ABSTRACT
This chapter considers a generalisation of the Jones-Kauffman polynomial. All the generalisations were constructed by using the following idea: one thinks of a virtual link diagram as a set of classical crossings provided with the information about how they are connected on the plane and one does not pay attention to virtual crossings. Thus, for instance, the generators of the fundamental group that correspond to arcs of the diagram may pass through virtual crossings, and all relations are taken only at classical crossings. The Kauffman construction for the Jones polynomial for virtual knots works just as well as in the case of classical knots. The invariance proof for the polynomial under classical Reidemeister moves is just the same as in the classical case; under purely virtual and semi virtual moves it is clearly invariant term-by-term.
