The Polynomials Of A Class Of Spatial Graphs And Corresponding Links

Topology is an important and fundamental branch of mathematics,it studies the properties of geometric objects that remain consistent under continuous deformation.Knot theory is an important field of research in topology.Classifying knots and links is an important topic of knot theory in the sense of isotopic,the classification of knots and links can be studied using knot polynomials.The Jones polynomial is an important knot polynomial.Space graph theory can be seen as a generalization of knot theory.Space graph polynomial is a tool for classifying spatial graphs.The Tutte polynomial is an important spatial graph polynomial.With the help of spatial graph theory,this article mainly studies the Jones polynomials of a certain class of alternating links.Firstly,we construct a new special type of spatial graphs,(4,n)graph,use the properties of Tutte polynomial to obtain the calculation formula to the Tutte polynomial of(4,n)graph,then find the links corresponding to(4,n)graph by the corresponding rules of projection graphs and symbol graphs,study its writhe number,with the help of the relationship between the Jones polynomial and the Tutte polynomial,the Jones polynomial of this type links is calculated.Secondly,on the basis,the(4,n)graph will be increased the number of non-multiple edges,we expand to the(A,n)graph,and study its Tutte polynomial,find the links corresponding to the(A,n)graph and its writhe number rule,obtain the calculation formula to the Jones polynomials of links corresponding to the graphs.Furthermore,we have a further extension of the(4,n)graph,construct the(4,m,n)graph,obtain the Tutte polynomial of the(4,m,n)graph using binary mathematical induction,then find their writhe number rule of links under different cases,and get the calculation formula to the Jones polynomials of links corresponding to the(4,m,n)graph,on this basis,the Jones polynomials of links corresponding to the(A,m,n)graph can be calculated by the same method.