The Jones Polynomials Of Several Classes Of 3-Tangles

Knot theory is an important part of topology,which mainly studies the properties of knots and links that remain constant under continuous deformation.A core content of knot theory is to study the classification of knots.Knot invariants are important tools to study knot classification.Knot polynomial is one of the common knot invariants,such as Alexander polynomial,Conway polynomial,and Jones polynomial.The discovery of Jones polynomial promotes the development of knot theory,makes a significant contribution to the study of the chirality of knots and links,and opens up channels of contact with many other branches of mathematics.Tangle is a basic way to construct knots and links.This thesis mainly studies the Jones polynomials of several special classes of oriented 3-tangles.Firstly,we choose an unusual orientation of 3-tangles,and then give the Jones polynomial of the concatenation of two oriented 3-tangles by combining the method of calculating 3-tangles polynomials proposed by Giller,the concept and properties of basis 3-tangles.Secondly,three classes of oriented 3-tangles are constructed by selecting different arcs for twisting and the concatenation of two 3-tangles.The formulas for calculating the Jones polynomials of these three classes of oriented 3-tangles are given through discussion on different cases.Finally,according to the characteristics of basis 3-tangles,the properties of the six closures of 3-tangles are discussed,and the Jones polynomials of the closures of 3-tangles are further given.On this basis,the calculation method of Jones polynomials for a class of links is given.