The HOMFLY Polynomial Of A Class Of Links

Knot theory is the branch of mathematics that studies how to embed circles Sl into three-dimensional Euclidean space R3.The knot polynomial refers to a class of knot invariant expressed by polynomials,such as the Alexander polynomial and the Jones polynomial.The HOMFLY polynomial is another important polynomial that calculates the knot invariant after the Jones polynomial.The HOMFLY polynomial is a bivariate polynomial invariant of knots.It can be converted into the Alexander polynomial and the Jones polynomial of the same link or knot by corresponding argument substitution method.In thesis paper,we will apply the definition and properties of the HOMFLY polynomial to study the HOMFLY polynomial of a special class of Brunnian links.Brunnian link is a special class of link in which the complement of any one component is a trivial link.Because any true-sublink of the Brunnian links is trivial,the simple and special structure of the link implies that it has very good properties compared to ordinary links.Thesis paper deals with the HOMFLY polynomial for a special class of n components of the orientations of Brunnian links B（σ1,σ2,…,σn）,when each unit element B（σ1）（i=1,…,n）has transverse half twists.On the one hand,the special case is calculated when σi is 0 and ±1.The recursion formula for skein relation is obtained by cutting the crossing of n components of the orientations Brunnian links B（0,…,0,0）and B（±1,…,±1,±1）.Then,the crossing of half twists σi which is even or odd in B（σi）are cut.All half twists σi which is even are eliminated.Eliminate σi which is odd to have only one half twist.The HOMFLY polynomial B（σi）is obtained by arranging the recursive formula of skein relation.On this basis,the recursive formula of the skein relation is obtained by cutting the crossing of n components of the orientations Brunnian links B（σ1,σ2,…,σn）.Combined with the HOMFLY polynomial property when the links are mirror images of each other,the calculation formula for different cross-over mode of half twists in unit elements B（σi）can be obtained.Finally,the calculation formula of the HOMFLY polynomial for a class of n components of the orientations of Brunnian links B（σ1,σ2,…,σn）with half twists is realized.