On invariants of virtual links
Virtual knot theory is a generalization of classical knot theory. Virtual links have diagrams which contain in addition to the classical crossings (where one arc passes over another) virtual crossings, that is points of self-intersections of the arcs in the diagram. The equivalence of virtual knots and links is defined combinatorically by generalized Reidemeister moves involving also the virtual crossings. In this paper the author generalizes classical link invariants (such as the knot quandle, the knot fundamental group, the Alexander module and coloring invariants) to virtual link invariants. His work generalizes a construction of a knot quandle for virtual knots by \textit{L. H. Kauffman} [Eur. J. Comb. 20, No. 7, 663-690 (1999; Zbl 0938.57006)] to a virtual quandle \(Q\). From \(Q\) a virtual Alexander module can be computed. That in turn leads to a definition of a virtual Alexander polynomial that has no analogue in the classical case (i.e. it vanishes on classical links).
scientific article; zbMATH DE number 1783465