A class of index polynomial invariants for virtual knots using flat virtual knots invariants

In this paper, a class of indices is constructed using a polynomial invariant [Formula: see text] for flat virtual knots. For any integer [Formula: see text] not less than [Formula: see text], the classical crossings of an oriented virtual knot diagram are classified into two types according to whether the intersection index of the classical crossings can be divided by [Formula: see text]. We assign an index to each classical crossing whose intersection index is not divisible by [Formula: see text], the index is related to the flat virtual knot diagram corresponding to the virtual knot diagram obtained by applying 0-smoothing at that classical crossing. We construct a class of index polynomials by using these indices and prove that they are invariants of virtual knots, and also discuss some of their properties.