ABSTRACT
This chapter presents a generalisation of the Jones–Kauffman polynomial for virtual knots by adding some "extra information" to it, namely, some objects connected with curves in 2-surfaces. It considers the minimality aspects in virtual knot theory and gives a proof of the generalised Murasugi theorem. Virtual equivalence and classical equivalence for classical knots coincide and the set of all classical knots is a subset of the set of all virtual knots. Thus, each invariant of virtual links generates some invariant of classical links. Recall that an atom is a two-dimensional connected closed manifold without boundary together with an embedded graph of valency four that divides the manifold into cells that admit a chessboard colouring. Atoms are considered up to the natural equivalence. In the case of an arbitrary atom one should replace embeddings by regular immersions. There might be many immersions for a given frame.
