Narrow Results

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is prove 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.

We study bridge presentations of virtual knots, and determine the virtual bridge numbers of pseudo-prime virtual knots with real crossing numbers less than 5, except two virtual knots.

In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.