unit roots must not be the zeros of the Jones polynomial of a class of pretzel knots or links,and their properties were summarized.