Properties And Distribution Of Zeros Of The Jones Polynomials Of Some Links

Many problems remain to be explored in knot theory.One of the most prominent ones is the classification of knots and links.To achieve this goal,a number of variables have been established,and the most famous of them is the Jones polynomial,which can distinguish many classes of links and their mirrors and was often used to solve problems in certain areas.Given the great importance of the Jones polynomial,some researchers began to study the properties and distribution of its zeros.Although substantial progress has been made through researches,there are still many problems to be further explored.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.This paper consists of three parts.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,so as to lay the foundation for understanding the following parts of the paper.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.Finally,based on the previous findings,an attempt was made to further explore the Jones polynomial of a particular class of 3-loop links and pretzel links,as well as the distribution of zeros.