A family of polynomial invariants for virtual knots

Abstract

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 affine index polynomial, a new invariant RD[t](h) of virtual knots, etc. In this paper, we introduce a family of new polynomial invariants H-polynomial HDn(t,h,ℓ)(n∈N⁎) of virtual knots, and we construct a series of examples of virtual knots which have the same RD[t](h) but different HDn(t,h,ℓ) for some n. Finally, we get the relationship between HDn(t,h,ℓ) and HD⁎n(t,h,ℓ), where D⁎ is obtained from D by interchanging over and under lines at the classical crossings.