Polynomial Invariants Of A Family Of Pretzel Links

Knot theory plays an important role in topology.One of the main problems is to classify knots and links equally.The polynomial invariant is an important method to study the equivalence classification of knot and link.Alexander polynomial and Jones polynomial are two relatively important representatives of knot polynomials.In this paper,we mainly study the polynomials of a special kind of pretzel link P(2K1,2K2,…,2Kn)with n-branches.(?)As shown in the figure,after choosing orientation for P(2K1,2K2,…,2Kn),the Seifert surface with better properties can be obtained and the Seifert matrix corresponding to it can be calculated.Then,using the Seifert matrix properties of Seifert surface,the calculation formula of Alexander polynomial of oriented pretzel links P(2K1,2K2,…,2K,)is derived.The pretzel links with the same Alexander polynomials are called a family of pretzel links.For the 3-branch chain with the same Alexander polynomials(i.e.the same family of pretzel links),by studying the calculation rules of its Kauffman bracket polynomial and writhe,the calculation formula of its polynomials is given,and then the Jones polynomial is used to show that the family of pretzel links has infinite different elements.