The Polynomial Of A Kind Of Alternative Links

Knot theory plays a very important role in topology.In knot theory,the Jones polynomial is an important invariant in the study of the classification of knots and links.Scholars have yielded many results in the research of the Jones polynomials of knots and links until now.The calculation formulas of the Jones polynomials of some special knots and links are given,and a series of results have also been obtained from the research on the zero point properties of the Jones polynomials of knots and links.A special kind of the Jones polynomial of alternative link A（c₁,c₂,…,c_n）is discussed in this paper.First of all,we consider a special case of c_i=0,which is A（0,0,…,0）,and find the law of twisting numbers of it by calculating its Chain polynomial and the Tutte polynomial.And then,we get the Jones polynomial of A（δ₁,0,δ₂,0,…,δ_n,0）.Secondly,in regard to alternative link A（δ₁,c1,δ₂,0）,we discuss two cases of odd and even number of intersections of c1,and calculate the Jones polynomial of A（δ₁,c₁,δ₂,0）by the same method.On the basis of above,we calculate the Tutte polynomial of A（c₁,c₂,…,c_n）and twist numbers by blocking for the orientated A（δ₁,c₁,δ₂,c₂,…,δ_n,c_n）,so we get the Jones polynomial of A（δ₁,c₁,δ₂,c₂,…,δ_n,c_n）.At last,for the Jones polynomial of A（δ₁,c₁,δ₂,c₂,…,δ_n,c_n）obtained above,we discuss the zeros point distribution of A（δ₁,k,δ₂,k,δ₃,k）when k tending ∞,and arbitrarily fix two parameters in k₁、k₂、k₃,we discuss the zeros point distribution of A（δ₁’,k₁’,δ₂’,k₂,δ₃’,k₃）when the remaining parameter tend to ∞.