Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

Abstract

We introduce two sequences of two-variable polynomials {L n K(t, `)} ∞n=1 and {F n K(t, `)} ∞n=1, expressed in terms of index value of a crossing and n-dwrithe value of a virtual knot K, where t and ` are variables. Basing on the fact that n-dwrithe is a flat virtual knot invariant we prove that L n K and F n K are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using L n K we give sufficient conditions when virtual knot does not admit cosmetic crossing change.