Abstract Link Diagrams and Virtual Knots.

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is proved that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated. [ABSTRACT FROM AUTHOR]