A Family Of Polynomial Invariants For Virtual Knots

As a generalization of the classical knots,virtual knots deal with the problem of circles embedded in a thickened orientable surface with arbitrary genus.In the study of virtual knots,the classification of virtual knots remains an important issue.In recent years,many polynomial invariants have appeared,such as:the affine index polynomial of virtual knots,extended square bracket polynomial,a new invariant R^[t]_D（h）of virtual knots,etc.We observe that R^[t]_D（h）sometimes fails to distinguish two virtual knots,to solve this prob-lem,we construct a family of new polynomial invariants H-polynomial Hⁿ_D（t,h,l）of virtual knots by affine index polynomial of flat virtual knots and n-writhe for any n∈N^*,where N^*is a set of positive integer.In particular,ifl=1,Hⁿ_D（t,h,l）=R^[t]_D（h）for any n.Then we give an example to show the process of the calculation,at the same time,we get the relationship between Hⁿ_D（t,h,l）,Hⁿ_-D（t,h,l）and Hⁿ_D^*（t,h,l）,where-D is obtained from D by reversing the orientation of D,D^*is obtained from D by interchanging over and under lines at the classical crossings.Finally,we construct some examples of virtual knots that can be distinguished by H-polynomial,i.e.they have the same R^[t]_D（h）,but the corresponding Hⁿ_D（t,h,l）are different for some n.